Preface

This book is titled Forecasting in Mathematics – Recent Advances, New Perspectives and Applications. Additionally, each time I work in the field of mathematical probability and statistics, I have the pleasure of tackling the knowledge, the theorems, the proofs, and the applications of the theory. In fact, each problem is like a riddle to be solved, a conquest to be won, and I am relieved and extremely happy when I find the solution. This proves two important facts: firstly, the power of mathematics and its models to deal with such problems and secondly the power of the human mind that is able to understand such problems and to tame such a wild concept that is randomness, probability, stochasticity, uncertainty, chaos, chance, and nondeterminism.

Mathematical probability and statistics are attractive, thriving, and respectable parts of mathematics. Some mathematicians and philosophers of science say that they are the gateway to mathematics' deepest mysteries. Moreover, mathematical probability and statistics denote an accumulation of mathematical discussions connected with the efforts to most efficiently collect and use numerical data subject to random or deterministic variations. In the twentieth century and the present time, the concept of probability and mathematical statistics has become one of the fundamental notions of modern science and philosophies of nature. This was accomplished after a long history of efforts by prominent and distinguished mathematicians and philosophers like the famous French Blaise Pascal and Pierre de Fermat, the Dutch Christiaan Huyghens, the Swiss Jakob Bernoulli, the German Carl Friedrich Gauss, the French Siméon-Denis Poisson, the English Thomas Bayes, the French Joseph Louis Lagrange and Pierre-Simon de Laplace, the English Karl Pearson and Ronald Aylmer Fisher, the Russian Andrey Nikolaevich Kolmogorov, the American John von Neumann, etc…

As a matter of fact, each time I read or meditate on these outstanding giants, I feel the respect, the admiration, and the esteem towards these magnificent men and giants of science who most of them were mathematicians, physicists, astronomers, statisticians, philosophers, etc... at the same time. They were, as we call them today: Universalists.

Moreover, this book develops methods for simulating simple or complicated processes or phenomena. If the computer can be made to imitate an experiment or a process, then by repeating the computer simulation with different data, we can draw statistical conclusions. Thus, a simulation of a spectrum of mathematical processes on computers was done. The result and accuracy of all the algorithms are truly amazing and delightful; hence, this confirms two complementary accomplishments: first the triumphs of the theoretical calculations already established using different theorems and second the power and success of modern computers to verify them.

To conclude, due to its universality, mathematics is the most positive and certain branch of science. It has been successfully called by philosophers the Esperanto of all sciences since it is the common, the logical, and the exact language of

understanding, capable of expressing accurately all scientific endeavors. Although probability and statistics are approximate sciences that deal with rough guesses, hypotheses tests, estimated computations, expected calculations, and uncertain results, they still keep in them the spirit of "exact" sciences through their numbers, proofs, figures, and graphs, since they remain a branch of mathematics. Surely, the pleasure of working and doing mathematics is everlasting. I hope that the reader will benefit from it and share the pleasure of examining the present book.

Sincerely, I am truly astonished by the power of probability and statistics to deal with deterministic or random data and phenomena, and this feeling and impression has never left me from the first time I was introduced to this branch of science and mathematics. I hope that in the present book I will convey and share this feeling with the reader. I hope also that they will discover and learn about the concepts and applications of probability and statistics paradigm.

> **Abdo Abou Jaoudé, Ph.D.** Notre Dame University-Louaizé, Zouk Mosbeh, Lebanon

**Chapter 1**

**Abstract**

*Abdo Abou Jaoude*

The Monte Carlo Techniques and

the Complex Probability Paradigm

The concept of mathematical probability was established in 1933 by Andrey Nikolaevich Kolmogorov by defining a system of five axioms. This system can be enhanced to encompass the imaginary numbers set after the addition of three novel axioms. As a result, any random experiment can be executed in the complex probabilities set **C** which is the sum of the real probabilities set **R** and the imaginary probabilities set **M**. We aim here to incorporate supplementary imaginary dimensions to the random experiment occurring in the "real" laboratory in **R** and therefore to compute all the probabilities in the sets **R**, **M**, and **C**. Accordingly, the probability in the whole set **C** ¼ **R** þ**M** is constantly equivalent to one independently of the distribution of the input random variable in **R**, and subsequently the output of the stochastic experiment in **R** can be determined absolutely in **C**. This is the consequence of the fact that the probability in **C** is computed after the subtraction of the chaotic factor from the degree of our knowledge of the nondeterministic experiment. We will apply this innovative paradigm to the well-known Monte Carlo techniques and to their random algorithms and procedures in a novel way.

**Keywords:** degree of our knowledge, chaotic factor, complex probability set,

*"Thus, joining the rigor of the demonstrations of science to the uncertainty of fate, and reconciling these two seemingly contradictory things, it can, taking its name from both, appropriately arrogate to itself this astonishing title: the geometry of*

*"You believe in the God who plays dice, and I in complete law and order."*

*"There is a certain Eternal Law, to wit, Reason, existing in the mind of God and*

*"Chance is the pseudonym of God when He did not want to sign."*

*Blaise Pascal*

*Anatole France*

*Saint Thomas Aquinas*

*Albert Einstein, Letter to Max Born*

probability norm, complex random vector, convergence probability,

divergence probability, simulation

*governing the whole universe."*

**1. Introduction**

*chance."*

#### **Chapter 1**

## The Monte Carlo Techniques and the Complex Probability Paradigm

*Abdo Abou Jaoude*

#### **Abstract**

The concept of mathematical probability was established in 1933 by Andrey Nikolaevich Kolmogorov by defining a system of five axioms. This system can be enhanced to encompass the imaginary numbers set after the addition of three novel axioms. As a result, any random experiment can be executed in the complex probabilities set **C** which is the sum of the real probabilities set **R** and the imaginary probabilities set **M**. We aim here to incorporate supplementary imaginary dimensions to the random experiment occurring in the "real" laboratory in **R** and therefore to compute all the probabilities in the sets **R**, **M**, and **C**. Accordingly, the probability in the whole set **C** ¼ **R** þ**M** is constantly equivalent to one independently of the distribution of the input random variable in **R**, and subsequently the output of the stochastic experiment in **R** can be determined absolutely in **C**. This is the consequence of the fact that the probability in **C** is computed after the subtraction of the chaotic factor from the degree of our knowledge of the nondeterministic experiment. We will apply this innovative paradigm to the well-known Monte Carlo techniques and to their random algorithms and procedures in a novel way.

**Keywords:** degree of our knowledge, chaotic factor, complex probability set, probability norm, complex random vector, convergence probability, divergence probability, simulation

#### **1. Introduction**

*"Thus, joining the rigor of the demonstrations of science to the uncertainty of fate, and reconciling these two seemingly contradictory things, it can, taking its name from both, appropriately arrogate to itself this astonishing title: the geometry of chance."*

*Blaise Pascal*

*"You believe in the God who plays dice, and I in complete law and order." Albert Einstein, Letter to Max Born*

*"Chance is the pseudonym of God when He did not want to sign."*

*Anatole France*

*"There is a certain Eternal Law, to wit, Reason, existing in the mind of God and governing the whole universe."*

*Saint Thomas Aquinas*

*"An equation has no meaning for me unless it expresses a thought of God." Srinivasa Ramanujan*

Calculating probabilities is the crucial task of classical probability theory. Adding supplementary dimensions to nondeterministic experiments will yield a deterministic expression of the theory of probability. This is the novel and original idea at the foundations of my complex probability paradigm. As a matter of fact, probability theory is a stochastic system of axioms in its essence; that means that the phenomena outputs are due to randomness and chance. Adding new imaginary dimensions to the nondeterministic phenomenon happening in the set **R** will lead to a deterministic phenomenon, and thus, a probabilistic experiment will have a certain output in the set **C** of complex probabilities. If the chaotic experiment becomes fully predictable, then we will be completely capable to foretell the output of random events that occur in the real world in all probabilistic processes. Accordingly, the task that has been achieved here was to extend the set **R** of random real probabilities to the deterministic set **C** ¼ **R** þ**M** of complex probabilities and this by incorporating the contributions of the set **M** which is the set of complementary imaginary probabilities to the set **R**. Consequently, since this extension reveals to be successful, an innovative paradigm of stochastic sciences and prognostic was put forward in which all nondeterministic phenomena in **R** was expressed deterministically in **C**. I coined this novel model by the term "the complex probability paradigm" that was initiated and established in my 14 earlier research works [1–14].

Regarding some applications of the novel established model and as a subsequent work, it can be applied to any nondeterministic experiments using Monte Carlo

The next figure displays the major aims and purposes of the complex probability

The simplicity of Kolmogorov's system of axioms may be surprising [1–14]. Let *E* be a collection of elements {*E*1, *E*2, … } called elementary events and let *F* be a set

**Axiom 3:** A nonnegative real number *Prob*(*A*), called the probability of *A*, is

**Axiom 5:** If *A* and *B* have no elements in common, the number assigned to their

*Prob*ð Þ¼ *A* ∪ *B Prob*ð Þþ *A Prob*ð Þ *B*

*Prob*ð Þ¼ *A* ∪ *B Prob*ð Þþ *A Prob*ð Þ� *B Prob*ð Þ *A* ∩ *B*

And we say also that *Prob*ð Þ¼ *A* ∩ *B Prob*ð Þ� *A Prob*ð Þ¼ *B=A Prob*ð Þ� *B Prob*ð Þ *A=B*

Moreover, we can generalize and say that for *N* disjoint (mutually exclusive) events *A*1, *A*2, … , *A <sup>j</sup>*, … , *AN* (for 1≤ *j*≤ *N*), we have the following additivity rule:

which is the conditional probability. If both *A* and *B* are independent then

Moreover, compared with existing literature, the major contribution of the current chapter is to apply the innovative complex probability paradigm to the techniques and concepts of the probabilistic Monte Carlo simulations and

**3.1 The original Andrey Nikolaevich Kolmogorov system of axioms**

of subsets of *E* called random events. The five axioms for a finite set *E* are:

assigned to each set *A* in *F*. We have always 0 ≤ *Prob*(*A*) ≤ 1.

hence, we say that *A* and *B* are disjoint; otherwise, we have

algorithms whether in the continuous or in the discrete cases.

*The diagram of the major aims of the complex probability paradigm.*

*The Monte Carlo Techniques and the Complex Probability Paradigm*

*DOI: http://dx.doi.org/10.5772/intechopen.93048*

algorithms.

**Figure 1.**

union is

**3**

paradigm (*CPP*) (**Figure 1**).

**3. The complex probability paradigm**

**Axiom 1:** *F* is a field of sets. **Axiom 2:** *F* contains the set *E*.

**Axiom 4:** *Prob*(*E*) equals 1.

*Prob*ð Þ¼ *A* ∩ *B Prob*ð Þ� *A Prob*ð Þ *B* .

#### **2. The purpose and the advantages of the current chapter**

The advantages and the purpose of the present chapter are to [15–39]:


**<sup>C</sup>** ð Þ¼ complex set **<sup>R</sup>** ð Þþ real set **<sup>M</sup>** imaginary set *:*

7.Prepare to apply this inventive paradigm to other topics in prognostics and to the field of stochastic processes. These will be the goals of my future research publications.

*The Monte Carlo Techniques and the Complex Probability Paradigm DOI: http://dx.doi.org/10.5772/intechopen.93048*

**Figure 1.**

*The diagram of the major aims of the complex probability paradigm.*

Regarding some applications of the novel established model and as a subsequent work, it can be applied to any nondeterministic experiments using Monte Carlo algorithms whether in the continuous or in the discrete cases.

Moreover, compared with existing literature, the major contribution of the current chapter is to apply the innovative complex probability paradigm to the techniques and concepts of the probabilistic Monte Carlo simulations and algorithms.

The next figure displays the major aims and purposes of the complex probability paradigm (*CPP*) (**Figure 1**).

#### **3. The complex probability paradigm**

#### **3.1 The original Andrey Nikolaevich Kolmogorov system of axioms**

The simplicity of Kolmogorov's system of axioms may be surprising [1–14]. Let *E* be a collection of elements {*E*1, *E*2, … } called elementary events and let *F* be a set of subsets of *E* called random events. The five axioms for a finite set *E* are:

**Axiom 1:** *F* is a field of sets.

**Axiom 2:** *F* contains the set *E*.

**Axiom 3:** A nonnegative real number *Prob*(*A*), called the probability of *A*, is assigned to each set *A* in *F*. We have always 0 ≤ *Prob*(*A*) ≤ 1.

**Axiom 4:** *Prob*(*E*) equals 1.

**Axiom 5:** If *A* and *B* have no elements in common, the number assigned to their union is

$$P\_{rob}(A \cup B) = P\_{rob}(A) + P\_{rob}(B)$$

hence, we say that *A* and *B* are disjoint; otherwise, we have

$$P\_{rob}(A \cup B) = P\_{rob}(A) + P\_{rob}(B) - P\_{rob}(A \cap B)$$

And we say also that *Prob*ð Þ¼ *A* ∩ *B Prob*ð Þ� *A Prob*ð Þ¼ *B=A Prob*ð Þ� *B Prob*ð Þ *A=B* which is the conditional probability. If both *A* and *B* are independent then *Prob*ð Þ¼ *A* ∩ *B Prob*ð Þ� *A Prob*ð Þ *B* .

Moreover, we can generalize and say that for *N* disjoint (mutually exclusive) events *A*1, *A*2, … , *A <sup>j</sup>*, … , *AN* (for 1≤ *j*≤ *N*), we have the following additivity rule: *Forecasting in Mathematics - Recent Advances, New Perspectives and Applications*

$$P\_{rob}\left(\bigcup\_{j=1}^{N}\mathcal{A}\_{j}\right) = \sum\_{j=1}^{N}P\_{rob}\left(\mathcal{A}\_{j}\right)$$

defined by Kolmogorov was hence expanded to encompass the set of imaginary

To summarize the novel paradigm, we state that in the real probability universe **R**, our degree of our certain knowledge is undesirably imperfect and hence unsatisfactory; thus, we extend our analysis to the set of complex numbers **C** which incorporates the contributions of both the set of real probabilities which is **R** and the complementary set of imaginary probabilities which is **M**. Afterward, this will yield an absolute and perfect degree of our knowledge in the probability universe **C** ¼ **R** þ**M**because *Pc* = 1 constantly. As a matter of fact, the work in the universe **C** of complex probabilities gives way to a sure forecast of any stochastic experiment, since in **C** we remove and subtract from the computed degree of our knowledge the measured chaotic factor. This will generate in the universe **C** a probability equal to 1 (*Pc*<sup>2</sup> <sup>¼</sup> *DOK* � *Chf* <sup>¼</sup> *DOK* <sup>þ</sup> *MChf* <sup>¼</sup> <sup>1</sup> <sup>¼</sup> *Pc*). Many applications taking into consideration numerous continuous and discrete probability distributions in my 14 previous research papers confirm this hypothesis and innovative paradigm. The extended Kolmogorov axioms (EKA) or the complex probability paradigm (CPP)

**3.3 A brief interpretation of the novel paradigm**

*DOI: http://dx.doi.org/10.5772/intechopen.93048*

*The Monte Carlo Techniques and the Complex Probability Paradigm*

can be shown and summarized in the next illustration (**Figure 2**).

**4.1 The divergence and convergence probabilities**

**4. The Monte Carlo techniques and the complex probability paradigm**

Let *RE* be the exact result of the stochastic phenomenon or of a multidimensional or simple integral that are not always possible to compute by probability theory ordinary procedures or by deterministic numerical means or by calculus [1–14]. And let *RA* be the phenomenon and integrals approximate results calculated by the

� � <sup>≤</sup>1 if *RA* <sup>≤</sup> *RE*

� � <sup>≤</sup> 1 if *RA* <sup>≥</sup> *RE*

<sup>1</sup> � <sup>1</sup> � *RA*

<sup>1</sup> <sup>þ</sup> <sup>1</sup> � *RA*

*RE*

*RE*

*RE* � � �

*RE* � � �

� �

⇔

� � if 0 <sup>≤</sup>*RA* <sup>≤</sup>*RE*

� � if *RE* <sup>≤</sup>*RA* <sup>≤</sup>2*RE*

�

� �

� and is always between

0≤*RA* ≤*RE RE* ≤*RA* ≤2*RE*

� <sup>¼</sup> <sup>1</sup> � *RA RE*

� � �.

� � �

The relative error in the Monte Carlo methods is Rel*:*Error <sup>¼</sup> *RE*�*RA*

0% and 100%. Therefore, the relative error is always between 0 and 1. Hence

*RE* � *RA RE*

*RE*

� � � � ¼ 8 >>><

>>>:

<sup>0</sup><sup>≤</sup> � *RE* � *RA RE*

Moreover, we define the real probability in the set **R** by

� <sup>¼</sup> <sup>1</sup> � <sup>1</sup> � *RA*

if 0≤ *RA* ≤*RE*

if *RE* ≤*RA* ≤2*RE*

� � � �

Additionally, the percent relative error is = 100% � *RE*�*RA*

0≤

8 >>><

>>>:

� � �

numbers.

**parameters**

techniques of Monte Carlo:

*RE* � *RA RE*

*Pr* <sup>¼</sup> <sup>1</sup> � *RE* � *RA*

� � � �

> *RA RE*

<sup>2</sup> � *RA RE*

*RE*

� � � � ≤1⇔

0≤

� � � �

¼

**5**

8 >><

>>:

And we say also that for *N* independent events *A*1, *A*2, … , *A <sup>j</sup>*, … , *AN* (for 1≤*j*≤ *N*), we have the following product rule

$$P\_{rob}\left(\bigcap\_{j=1}^N \mathcal{A}\_j\right) = \prod\_{j=1}^N P\_{rob}\left(\mathcal{A}\_j\right)$$

#### **3.2 Adding the imaginary part M**

Now, we can add to this system of axioms an imaginary part such that:

**Axiom 6:** Let *Pm* ¼ *i* � ð Þ 1 � *Pr* be the probability of an associated complementary event in **M** (the imaginary part) to the event *A* in **R** (the real part). It follows that *Pr* <sup>þ</sup> *Pm=<sup>i</sup>* <sup>¼</sup> 1 where *<sup>i</sup>* is the imaginary number with *<sup>i</sup>* <sup>¼</sup> ffiffiffiffiffiffi �<sup>1</sup> <sup>p</sup> or *<sup>i</sup>* <sup>2</sup> ¼ �1.

**Axiom 7:** We construct the complex number or vector *Z* ¼ *Pr* þ *Pm* ¼ *Pr* þ *i*ð Þ 1 � *Pr* having a norm j j *Z* such that

$$|Z|^2 = P\_r^2 + \left(P\_m/i\right)^2.$$

**Axiom 8:** Let *Pc* denotes the probability of an event in the complex probability universe **C** where **C** ¼ **R** þ**M**. We say that *Pc* is the probability of an event *A* in **R** with its associated event in **M** such that

$$\text{Pc}^2 = \left(P\_r + P\_m/i\right)^2 = |Z|^2 - 2\text{iP}\_r P\_m \text{ and is always equal to } \mathbf{1}.$$

We can see that by taking into consideration the set of imaginary probabilities, we added three new and original axioms, and consequently the system of axioms

**Figure 2.** *The* EKA *or the* CPP *diagram.*

defined by Kolmogorov was hence expanded to encompass the set of imaginary numbers.

#### **3.3 A brief interpretation of the novel paradigm**

To summarize the novel paradigm, we state that in the real probability universe **R**, our degree of our certain knowledge is undesirably imperfect and hence unsatisfactory; thus, we extend our analysis to the set of complex numbers **C** which incorporates the contributions of both the set of real probabilities which is **R** and the complementary set of imaginary probabilities which is **M**. Afterward, this will yield an absolute and perfect degree of our knowledge in the probability universe **C** ¼ **R** þ**M**because *Pc* = 1 constantly. As a matter of fact, the work in the universe **C** of complex probabilities gives way to a sure forecast of any stochastic experiment, since in **C** we remove and subtract from the computed degree of our knowledge the measured chaotic factor. This will generate in the universe **C** a probability equal to 1 (*Pc*<sup>2</sup> <sup>¼</sup> *DOK* � *Chf* <sup>¼</sup> *DOK* <sup>þ</sup> *MChf* <sup>¼</sup> <sup>1</sup> <sup>¼</sup> *Pc*). Many applications taking into consideration numerous continuous and discrete probability distributions in my 14 previous research papers confirm this hypothesis and innovative paradigm. The extended Kolmogorov axioms (EKA) or the complex probability paradigm (CPP) can be shown and summarized in the next illustration (**Figure 2**).

#### **4. The Monte Carlo techniques and the complex probability paradigm parameters**

#### **4.1 The divergence and convergence probabilities**

Let *RE* be the exact result of the stochastic phenomenon or of a multidimensional or simple integral that are not always possible to compute by probability theory ordinary procedures or by deterministic numerical means or by calculus [1–14]. And let *RA* be the phenomenon and integrals approximate results calculated by the techniques of Monte Carlo:

The relative error in the Monte Carlo methods is Rel*:*Error <sup>¼</sup> *RE*�*RA RE* � � � � � � <sup>¼</sup> <sup>1</sup> � *RA RE* � � � � � �.

Additionally, the percent relative error is = 100% � *RE*�*RA RE* � � � � � � and is always between 0% and 100%. Therefore, the relative error is always between 0 and 1. Hence

$$0 \le \left| \frac{R\_E - R\_A}{R\_E} \right| \le 1\\ \Leftrightarrow \begin{cases} \quad 0 \le \left( \frac{R\_E - R\_A}{R\_E} \right) \le 1 & \text{if } R\_A \le R\_E\\ \quad 0 \le \left( \frac{R\_E - R\_A}{R\_E} \right) \le 1 & \text{if } R\_A \ge R\_E \end{cases}\\ \Leftrightarrow \begin{cases} \quad 0 \le R\_A \le R\_E\\ \quad R\_E \le R\_A \le 2R\_E \end{cases}$$

Moreover, we define the real probability in the set **R** by

$$P\_r = \mathbf{1} - \left| \frac{R\_E - R\_A}{R\_E} \right| = \mathbf{1} - \left| \mathbf{1} - \frac{R\_A}{R\_E} \right| = \begin{cases} \mathbf{1} - \left( \mathbf{1} - \frac{R\_A}{R\_E} \right) & \text{if } \ 0 \le R\_A \le R\_E \\\\ \mathbf{1} + \left( \mathbf{1} - \frac{R\_A}{R\_E} \right) & \text{if } \ R\_E \le R\_A \le 2R\_E \\\\ 2 - \frac{R\_A}{R\_E} & \text{if } \ R\_E \le R\_A \le 2R\_E \end{cases}$$

= 1 � the relative error in the Monte Carlo method. = probability of Monte Carlo method convergence in **R**. And therefore,

$$P\_m = i(1 - P\_r) = i\left\{ \mathbf{1} - \left[ \mathbf{1} - \left| \frac{R\_E - R\_A}{R\_E} \right| \right] \right\} = i \left\{ \mathbf{1} - \left[ \mathbf{1} - \left| \mathbf{1} - \frac{R\_A}{R\_E} \right| \right] \right\} = i \left| \mathbf{1} - \frac{R\_A}{R\_E} \right|$$

$$= \begin{cases} i \left( \mathbf{1} - \frac{R\_A}{R\_E} \right) & \text{if } \ 0 \le R\_A \le R\_E \\\ -i \left( \mathbf{1} - \frac{R\_A}{R\_E} \right) & \text{if } \ R\_E \le R\_A \le 2R\_E \end{cases} = \begin{cases} i \left( \mathbf{1} - \frac{R\_A}{R\_E} \right) & \text{if } \ 0 \le R\_A \le R\_E \\\ i \left( \frac{R\_A}{R\_E} - 1 \right) & \text{if } \ R\_E \le R\_A \le 2R\_E \end{cases}$$

= probability of Monte Carlo method divergence in the imaginary complementary probability set **M** since it is the imaginary complement of *Pr*.

Consequently,

$$P\_m/i = \mathbf{1} - P\_r = \left| \mathbf{1} - \frac{R\_A}{R\_E} \right| = \begin{cases} \mathbf{1} - \frac{R\_A}{R\_E} & \text{if } \ \mathbf{0} \le R\_A \le R\_E \\\\ \frac{R\_A}{R\_E} - \mathbf{1} & \text{if } \ R\_E \le R\_A \le 2R\_E \end{cases}$$

= the relative error in the Monte Carlo method.

= probability of Monte Carlo method divergence in **R** since it is the real complement of *Pr*.

In the case where 0<sup>≤</sup> *RA* <sup>≤</sup>*RE* ) <sup>0</sup><sup>≤</sup> *RA RE* ≤1 ) 0≤*Pr* ≤ 1 and we deduce also that <sup>0</sup><sup>≤</sup> <sup>1</sup> � *RA RE* � �≤<sup>1</sup> ) <sup>0</sup>≤*Pm=i*≤1 and ) <sup>0</sup><sup>≤</sup> *Pm* <sup>≤</sup>*i*.

And in the case where *RE* <sup>≤</sup>*RA* <sup>≤</sup> <sup>2</sup>*RE* ) <sup>1</sup><sup>≤</sup> *RA RE* <sup>≤</sup><sup>2</sup> ) <sup>0</sup><sup>≤</sup> <sup>2</sup> � *RA RE* � �≤<sup>1</sup> )

0≤*Pr* ≤1 and we deduce also that 0 ≤ *RA RE* � 1 � �≤<sup>1</sup> ) <sup>0</sup>≤*Pm=i*≤1 and ) <sup>0</sup>≤*Pm* <sup>≤</sup>*i*. Consequently, if *RA* ¼ 0 or *RA* ¼ 2*RE* that means before the beginning of the

simulation, then

$$P\_r = P\_{rb} \text{ (convergence) in } \mathcal{R} = 0$$

$$P\_m = P\_{mb} \text{ (divergence) in } \mathcal{M} = i$$

$$P\_m/i = P\_{mb} \text{ (divergence) in } \mathcal{R} = 1$$

And if *RA* ¼ *RE* that means at the end of Monte Carlo simulation, then

$$P\_r = P\_{rob} \text{ (convergence) in } \mathcal{R} = 1$$

$$P\_m = P\_{rob} \text{ (divergence) in } \mathcal{M} = 0$$

$$P\_m/i = P\_{rob} \text{ (divergence) in } \mathcal{R} = 0$$

**4.2 The complex random vector** *Z* **in C** ¼ **R** þ**M**

$$\text{We have } \begin{aligned} Z &= P\_r + P\_m = \begin{cases} \frac{R\_A}{R\_E} + i \left( 1 - \frac{R\_A}{R\_E} \right) & \text{if } \ 0 \le R\_A \le R\_E\\ \left( 2 - \frac{R\_A}{R\_E} \right) + i \left( \frac{R\_A}{R\_E} - 1 \right) & \text{if } \ R\_E \le R\_A \le 2R\_E \end{cases} \\ &= \text{Re}\left( Z \right) + i \text{Im}(Z) \end{aligned}$$

.

where

and

and

*Pm* <sup>¼</sup> *<sup>i</sup>* <sup>1</sup> � *RA*

If *RA* <sup>¼</sup> *RE*

8 >><

>>:

and

*RA RE*

<sup>2</sup> � *RA RE*

*Pr* ¼

Re ð Þ¼ *Z Pr* ¼

*DOI: http://dx.doi.org/10.5772/intechopen.93048*

*Pr* <sup>¼</sup> *RA RE*

*RE* � �

therefore *Z* ¼ 0 þ *i* ¼ *i*.

*Pm* ¼

¼

therefore *Z* ¼ 0*:*5 þ 0*:*5*i*.

*Pr* ¼

**7**

8 >>><

>>>:

8 >>><

>>>:

*RA RE*

8 >><

>>:

<sup>2</sup> � *RA RE*

<sup>2</sup> or *RA* <sup>¼</sup> <sup>3</sup>*RE*

¼ *i*ð Þ¼ 2 � 1 *i*

Imð Þ¼ *Z Pm=i* ¼

*RA RE*

*The Monte Carlo Techniques and the Complex Probability Paradigm*

8 >><

>>:

<sup>1</sup> � *RA RE*

*RA RE*

8 >><

>>:

¼ 0 *RE*

<sup>¼</sup> *<sup>i</sup>* <sup>1</sup> � <sup>0</sup>

if 0 ≤*RA* ≤ *RE*

if *RE* ≤*RA* ≤ 2*RE*

*<sup>i</sup>* <sup>1</sup> � *RA RE*

*<sup>i</sup>* <sup>1</sup> � *RE* 2*RE* � �

*<sup>i</sup>* <sup>3</sup>*RE* 2*RE* � 1 � �

If *RA* ¼ *RE* (at the simulation end), then

<sup>¼</sup> *RE RE*

> <sup>¼</sup> <sup>2</sup> � *RE RE*

*<sup>i</sup> RA RE* � 1

*RE* � �

<sup>2</sup> � *RA RE* if 0≤ *RA* ≤*RE*

if *RE* ≤*RA* ≤2*RE*

if 0 ≤*RA* ≤ *RE*

That means that the complex random vector *Z* is the sum in **C** of the real probability of convergence in **R** and of the imaginary probability of divergence in **M**.

*RE*

<sup>¼</sup> *<sup>i</sup>*ð Þ¼ <sup>1</sup> � <sup>0</sup> *<sup>i</sup>* or *Pm* <sup>¼</sup> *<sup>i</sup> RA*

<sup>2</sup> (at the middle of the simulation), then

*RE* 2*RE*

<sup>2</sup> � <sup>3</sup>*RE* 2*RE*

¼ 0*:*5*i* if 0 ≤*RA* ≤*RE*

¼ 0*:*5*i* if *RE* ≤ *RA* ≤2*RE*

¼ 1 if 0≤ *RA* ≤*RE*

¼ 2 � 1 ¼ 1 if *RE* ≤*RA* ≤ 2*RE*

<sup>¼</sup> <sup>2</sup> � <sup>2</sup>*RE RE*

� 1 if *RE* ≤*RA* ≤2*RE*

If *RA* ¼ 0 or *RA* ¼ 2*RE* (before the simulation begins), then

<sup>¼</sup> 0 or *Pr* <sup>¼</sup> <sup>2</sup> � *RA*

¼

� � if 0<sup>≤</sup> *RA* <sup>≤</sup>*RE*

� � if *RE* <sup>≤</sup> *RA* <sup>≤</sup>2*RE*

8 >><

>>:

¼ the real part of *Z*

¼ the imaginary part of *Z:*

¼ 2 � 2 ¼ 0*:*

<sup>¼</sup> *<sup>i</sup>* <sup>2</sup>*RE RE* � 1 � �

⇔*Pr* ¼ 0*:*5

*RE* � 1 � �

¼ 0*:*5 if 0≤*RA* ≤*RE*

¼ 0*:*5 if *RE* ≤ *RA* ≤2*RE*

⇔*Pm* ¼ 0*:*5*i*

⇔*Pr* ¼ 1

*The Monte Carlo Techniques and the Complex Probability Paradigm DOI: http://dx.doi.org/10.5772/intechopen.93048*

where

$$\text{Re}\,(Z) = P\_r = \begin{cases} \frac{R\_A}{R\_E} & \text{if } \; 0 \le R\_A \le R\_E \\\\ 2 - \frac{R\_A}{R\_E} & \text{if } \; R\_E \le R\_A \le 2R\_E \end{cases} = \text{the real part of } Z = \text{"the real part of } Z = \text{"} $$

and

$$\operatorname{Im}(Z) = P\_m/i = \begin{cases} 1 - \frac{R\_A}{R\_E} & \text{if } \ 0 \le R\_A \le R\_E \\\ R\_A & = \text{the imaginary part of } Z. \end{cases} = \text{the imaginary part of } Z.$$

That means that the complex random vector *Z* is the sum in **C** of the real probability of convergence in **R** and of the imaginary probability of divergence in **M**.

If *RA* ¼ 0 or *RA* ¼ 2*RE* (before the simulation begins), then

$$P\_r = \frac{R\_A}{R\_E} = \frac{0}{R\_E} = 0 \text{ or } P\_r = 2 - \frac{R\_A}{R\_E} = 2 - \frac{2R\_E}{R\_E} = 2 - 2 = 0.1$$

and

$$\begin{split} P\_m &= i \left( \mathbf{1} - \frac{R\_A}{R\_E} \right) = i \left( \mathbf{1} - \frac{\mathbf{0}}{R\_E} \right) = i(\mathbf{1} - \mathbf{0}) = i \text{ or } P\_m = i \left( \frac{R\_A}{R\_E} - \mathbf{1} \right) = i \left( \frac{2R\_E}{R\_E} - \mathbf{1} \right) \\ &= i(\mathbf{2} - \mathbf{1}) = i \end{split}$$

therefore *Z* ¼ 0 þ *i* ¼ *i*. If *RA* <sup>¼</sup> *RE* <sup>2</sup> or *RA* <sup>¼</sup> <sup>3</sup>*RE* <sup>2</sup> (at the middle of the simulation), then

$$P\_r = \begin{cases} \begin{array}{c} R\_A \\ \frac{R\_E}{R\_E} \end{array} & \text{if } \ 0 \le R\_A \le R\_E \\\ 2 - \frac{R\_A}{R\_E} & \text{if } \ R\_E \le R\_A \le 2R\_E \end{cases} = \begin{cases} \begin{array}{c} \frac{R\_E}{2R\_E} = 0.5 & \text{if } \ 0 \le R\_A \le R\_E \\\ 2 - \frac{3R\_E}{2R\_E} = 0.5 & \text{if } \ R\_E \le R\_A \le 2R\_E \end{cases} \end{cases} \Leftrightarrow P\_r = 0.5$$

and

$$\begin{aligned} \;^1P\_m &= \begin{cases} i\left(1 - \frac{R\_A}{R\_E}\right) & \text{if } \; \mathbf{0} \le R\_A \le R\_E\\ i\left(\frac{R\_A}{R\_E} - 1\right) & \text{if } \; \; R\_E \le R\_A \le 2R\_E \end{cases} \\\ &= \begin{cases} i\left(1 - \frac{R\_E}{2R\_E}\right) = 0.5i & \text{if } \; \mathbf{0} \le R\_A \le R\_E\\ i\left(\frac{3R\_E}{2R\_E} - 1\right) = 0.5i & \text{if } \; \; R\_E \le R\_A \le 2R\_E \end{cases} \end{aligned} \Leftrightarrow \mathbf{P}\_m = 0.5i$$

therefore *Z* ¼ 0*:*5 þ 0*:*5*i*.

If *RA* ¼ *RE* (at the simulation end), then

$$P\_r = \begin{cases} \frac{R\_A}{R\_E} = \frac{R\_E}{R\_E} = 1 & \text{if } \ 0 \le R\_A \le R\_E\\ 2 - \frac{R\_A}{R\_E} = 2 - \frac{R\_E}{R\_E} = 2 - 1 = 1 & \text{if } \ R\_E \le R\_A \le 2R\_E \end{cases} \Leftrightarrow P\_r = 1$$

and

$$\begin{split} P\_m &= \begin{cases} i\left(\mathbf{1} - \frac{R\_A}{R\_E}\right) & \text{if } \ 0 \le R\_A \le R\_E\\ i\left(\frac{R\_A}{R\_E} - \mathbf{1}\right) & \text{if } \ R\_E \le R\_A \le 2R\_E \end{cases} &= \begin{cases} i\left(\mathbf{1} - \frac{R\_E}{R\_E}\right) & \text{if } \ 0 \le R\_A \le R\_E\\ i\left(\frac{R\_E}{R\_E} - \mathbf{1}\right) & \text{if } \ R\_E \le R\_A \le 2R\_E \end{cases} \\ &= \begin{cases} 0 & \text{if } \ 0 \le R\_A \le R\_E\\ 0 & \text{if } \ R\_E \le R\_A \le 2R\_E \end{cases} \\ \Leftrightarrow P\_m &= \mathbf{0} \end{split}$$

⇔

(

*DOI: http://dx.doi.org/10.5772/intechopen.93048*

**4.4 The chaotic factor,** *Chf*

*RA RE*

8 >>><

>>>:

<sup>2</sup> ¼ �1 then

*Chf* ¼

*RE* � �

> *RE* � 1 � �

In addition, if *Chf* ¼ 0 then

*RE* � �

�

*RE* � 1 � �

<sup>2</sup> � *RA RE*

> 8 >>>><

> >>>>:

We have

Since *i*

�<sup>2</sup> *RA RE* � � <sup>1</sup> � *RA*

�2 2 � *RA RE* � � *RA*

and vice versa.

⇔

⇔

**9**

8 >>>><

>>>>:

8 >>>><

>>>>:

simulation.

�<sup>2</sup> *RA RE* � � <sup>1</sup> � *RA*

�2 2 � *RA RE* � � *RA*

*Chf* ¼ 2*iPrPm*

¼ 2*i* �

*RA* ¼ 0 OR *RA* ¼ *RE* if 0 ≤*RA* ≤*RE RA* ¼ 2*RE* OR *RA* ¼ *RE* if *RE* ≤ *RA* ≤2*RE*

*The Monte Carlo Techniques and the Complex Probability Paradigm*

simulation after the convergence of the method to the exact result.

if 0 ≤*RA* ≤*RE*

if *RE* ≤*RA* ≤ 2*RE*

� � <sup>1</sup> � *RA*

*RE* � � *RA*

From *CPP* we have that �0*:*5≤*Chf* ≤ 0, and then if *Chf* ¼ �0*:*5

¼ 0 if 0≤*RA* ≤*RE*

¼ 0 if *RE* ≤*RA* ≤2*RE*

And, conversely, if *RA* <sup>¼</sup> 0 OR *RA* <sup>¼</sup> *RE* if 0 <sup>≤</sup>*RA* <sup>≤</sup>*RE*

¼ �0*:*5 if 0≤*RA* ≤*RE*

¼ �0*:*5 if *RE* ≤*RA* ≤2*RE*

That means that *Chf* is minimum when the approximate result *RA* is equal to half of the exact result *RE* if 0 ≤*RA* ≤*RE* or when the approximate result is equal to three times the half of the exact result if *RE* ≤ *RA* ≤2*RE*, which means at the middle of the

⇔

*RA* ¼ 2*RE* OR *RA* ¼ *RE* if *RE* ≤*RA* ≤2*RE*

8 < :

*RE*

*RE* � 1

�<sup>2</sup> *RA RE*

�2 2 � *RA*

That means that *DOK* is maximum when the approximate result *RA* is equal to 0 or 2*RE* (before the beginning of the simulation) and when it is equal to the exact result *RE* (at the end of the simulation). We can deduce that we have perfect and total knowledge of the stochastic experiment before the beginning of Monte Carlo simulation since no randomness was introduced yet, as well as at the end of the

�

8 >>>><

>>>>:

*<sup>i</sup>* <sup>1</sup> � *RA RE*

*<sup>i</sup> RA RE* � 1

� � if 0 <sup>≤</sup>*RA* <sup>≤</sup> *RE*

� � if *RE* <sup>≤</sup>*RA* <sup>≤</sup>2*RE*

⇔

(

� � if 0 <sup>≤</sup>*RA* <sup>≤</sup> *RE*

� � if *RE* <sup>≤</sup>*RA* <sup>≤</sup>2*RE*

*RA* ¼ *RE=*2 if 0≤*RA* ≤*RE RA* ¼ 3*RE=*2 if *RE* ≤*RA* ≤2*RE*

*RA* ¼ 0 OR *RA* ¼ *RE* if 0≤*RA* ≤*RE*

*RA* ¼ 2*RE* OR *RA* ¼ *RE* if *RE* ≤*RA* ≤2*RE*

, then *Chf* ¼ 0.

and vice versa*:*

therefore *Z* ¼ 1 þ 0*i* ¼ 1.

#### **4.3 The degree of our knowledge,** *DOK*

We have

*DOK* <sup>¼</sup> j j *<sup>Z</sup>* <sup>2</sup> <sup>¼</sup> *<sup>P</sup>*<sup>2</sup> *<sup>r</sup>* <sup>þ</sup> ð Þ *Pm=<sup>i</sup>* <sup>2</sup> <sup>¼</sup> *RA RE* � �<sup>2</sup> if 0≤*RA* ≤*RE* <sup>2</sup> � *RA RE* � �<sup>2</sup> if *RE* ≤*RA* ≤2*RE* 8 >>< >>: þ <sup>1</sup> � *RA RE* � �<sup>2</sup> if 0≤*RA* ≤*RE RA RE* � 1 � �<sup>2</sup> if *RE* ≤*RA* ≤2*RE* 8 >>< >>: ¼ *RA RE* � �<sup>2</sup> <sup>þ</sup> <sup>1</sup> � *RA RE* � �<sup>2</sup> if 0≤*RA* ≤*RE* <sup>2</sup> � *RA RE* � �<sup>2</sup> <sup>þ</sup> *RA RE* � 1 � �<sup>2</sup> if *RE* ≤*RA* ≤2*RE* 8 >>< >>: ¼ 2 *RA RE* � �<sup>2</sup> � 2 *RA RE* � � þ 1 if 0≤*RA* ≤*RE* 2 *RA RE* � �<sup>2</sup> � <sup>6</sup> *RA RE* � � þ 5 if *RE* ≤*RA* ≤2*RE* 8 >>>< >>>:

From *CPP* we have that 0*:*5≤ *DOK* ≤ 1, then if *DOK* = 0.5

$$\Leftrightarrow \begin{cases} 2\left(\frac{R\_A}{R\_E}\right)^2 - 2\left(\frac{R\_A}{R\_E}\right) + \mathbf{1} = \mathbf{0}.5 & \text{if } \ \mathbf{0} \le R\_A \le R\_E \\\ 2\left(\frac{R\_A}{R\_E}\right)^2 - 6\left(\frac{R\_A}{R\_E}\right) + \mathbf{5} = \mathbf{0}.5 & \text{if } \ R\_E \le R\_A \le 2R\_E \end{cases}$$

then solving the second-degree equations for *RA RE* gives

$$\begin{cases} \frac{R\_A}{R\_E} = 1/2 & \text{if } \ 0 \le R\_A \le R\_E\\ \frac{R\_A}{R\_E} = 3/2 & \text{if } \ R\_E \le R\_A \le 2R\_E \end{cases} \Leftrightarrow \begin{cases} \ R\_A = R\_E/2 & \text{if } \ 0 \le R\_A \le R\_E\\ R\_A = 3R\_E/2 & \text{if } \ R\_E \le R\_A \le 2R\_E \end{cases} \text{ and vice versa.}$$

That means that *DOK* is minimum when the approximate result *RA* is equal to half of the exact result *RE* if 0≤*RA* ≤*RE* or when the approximate result is equal to three times the half of the exact result if *RE* ≤*RA* ≤2*RE*, which means at the middle of the simulation.

In addition, if *DOK* ¼ 1, then

$$\Leftrightarrow \begin{cases} 2\left(\frac{R\_A}{R\_E}\right)^2 - 2\left(\frac{R\_A}{R\_E}\right) + 1 = 1 & \text{if } 0 \le R\_A \le R\_E\\ 2\left(\frac{R\_A}{R\_E}\right)^2 - 6\left(\frac{R\_A}{R\_E}\right) + 5 = 1 & \text{if } R\_E \le R\_A \le 2R\_E \end{cases} \Leftrightarrow \begin{cases} \left(\frac{R\_A}{R\_E}\right)^2 - \left(\frac{R\_A}{R\_E}\right) = 0 & \text{if } 0 \le R\_A \le R\_E\\ 2\left(\frac{R\_A}{R\_E}\right)^2 - 6\left(\frac{R\_A}{R\_E}\right) + 4 = 0 & \text{if } R\_E \le R\_A \le 2R\_E \end{cases} $$

*The Monte Carlo Techniques and the Complex Probability Paradigm DOI: http://dx.doi.org/10.5772/intechopen.93048*

$$\begin{cases} R\_A = 0 \text{ OR } R\_A = R\_E & \text{if } \ 0 \le R\_A \le R\_E \\\\ R\_A = 2R\_E \text{ OR } R\_A = R\_E & \text{if } \ R\_E \le R\_A \le 2R\_E \end{cases} \text{ and vice versa.}$$

That means that *DOK* is maximum when the approximate result *RA* is equal to 0 or 2*RE* (before the beginning of the simulation) and when it is equal to the exact result *RE* (at the end of the simulation). We can deduce that we have perfect and total knowledge of the stochastic experiment before the beginning of Monte Carlo simulation since no randomness was introduced yet, as well as at the end of the simulation after the convergence of the method to the exact result.

#### **4.4 The chaotic factor,** *Chf*

We have

$$\begin{aligned} \text{Cbf} &= 2 \text{iP}\_{} \text{P}\_{m} \\ &= 2i \times \begin{cases} \frac{R\_{A}}{R\_{E}} & \text{if } \ 0 \le R\_{A} \le R\_{E} \\\\ 2 - \frac{R\_{A}}{R\_{E}} & \text{if } \ R\_{E} \le R\_{A} \le 2R\_{E} \end{cases} \times \begin{cases} i \left( 1 - \frac{R\_{A}}{R\_{E}} \right) & \text{if } \ 0 \le R\_{A} \le R\_{E} \\\\ i \left( \frac{R\_{A}}{R\_{E}} - 1 \right) & \text{if } \ R\_{E} \le R\_{A} \le 2R\_{E} \end{cases} \end{aligned}$$

Since *i* <sup>2</sup> ¼ �1 then

$$Clf = \begin{cases} -2\left(\frac{R\_A}{R\_E}\right)\left(1 - \frac{R\_A}{R\_E}\right) & \text{if } \ 0 \le R\_A \le R\_E \\\\ -2\left(2 - \frac{R\_A}{R\_E}\right)\left(\frac{R\_A}{R\_E} - 1\right) & \text{if } \ R\_E \le R\_A \le 2R\_E \end{cases}$$

From *CPP* we have that �0*:*5≤*Chf* ≤ 0, and then if *Chf* ¼ �0*:*5

$$\Leftrightarrow \begin{cases} & -2\left(\frac{R\_A}{R\_E}\right)\left(1-\frac{R\_A}{R\_E}\right) = -0.5 & \text{if } \ 0 \le R\_A \le R\_E\\ & \Leftrightarrow \begin{cases} & R\_A = R\_E/2\\ & R\_A = 1 \end{cases} \Leftrightarrow \begin{cases} & R\_A = R\_E/2\\ & R\_A = 3R\_E/2 \end{cases} \text{ if } \ R\_E \le R\_A \le 2R\_E \end{cases} $$

and vice versa.

That means that *Chf* is minimum when the approximate result *RA* is equal to half of the exact result *RE* if 0 ≤*RA* ≤*RE* or when the approximate result is equal to three times the half of the exact result if *RE* ≤ *RA* ≤2*RE*, which means at the middle of the simulation.

In addition, if *Chf* ¼ 0 then

$$\Leftrightarrow \begin{cases} & -2\left(\frac{R\_A}{R\_E}\right)\left(1-\frac{R\_A}{R\_E}\right) = 0 \qquad \text{if} \quad 0 \le R\_A \le R\_E\\ & \Leftrightarrow \begin{cases} & R\_A = 0 \text{ OR } R\_A = R\_E \qquad \text{if } 0 \le R\_A \le R\_E\\ & \Leftrightarrow \begin{cases} & R\_A = 2R\_E \text{ OR } R\_A = R\_E \end{cases} \qquad \text{if } 0 \le R\_A \le R\_E \end{cases} \qquad \Leftrightarrow \begin{cases} & R\_A = 2R\_E \text{ OR } R\_A = R\_E\\ & R\_A = 2R\_E \text{ OR } R\_A = R\_E \quad \text{if } 0 \le R\_A \le R\_E \end{cases} \qquad \text{if } 0 \le R\_A \le R\_E \text{ OR } R\_A \le R\_E \end{cases}$$

And, conversely, if *RA* <sup>¼</sup> 0 OR *RA* <sup>¼</sup> *RE* if 0 <sup>≤</sup>*RA* <sup>≤</sup>*RE RA* ¼ 2*RE* OR *RA* ¼ *RE* if *RE* ≤*RA* ≤2*RE* � , then *Chf* ¼ 0.

That means that *Chf* is equal to 0 when the approximate result *RA* is equal to 0 or 2*RE* (before the beginning of the simulation) and when it is equal to the exact result *RE* (at the end of the simulation).

equal to the exact result *RE* (at the end of the simulation). We can deduce that the magnitude of the chaos in the stochastic experiment is null before the beginning of Monte Carlo simulation since no randomness was introduced yet, as well as at the end of the simulation after the convergence of the method to the exact result when

�

8 >>>><

>>>>:

⇔

⇔*RA* ¼ *RE* for 0 ≤ ∀*RA* ≤2*RE* continuously in the probability set **C** ¼ **R** þ**M**.

This is due to the fact that in **C**, we have subtracted in the equation above the chaotic factor *Chf* from our knowledge *DOK*, and therefore we have eliminated chaos caused and introduced by all the random variables and the stochastic fluctuations that lead to approximate results in the Monte Carlo simulation in **R**. Therefore, since in **C** we have always *RA* ¼ *RE*, then the Monte Carlo simulation which is a stochastic method by nature in **R** becomes after applying the *CPP* a deterministic method in **C** since the probability of convergence of any random experiment in **C** is constantly and permanently equal to 1 for any iterations

�

�<sup>2</sup> *RA RE* � � <sup>1</sup> � *RA*

�2 2 � *RA RE* � � *RA*

<sup>⇔</sup>*Pc*<sup>2</sup> <sup>¼</sup> 1 for 0 ≤ ∀*RA* <sup>≤</sup> <sup>2</sup>*RE*

*RE*

*RA* ¼ *RE* if 0 ≤*RA* ≤ *RE RA* ¼ *RE* if *RE* ≤*RA* ≤2*RE*

*RE* � 1

� � if 0≤*RA* <sup>≤</sup>*RE*

� � if *RE* <sup>≤</sup>*RA* <sup>≤</sup>2*RE*

¼ Re ð Þþ *Z i*Imð Þ *Z*

randomness has finished its task in the stochastic Monte Carlo method and

**4.6 The probability** *Pc* **in the probability set C** ¼ **R** þ**M**

*The Monte Carlo Techniques and the Complex Probability Paradigm*

*DOI: http://dx.doi.org/10.5772/intechopen.93048*

þ 1 if 0≤*RA* ≤*RE*

þ 5 if *RE* ≤*RA* ≤2*RE*

⇔*Pc* ¼ 1 = probability of convergence in **C**; therefore,

¼ 1 if 0≤ *RA* ≤*RE*

¼ 1 if *RE* ≤*RA* ≤2*RE*

**4.7 The rates of change of the probabilities in R, M, and C**

<sup>þ</sup> *<sup>i</sup>* <sup>1</sup> � *RA*

<sup>þ</sup> *<sup>i</sup> RA RE* � 1

*RE*

� � if 0 <sup>≤</sup>*RA* <sup>≤</sup>*RE*

� � if *RE* <sup>≤</sup>*RA* <sup>≤</sup> <sup>2</sup>*RE*

*RA RE*

8 >>>><

>>>>:

<sup>2</sup> � *RA RE* � �

<sup>¼</sup> 1 if 0<sup>≤</sup> *RA* <sup>≤</sup>*RE* 1 if *RE* ≤ *RA* ≤2*RE*

experiment.

We have

2 *RA RE* � �<sup>2</sup>

2 *RA RE* � �<sup>2</sup>

*Pc* ¼

number *N*.

Since

*Z* ¼ *Pr* þ *Pm* ¼

Then

**11**

¼

8 >>>><

>>>>:

*Pc*<sup>2</sup> <sup>¼</sup> *DOK* � *Chf* <sup>¼</sup> *DOK* <sup>þ</sup> *MChf*

� <sup>2</sup> *RA RE* � �

� <sup>6</sup> *RA RE* � �

�

*RA RE*

8 >>><

>>>:

<sup>2</sup> � *RA RE*

#### **4.5 The magnitude of the chaotic factor,** *MChf*

We have

$$\begin{aligned} \text{MChf} &= |\text{Ch}f| = -2i P\_r P\_m \\ &= -2i \times \begin{cases} \frac{R\_A}{R\_E} & \text{if } \ 0 \le R\_A \le R\_E \\\\ 2 - \frac{R\_A}{R\_E} & \text{if } \ R\_E \le R\_A \le 2R\_E \end{cases} \times \begin{cases} i \left(1 - \frac{R\_A}{R\_E}\right) & \text{if } \ 0 \le R\_A \le R\_E \\\\ i \left(\frac{R\_A}{R\_E} - 1\right) & \text{if } \ R\_E \le R\_A \le 2R\_E \end{cases} \end{aligned}$$

Since *i* <sup>2</sup> ¼ �1 then

$$\text{MChf} = \begin{cases} \quad 2\left(\frac{R\_A}{R\_E}\right)\left(1 - \frac{R\_A}{R\_E}\right) & \text{if} \quad 0 \le R\_A \le R\_E \\\\ \quad 2\left(2 - \frac{R\_A}{R\_E}\right)\left(\frac{R\_A}{R\_E} - 1\right) & \text{if} \quad R\_E \le R\_A \le 2R\_E \end{cases}$$

From *CPP* we have that 0 ≤ *MChf* ≤0*:*5, and then if *MChf* ¼ 0*:*5

$$\Leftrightarrow \begin{cases} \begin{array}{l} 2\left(\frac{R\_A}{R\_E}\right)\left(1-\frac{R\_A}{R\_E}\right) = 0.5\\\\ 2\left(2-\frac{R\_A}{R\_E}\right)\left(\frac{R\_A}{R\_E}-1\right) = 0.5 \end{array} \Leftrightarrow \begin{cases} \begin{array}{l} \begin{array}{l} R\_A = R\_E/2 \end{array} \quad \text{if} \quad 0 \leq R\_A \leq R\_E\\\\ R\_A = 3R\_E/2 \quad \text{if} \quad R\_E \leq R\_A \end{cases} \end{cases} \Leftrightarrow \begin{cases} \begin{array}{l} \begin{array}{l} \begin{array}{l} R\_A = R\_E/2 \end{array} \quad \text{if} \quad 0 \leq R\_A \leq R\_E \end{array} \end{cases} \Leftrightarrow \begin{cases} \begin{array}{l} \begin{array}{l} \begin{array}{l} \begin{array}{l} \begin{array}{l} \begin{array}{l} \end{array} \quad R\_A \leq R\_E \end{array} \end{cases} \end{cases} \end{cases}$$

and vice versa.

That means that *MChf* is maximum when the approximate result *RA* is equal to half of the exact result *RE* if 0≤ *RA* ≤*RE* or when the approximate result is equal to three times the half of the exact result if *RE* ≤*RA* ≤2*RE*, which means at the middle of the simulation. This implies that the magnitude of the chaos (*MChf*) introduced by the random variables used in Monte Carlo method is maximum at the halfway of the simulation.

In addition, if *MChf* ¼ 0, then

$$\Leftrightarrow \begin{cases} & 2\left(\frac{R\_A}{R\_E}\right)\left(1-\frac{R\_A}{R\_E}\right) = 0 \qquad \text{if} \; 0 \le R\_A \le R\_E\\ & 2\left(2-\frac{R\_A}{R\_E}\right)\left(\frac{R\_A}{R\_E}-1\right) = 0 \quad \text{if} \; \; R\_E \le R\_A \le 2R\_E \end{cases} \Leftrightarrow \begin{cases} & R\_A = 0 \quad \text{OR} \; \; R\_A = R\_E \qquad \text{if} \; \; 0 \le R\_A \le R\_E\\ & R\_A = 2R\_E \; \text{OR} \; \; R\_A = R\_E \quad \text{if} \; \; R\_E \le R\_A \le 2R\_E \end{cases} $$

And, conversely, if *RA* <sup>¼</sup> 0 OR *RA* <sup>¼</sup> *RE* if 0<sup>≤</sup> *RA* <sup>≤</sup>*RE RA* ¼ 2*RE* OR *RA* ¼ *RE* if *RE* ≤ *RA* ≤2*RE* � , then *MChf* ¼ 0.

That means that *MChf* is minimum and is equal to 0 when the approximate result *RA* is equal to 0 or 2*RE* (before the beginning of the simulation) and when it is

#### *The Monte Carlo Techniques and the Complex Probability Paradigm DOI: http://dx.doi.org/10.5772/intechopen.93048*

equal to the exact result *RE* (at the end of the simulation). We can deduce that the magnitude of the chaos in the stochastic experiment is null before the beginning of Monte Carlo simulation since no randomness was introduced yet, as well as at the end of the simulation after the convergence of the method to the exact result when randomness has finished its task in the stochastic Monte Carlo method and experiment.

### **4.6 The probability** *Pc* **in the probability set C** ¼ **R** þ**M**

We have

$$\begin{split} &Pc^{2} = DOK - G\mathbf{f}' = DOK + M\mathbf{C}\mathbf{f}' \\ &= \begin{cases} 2\left(\frac{R\_{A}}{R\_{E}}\right)^{2} - 2\left(\frac{R\_{A}}{R\_{E}}\right) + 1 & \text{if } 0 \le R\_{A} \le R\_{E} \\\\ 2\left(\frac{R\_{A}}{R\_{E}}\right)^{2} - 6\left(\frac{R\_{A}}{R\_{E}}\right) + 5 & \text{if } R\_{E} \le R\_{A} \le 2R\_{E} \end{cases} - \begin{cases} -2\left(\frac{R\_{A}}{R\_{E}}\right)\left(1 - \frac{R\_{A}}{R\_{E}}\right) & \text{if } 0 \le R\_{A} \le R\_{E} \\\\ -2\left(2 - \frac{R\_{A}}{R\_{E}}\right)\left(\frac{R\_{A}}{R\_{E}} - 1\right) & \text{if } R\_{E} \le R\_{A} \le 2R\_{E} \\\\ 1 & \text{if } R\_{E} \le R\_{A} \le 2R\_{E} \end{cases} \\ & \Leftrightarrow \mathbf{P}c^{2} = \mathbf{1} \quad \text{for} \quad \mathbf{0} \le \mathbf{V}R\_{A} \le 2R\_{E} \end{split}$$

⇔*Pc* ¼ 1 = probability of convergence in **C**; therefore,

$$Pc = \begin{cases} \begin{array}{l} R\_A = 1 \\ \frac{R\_E}{R\_E} = 1 \\\\ 2 - \frac{R\_A}{R\_E} = 1 \end{array} & \text{if } \begin{array}{l} 0 \le R\_A \le R\_E \\\\ R\_E \le R\_A \le 2R\_E \end{array} \\ \end{cases} \Leftrightarrow \begin{cases} R\_A = R\_E & \text{if } \ 0 \le R\_A \le R\_E \\\ R\_A = R\_E & \text{if } \ R\_E \le R\_A \le 2R\_E \end{cases}$$

⇔*RA* ¼ *RE* for 0 ≤ ∀*RA* ≤2*RE* continuously in the probability set **C** ¼ **R** þ**M**. This is due to the fact that in **C**, we have subtracted in the equation above the chaotic factor *Chf* from our knowledge *DOK*, and therefore we have eliminated chaos caused and introduced by all the random variables and the stochastic fluctuations that lead to approximate results in the Monte Carlo simulation in **R**. Therefore, since in **C** we have always *RA* ¼ *RE*, then the Monte Carlo simulation which is a stochastic method by nature in **R** becomes after applying the *CPP* a deterministic method in **C** since the probability of convergence of any random experiment in **C** is constantly and permanently equal to 1 for any iterations number *N*.

#### **4.7 The rates of change of the probabilities in R, M, and C**

Since

$$Z = P\_r + P\_m = \begin{cases} \frac{R\_A}{R\_E} + i \left( 1 - \frac{R\_A}{R\_E} \right) & \text{if } \ 0 \le R\_A \le R\_E \\\\ \left( 2 - \frac{R\_A}{R\_E} \right) + i \left( \frac{R\_A}{R\_E} - 1 \right) & \text{if } \ R\_E \le R\_A \le 2R\_E \end{cases} = \text{Re}\left( Z \right) + i \text{Im}(Z)$$

Then

$$\frac{dZ}{dR\_A} = \frac{dP\_r}{dR\_A} + \frac{dP\_m}{dR\_A} = \begin{cases} \quad & \frac{d}{dR\_A} \left[ \frac{R\_A}{R\_E} + i \left( 1 - \frac{R\_A}{R\_E} \right) \right] & \text{if } \ 0 \le R\_A \le R\_E \\\\ \quad & \frac{d}{dR\_A} \left[ \left( 2 - \frac{R\_A}{R\_E} \right) + i \left( \frac{R\_A}{R\_E} - 1 \right) \right] & \text{if } \ R\_E \le R\_A \le 2R\_E \\\\ \quad & \left( \frac{d}{dR\_A} \left[ \frac{R\_A}{R\_E} \right] + \frac{d}{dR\_A} \left[ i \left( 1 - \frac{R\_A}{R\_E} \right) \right] & \text{if } \ 0 \le R\_A \le R\_E \\\\ \quad & \frac{d}{dR\_A} \left[ 2 - \frac{R\_A}{R\_E} \right] + \frac{d}{dR\_A} \left[ i \left( \frac{R\_A}{R\_E} - 1 \right) \right] & \text{if } R\_E \le R\_A \le 2R\_E \\\\ \quad & \left( \frac{1}{R\_E} - \frac{i}{R\_E} = \frac{1}{R\_E} (1 - i) \qquad \text{if } \ 0 \le R\_A \le R\_E \\\\ \quad & -\frac{1}{R\_E} + \frac{i}{R\_E} = \frac{1}{R\_E} (i - 1) \quad \text{if } \ R\_E \le R\_A \le 2R\_E \end{cases}$$

Additionally,

ment converges to *RE*.

*dZ dRA* � � � �

� � � �

*DOI: http://dx.doi.org/10.5772/intechopen.93048*

2

¼

⇔ *dZ dRA* � � � �

> ⇔ *dZ dRA* � � � �

**5. The new paradigm parameter evaluation**

The random vector of complex probability

**13**

<sup>¼</sup> *dPr dRA* � �<sup>2</sup>

> 8 >><

> >>:

� � � �

2 ¼ 1 *R*2 *E* þ 1 *R*2 *E* ¼ 2 *R*2 *E*

1 *RE* � �<sup>2</sup>

*The Monte Carlo Techniques and the Complex Probability Paradigm*

� <sup>1</sup> *RE* � �<sup>2</sup>

> � � � � ¼

þ 1 *i dPm dRA* � �<sup>2</sup>

þ � <sup>1</sup> *RE* � �<sup>2</sup>

> <sup>þ</sup> <sup>1</sup> *RE* � �<sup>2</sup>

ffiffi 2 p *RE*

Furthermore, since *Pc*<sup>2</sup> <sup>¼</sup> *DOK* � *Chf* <sup>¼</sup> *DOK* <sup>þ</sup> *MChf* <sup>¼</sup> 1, then *Pc* ¼ 1 = probability of convergence in **C**, and consequently

> *d Pc* ð Þ *dRA*

We can infer from what has been developed earlier the following:

The real complementary probability of divergence *Pm*ð Þ *<sup>N</sup> <sup>=</sup><sup>i</sup>* <sup>¼</sup> *RE*�*RA*ð Þ *<sup>N</sup>*

� � � �

The real probability of convergence *Pr*ð Þ¼ *<sup>N</sup>* <sup>1</sup> � *RE*�*RA*ð Þ *<sup>N</sup>*

*Z N*ð Þ¼ *Pr*ð Þþ *<sup>N</sup> Pm*ð Þ¼ *<sup>N</sup>* <sup>1</sup> � *RE* � *RA*ð Þ *<sup>N</sup>*

<sup>¼</sup> *dPr dRA* � �<sup>2</sup>

¼ constant >0 if *RE* >0;

that means that the module of the slope of the complex probability vector *Z* in **C** or of its rate of change is constant and positive and it depends only on *RE*; hence, we have a constant increase in Re ð Þ *Z* and a constant decrease in Imð Þ *Z* as functions of the iterations number *N* and as *Z* goes from (0, *i*) at *N* = 0 till (1,0) at the simulation end; hence, till Re ð Þ¼ *Z Pr* reaches the value 1 that means till the random experi-

> <sup>¼</sup> *<sup>d</sup>*ð Þ<sup>1</sup> *dRA*

which means that *Pc* is constantly equal to 1 for every value of *RA*, of *RE*, and of the iterations number *N*, which means for any stochastic experiment and for any simulation of Monte Carlo method. So, we conclude that in **C** ¼ **R** þ**M**, we have complete and perfect knowledge of the random experiment which has become now a deterministic one since the extension in the complex probability plane **C** defined by the *CPP* axioms has changed all stochastic variables to deterministic variables.

We have 0≤ *N* ≤ *NC* where *N* = 0 corresponds to the instant before the beginning of the random experiment when *RA*ð Þ¼ *N* ¼ 0 0 or ¼ 2*RE* and where *N* ¼ *NC* (iterations number needed for the method convergence) corresponds to the instant at the end of the random experiments and Monte Carlo methods when *RA*ð Þ! *N* ¼ *NC RE*. The imaginary complementary probability of divergence *Pm*ð Þ¼ *<sup>N</sup> <sup>i</sup> RE*�*RA*ð Þ *<sup>N</sup>*

*RE*

� �

� � � �

þ *i*

� � � �

¼ 0,

*RE*

� � �.

*RE*

� � �.

� � � � � � �.

� � �

*RE*

� � �

*RE* � *RA*ð Þ *N RE*

� � � þ

if 0 ≤*RA* ≤*RE*

if *RE* ≤ *RA* ≤2*RE*

for 0 ≤ ∀*RA* ≤2*RE*

*d P*ð Þ *<sup>m</sup>=i dRA* � �<sup>2</sup>

Therefore,

$$\text{Re}\left[\frac{dZ}{dR\_A}\right] = \frac{dP\_r}{dR\_A} = \begin{cases} +\frac{1}{R\_E} & \text{if } \ 0 \le R\_A \le R\_E \\\\ -\frac{1}{R\_E} & \text{if } \ R\_E \le R\_A \le 2R\_E \\\\ \text{constant} > 0 & \text{if } \ 0 \le R\_A \le R\_E \text{ and } R\_E > 0 \end{cases}$$

$$= \begin{cases} \text{constant} > 0 & \text{if } \ R\_E \le R\_A \le 2R\_E \text{ and } R\_E > 0 \end{cases}$$

That means that the slope of the probability of convergence in **R** or its rate of change is constant and positive if 0≤ *RA* ≤*RE*, and constant and negative if *RE* ≤*RA* ≤2*RE*, and it depends only on *RE*; hence, we have a constant increase in *Pr* (the convergence probability) as a function of the iterations number *N* as *RA* increases from 0 to *RE* and as *RA* decreases from 2*RE* to *RE* till *Pr* reaches the value 1 that means till the random experiment converges to *RE*:

$$\operatorname{Im}\left[\frac{dZ}{dR\_A}\right] = \frac{1}{i}\frac{dP\_m}{dR\_A} = \frac{d(P\_m/i)}{dR\_A} = \begin{cases} -\frac{1}{R\_E} & \text{if } \ 0 \le R\_A \le R\_E \\\\ +\frac{1}{R\_E} & \text{if } \ R\_E \le R\_A \le 2R\_E \\\\ \text{constant} > 0 & \text{if } \ R\_E \le R\_E \text{ and } R\_E > 0 \end{cases}$$

$$= \begin{cases} \text{constant} < 0 & \text{if } \ 0 \le R\_A \le R\_E \text{ and } R\_E > 0 \\\\ \text{constant} > 0 & \text{if } \ R\_E \le R\_A \le 2R\_E \text{ and } R\_E > 0 \end{cases}$$

That means that the slopes of the probabilities of divergence in **R** and **M** or their rates of change are constant and negative if 0 ≤*RA* ≤*RE* and constant and positive if *RE* ≤*RA* ≤2*RE* and they depend only on *RE*; hence, we have a constant decrease in *Pm=i* and *Pm* (the divergence probabilities) as functions of the iterations number *N* as *RA* increases from 0 to *RE* and as *RA* decreases from 2*RE* to *RE* till *Pm=i* and *Pm* reach the value 0 that means till the random experiment converges to *RE*.

*The Monte Carlo Techniques and the Complex Probability Paradigm DOI: http://dx.doi.org/10.5772/intechopen.93048*

Additionally,

� � � �

$$\left|\frac{dZ}{dR\_A}\right|^2 = \left[\frac{dP\_r}{dR\_A}\right]^2 + \left[\frac{1}{i}\frac{dP\_m}{dR\_A}\right]^2 = \left[\frac{dP\_r}{dR\_A}\right]^2 + \left[\frac{d(P\_m/i)}{dR\_A}\right]^2$$

$$= \begin{cases} \left(\frac{1}{R\_E}\right)^2 + \left(-\frac{1}{R\_E}\right)^2 & \text{if } \ 0 \le R\_A \le R\_E \\\\ \left(-\frac{1}{R\_E}\right)^2 + \left(\frac{1}{R\_E}\right)^2 & \text{if } \ R\_E \le R\_A \le 2R\_E \end{cases}$$

$$\Leftrightarrow \left|\frac{dZ}{dR\_A}\right|^2 = \frac{1}{R\_E^2} + \frac{1}{R\_E^2} = \frac{2}{R\_E^2} \quad \text{for} \ 0 \le \forall R\_A \le 2R\_E$$

$$\Leftrightarrow \left|\frac{dZ}{dR\_A}\right| = \frac{\sqrt{2}}{R\_E} = \text{constant} > 0 \quad \text{if} \ R\_E > 0;$$

that means that the module of the slope of the complex probability vector *Z* in **C** or of its rate of change is constant and positive and it depends only on *RE*; hence, we have a constant increase in Re ð Þ *Z* and a constant decrease in Imð Þ *Z* as functions of the iterations number *N* and as *Z* goes from (0, *i*) at *N* = 0 till (1,0) at the simulation end; hence, till Re ð Þ¼ *Z Pr* reaches the value 1 that means till the random experiment converges to *RE*.

Furthermore, since *Pc*<sup>2</sup> <sup>¼</sup> *DOK* � *Chf* <sup>¼</sup> *DOK* <sup>þ</sup> *MChf* <sup>¼</sup> 1, then *Pc* ¼ 1 = probability of convergence in **C**, and consequently

$$\frac{d(Pc)}{dR\_A} = \frac{d(\mathbf{1})}{dR\_A} = \mathbf{0}\_\*$$

which means that *Pc* is constantly equal to 1 for every value of *RA*, of *RE*, and of the iterations number *N*, which means for any stochastic experiment and for any simulation of Monte Carlo method. So, we conclude that in **C** ¼ **R** þ**M**, we have complete and perfect knowledge of the random experiment which has become now a deterministic one since the extension in the complex probability plane **C** defined by the *CPP* axioms has changed all stochastic variables to deterministic variables.

#### **5. The new paradigm parameter evaluation**

We can infer from what has been developed earlier the following:

The real probability of convergence *Pr*ð Þ¼ *<sup>N</sup>* <sup>1</sup> � *RE*�*RA*ð Þ *<sup>N</sup> RE* � � � � � �.

We have 0≤ *N* ≤ *NC* where *N* = 0 corresponds to the instant before the beginning of the random experiment when *RA*ð Þ¼ *N* ¼ 0 0 or ¼ 2*RE* and where *N* ¼ *NC* (iterations number needed for the method convergence) corresponds to the instant at the end of the random experiments and Monte Carlo methods when *RA*ð Þ! *N* ¼ *NC RE*.

The imaginary complementary probability of divergence *Pm*ð Þ¼ *<sup>N</sup> <sup>i</sup> RE*�*RA*ð Þ *<sup>N</sup> RE* � � � � � �.

The real complementary probability of divergence *Pm*ð Þ *<sup>N</sup> <sup>=</sup><sup>i</sup>* <sup>¼</sup> *RE*�*RA*ð Þ *<sup>N</sup> RE* � � � � � �. The random vector of complex probability

$$Z(N) = P\_r(N) + P\_m(N) = \left[1 - \left|\frac{R\_E - R\_A(N)}{R\_E}\right|\right] + i \left|\frac{R\_E - R\_A(N)}{R\_E}\right|$$

The degree of our knowledge

$$\begin{split} D\text{OK}(N) &= |Z(N)|^2 = P\_r^2(N) + [P\_m(N)/i]^2 = \left[\mathbf{1} - \left|\frac{R\_E - R\_A(N)}{R\_E}\right|\right]^2 + \left[\left|\frac{R\_E - R\_A(N)}{R\_E}\right|\right]^2 \\ &= \mathbf{1} + 2i P\_r(N) P\_m(N) = \mathbf{1} - 2P\_r(N)[\mathbf{1} - P\_r(N)] = \mathbf{1} - 2P\_r(N) + 2P\_r^2(N) \\ &= \mathbf{1} - 2\left|\frac{R\_E - R\_A(N)}{R\_E}\right| + 2\left[\frac{R\_E - R\_A(N)}{R\_E}\right]^2. \end{split}$$

*DOK N*ð Þ is equal to 1 when *Pr*ð Þ¼ *N Pr*ð Þ¼ 0 0 and when *Pr*ð Þ¼ *N Pr*ð Þ¼ *NC* 1. The Chaotic factor

$$\begin{aligned} Chf(N) &= 2i P\_r(N) P\_m(N) = -2P\_r(N)[1 - P\_r(N)] = -2P\_r(N) + 2P\_r^2(N) \\ &= -2\left|\frac{R\_E - R\_A(N)}{R\_E}\right| + 2\left[\frac{R\_E - R\_A(N)}{R\_E}\right]^2 \end{aligned}$$

*Chf N*ð Þ is null when *Pr*ð Þ¼ *N Pr*ð Þ¼ 0 0 and when *Pr*ð Þ¼ *N Pr*ð Þ¼ *NC* 1. The magnitude of the chaotic factor *MChf*

$$\begin{aligned} \text{MClf}(N) &= |\text{Clf}(N)| = -2\text{i}P\_r(N)P\_m(N) = 2P\_r(N)[1 - P\_r(N)] = 2P\_r(N) - 2P\_r^2(N) \\\\ &= 2\left|\frac{R\_E - R\_A(N)}{R\_E}\right| - 2\left[\frac{R\_E - R\_A(N)}{R\_E}\right]^2 \end{aligned}$$

*MChf N*ð Þ is null when *Pr*ð Þ¼ *N Pr*ð Þ¼ 0 0 and when *Pr*ð Þ¼ *N Pr*ð Þ¼ *NC* 1. At any iteration number *N* 0≤ ∀*N* ≤ *NC*, the probability calculated in the set **C** of complex probabilities is as follows:

$$\begin{aligned} \text{Pr}^2(N) &= \left[P\_r(N) + P\_m(N)/i\right]^2 = \left|Z(N)\right|^2 - 2\text{iP}\_r(N)P\_m(N) = \text{DOK}(N) - \text{Cbf}(N) \\\\ &= \text{DOK}(N) + \text{MChf}(N) = \textbf{1} \end{aligned}$$

then

$$\left[\text{Pc}^2(\text{N}) = \left[P\_r(\text{N}) + P\_m(\text{N})/i\right]^2 = \left\{P\_r(\text{N}) + \left[\mathbf{1} - P\_r(\text{N})\right]\right\}^2 = \mathbf{1}^2 = \mathbf{1}\\ \Leftrightarrow \text{Pc}(\text{N}) = \mathbf{1}\\ \text{(continuously)}.$$

Thus, the prediction in the set **C** of the probabilities of convergence of the random Monte Carlo methods is always certain.

Let us consider afterward a multidimensional integral and a stochastic experiment to simulate the Monte Carlo procedures and to quantify, to draw, as well as to visualize all the prognostic and *CPP* parameters.

**7. Simulation of the new paradigm**

*The Monte Carlo Techniques and the Complex Probability Paradigm*

*DOI: http://dx.doi.org/10.5772/intechopen.93048*

**15**

Note that all the numerical values found in the simulations of the new paradigm for any iteration cycles *N* were computed using the 64-bit MATLAB version 2020 software and compared to the values found by Microsoft Visual C++ programs. Additionally, the reader should be careful of the truncation and rounding errors since we represent all numerical values by at most five significant digits and since we are using Monte Carlo techniques of simulation and integration which yield approximate results under the influence of stochastic aspects and variations. We have considered for this purpose a high-capacity computer system: a workstation computer with

parallel microprocessors, a 64-bit operating system, and a 64-GB RAM.

#### **6. The flowchart of the prognostic model of Monte Carlo techniques and** *CPP*

The flowchart that follows illustrates all the procedures of the elaborated prognostic model of *CPP*.

*The Monte Carlo Techniques and the Complex Probability Paradigm DOI: http://dx.doi.org/10.5772/intechopen.93048*

#### **7. Simulation of the new paradigm**

Note that all the numerical values found in the simulations of the new paradigm for any iteration cycles *N* were computed using the 64-bit MATLAB version 2020 software and compared to the values found by Microsoft Visual C++ programs. Additionally, the reader should be careful of the truncation and rounding errors since we represent all numerical values by at most five significant digits and since we are using Monte Carlo techniques of simulation and integration which yield approximate results under the influence of stochastic aspects and variations. We have considered for this purpose a high-capacity computer system: a workstation computer with parallel microprocessors, a 64-bit operating system, and a 64-GB RAM.

#### **7.1 The continuous random case: a four-dimensional multiple integral**

The Monte Carlo technique of integration can be summarized by the following equation:

⇔ ð 4*=*3

0

**Figure 5.**

**17**

**Figure 4.**

ð 4*=*3

ð 4*=*3

ð 4*=*3

*xyzw:dxdydzdw* ffi ½ � ð Þ� <sup>4</sup>*=*<sup>3</sup> � <sup>0</sup> ð Þ� <sup>4</sup>*=*<sup>3</sup> � <sup>0</sup> ð Þ� <sup>4</sup>*=*<sup>3</sup> � <sup>0</sup> ð Þ <sup>4</sup>*=*<sup>3</sup> � <sup>0</sup>

*z jw <sup>j</sup>* ¼ *RA*

Furthermore, the four figures (**Figures 3**–**6**) illustrate and prove the increasing

*N*

X*N j*¼1 *x jy <sup>j</sup> z jw <sup>j</sup>*

0

<sup>¼</sup> <sup>256</sup>*=*<sup>81</sup> *N*

with 1≤ *N* ≤ *NC* after applying Monte Carlo method.

*The Monte Carlo Techniques and the Complex Probability Paradigm*

*DOI: http://dx.doi.org/10.5772/intechopen.93048*

*The increasing convergence of the Monte Carlo method up to* N *= 500 iterations.*

*The increasing convergence of the Monte Carlo method up to* N *= 100 iterations.*

X*N j*¼1 *x jy <sup>j</sup>*

convergence of Monte Carlo simulation and technique to the exact result

0

0

$$\begin{aligned} \label{eq:1} \quad & \displaystyle \int\_{a\_1}^{b\_1} \int \dots \int\_{a\_n}^{b\_n} f(\mathbf{x}\_1, \mathbf{x}\_2, \dots, \mathbf{x}\_n) \, d\mathbf{x}\_1 d\mathbf{x}\_2 \dots d\mathbf{x}\_n \cong \\\\ \cline{-6.1.1} \quad & \displaystyle \frac{[(b\_1 - a\_1) \times (b\_2 - a\_2) \times \dots \times (b\_n - a\_n)]}{N} \sum\_{j=1}^N f\left(\mathbf{x}\_{1j}, \mathbf{x}\_{2j}, \dots, \mathbf{x}\_{nj}\right). \end{aligned}$$

Let us consider here the multidimensional integral of the following function:

$$\int\limits\_{0}^{4/3} \int\limits\_{0}^{\cdot} \int\limits\_{0}^{\cdot} \int\limits\_{0}^{\cdot} \int\limits\_{0}^{\cdot} \text{xyz}\,d\text{zd}d\text{ydz}d\text{w} = \int\limits\_{0}^{4/3} \int\limits\_{0}^{\cdot} \left[\frac{\text{x}^{2}}{2}\right]\_{0}^{4/3} \text{yz}\,d\text{ydz}d\text{w} = \int\limits\_{0}^{4/3} \int\limits\_{0}^{4/3} \left[\frac{16}{18}\text{yz}\,d\text{ydz}d\text{w}\right]$$

$$= \frac{8}{9} \int\limits\_{0}^{4/3} \left[\frac{\text{y}^{2}}{2}\right]\_{0}^{4/3} \text{zw}\,d\text{zd}d\text{w} = \frac{8}{9} \int\limits\_{0}^{4/3} \left[\frac{16}{18}\text{zw}\,d\text{zd}d\text{w} = \frac{64}{81} \int\limits\_{0}^{2} \left[\frac{\text{z}^{2}}{2}\right]\_{0}^{4/3} \text{w}\,d\text{w} = \frac{64}{81} \int\limits\_{0}^{4/3} \frac{16}{18}\text{w}\,d\text{w}$$

$$= \frac{512}{729} \left[\frac{\text{w}^{2}}{2}\right]\_{0}^{4/3} = \frac{512}{729} \times \frac{16}{18} = \frac{512}{729} \times \frac{8}{9} = \frac{4}{6}, 096$$

⇔*RE* ¼ 0*:*62429507696997411 … by the deterministic methods of calculus.

⇔*f x*ð Þ¼ , *y*, *z*, *w xyzw*, where *x*, *y*, *z*, and *w* follow a discrete uniform distribution *U* such that

$$x \mapsto U(0, 4/3), \\ y \mapsto U(0, 4/3), \\ z \mapsto U(0, 4/3), \\ w \mapsto U(0, 4/3)$$

**Figure 3.** *The increasing convergence of the Monte Carlo method up to* N *= 50 iterations.*

*The Monte Carlo Techniques and the Complex Probability Paradigm DOI: http://dx.doi.org/10.5772/intechopen.93048*

$$\stackrel{\text{def}}{\approx} \int\_0^{4/3} \int\_0^{4/3} \int\_0^{4/3} \int\_0^{4} \mathbf{x} \rho z \nu \, dx dy dz dw \cong \frac{\left[ (4/3 - 0) \times (4/3 - 0) \times (4/3 - 0) \times (4/3 - 0) \right]}{N} \sum\_{j=1}^N x\_j y\_j z\_j w\_j$$

$$= \frac{256/81}{N} \sum\_{j=1}^N x\_j y\_j z\_j w\_j = R\_A$$

with 1≤ *N* ≤ *NC* after applying Monte Carlo method.

Furthermore, the four figures (**Figures 3**–**6**) illustrate and prove the increasing convergence of Monte Carlo simulation and technique to the exact result

**Figure 4.** *The increasing convergence of the Monte Carlo method up to* N *= 100 iterations.*

**Figure 5.** *The increasing convergence of the Monte Carlo method up to* N *= 500 iterations.*

**7.2 The discrete random case: the matching birthday problem**

*The Monte Carlo Techniques and the Complex Probability Paradigm*

*DOI: http://dx.doi.org/10.5772/intechopen.93048*

*<sup>P</sup>* all *<sup>n</sup>* birthdays are different � � <sup>¼</sup> <sup>365</sup>

*<sup>P</sup>* matching birthdays � � <sup>¼</sup>

364 <sup>365</sup> � � �

matching birthdays for a selected number of people *n* (**Table 1**).

*<sup>P</sup>* matching birthdays � � <sup>¼</sup> 1 by the famous pigeonhole principle.

<sup>1</sup> � <sup>365</sup> <sup>365</sup> � � �

iterations. Consequently, we have lim *<sup>N</sup>*!þ<sup>∞</sup>

<sup>1</sup> � *RE*�*RE RE* � � �

**19**

� �

Carlo technique as *N* ! þ∞.

have the same birthday.

An interesting problem that can be solved using simulation is *the famous birthday problem*. Suppose that in a room of *n* persons, each of the 365 days of the year (not a leap year) is equally likely to be someone's birthday. It can be proved from the theory of probability and contrary to intuition that only 23 persons need to be present for the probability to be better than fifty-fifty that at least two of them will

Many people are interested in checking the theoretical proof of this statement, so we will demonstrate it briefly before doing the problem simulation. After someone is asked about his or her birthday, the probability that the next person asked will not have the same birthday is 364/365. The probability that the third person's birthday will not match those of the first two people are 363/365. It is well-known that the probability of two independent and successive events happening is the product of the probability of the separate events. In general, the probability that the *n*th person asked will have a birthday different from that of anyone already asked is

<sup>365</sup> � � �

�

The probability that the *n*th person asked will provide a match is 1 minus this value:

<sup>¼</sup> <sup>1</sup> � ð Þ� <sup>365</sup> ð Þ� <sup>364</sup> ð Þ� <sup>363</sup> … � ½ � <sup>365</sup> � ð Þ *<sup>n</sup>* � <sup>1</sup>

which shows that with 23 persons, the chances are 50.7%; with 55 persons, the chances are 98.6% or almost theoretically certain that at least two out of 55 people will have the same birthday. The table gives the theoretical probabilities of

Without using the probability theory, we can write a routine that uses the random number generator to compute the approximate chances for groups of *n* persons. Obviously, what is needed here is to choose *n* random integers from the set of integers {1, 2, 3, … , 365} and to check whether there is a match. When we repeat this experiment a large number of times, we can calculate afterward the probability of at least one match in any gathering of *n* persons. Note that if *n* ≥366, then

Furthermore, the four figures (**Figures 8**–**11**) illustrate and prove the increasing

*Pr*ð Þ¼ *N* lim

� <sup>¼</sup> <sup>1</sup> � <sup>0</sup> <sup>¼</sup> 1 which is equal to the probability of convergence of Monte

Moreover, **Figure 12** shows undoubtedly and graphically the relation of all the parameters of the complex probability paradigm (*Chf*, *RA*, *Pr*, *MChf*, *RE*, *DOK*, *Pm=i*, *Pc*) to the Monte Carlo technique after applying *CPP* to this problem of matching birthday.

convergence of Monte Carlo simulation and technique to the exact result *RE* ¼ 0*:*706316242719 … for *n* = 30 and for *N* = 50, 100, 500, and *N* ¼ *NC* ¼ 750, 000

363 <sup>365</sup> � � � … �

364 <sup>365</sup> � � �

365 � ð Þ *n* � 1 <sup>365</sup> � �

363 <sup>365</sup> � � � …

365 � ð Þ *n* � 1 <sup>365</sup> � �

<sup>365</sup>*<sup>n</sup>* <sup>¼</sup> *RE*

*<sup>N</sup>*!þ<sup>∞</sup> <sup>1</sup> � *RE*�*RA*ð Þ *<sup>N</sup>*

� � �

*RE*

n o <sup>¼</sup>

� � �

**Figure 6.**

*The increasing convergence of the Monte Carlo method up to* N *= 100,000 iterations.*

**Figure 7.** *The* CPP *parameters and the Monte Carlo method for a multiple integral.*

*RE* ¼ 0*:*62429507696997411 … for *N* = 50, 100, 500, and *N* ¼ *NC* ¼ 100, 000 iterations. Consequently, we have lim*<sup>N</sup>*!þ<sup>∞</sup> *Pr*ð Þ¼ *<sup>N</sup>* lim*<sup>N</sup>*!þ<sup>∞</sup> <sup>1</sup> � *RE*�*RA*ð Þ *<sup>N</sup> RE* � � � � � � n o <sup>¼</sup> <sup>1</sup> � *RE*�*RE RE* � � � � � � <sup>¼</sup> <sup>1</sup> � <sup>0</sup> <sup>¼</sup> 1 which is equal to the probability of convergence of Monte Carlo technique as *N* ! þ∞.

Moreover, **Figure 7** shows undoubtedly and graphically the relation of all the parameters of the complex probability paradigm (*Chf*, *RA*, *Pr*, *MChf*, *RE*, *DOK*, *Pm=i*, *Pc*) to the Monte Carlo technique after applying *CPP* to this four-dimensional integral.

#### **7.2 The discrete random case: the matching birthday problem**

An interesting problem that can be solved using simulation is *the famous birthday problem*. Suppose that in a room of *n* persons, each of the 365 days of the year (not a leap year) is equally likely to be someone's birthday. It can be proved from the theory of probability and contrary to intuition that only 23 persons need to be present for the probability to be better than fifty-fifty that at least two of them will have the same birthday.

Many people are interested in checking the theoretical proof of this statement, so we will demonstrate it briefly before doing the problem simulation. After someone is asked about his or her birthday, the probability that the next person asked will not have the same birthday is 364/365. The probability that the third person's birthday will not match those of the first two people are 363/365. It is well-known that the probability of two independent and successive events happening is the product of the probability of the separate events. In general, the probability that the *n*th person asked will have a birthday different from that of anyone already asked is

$$P(\text{all } n \text{ birthdays are different}) = \begin{pmatrix} 365\\ 365 \end{pmatrix} \times \begin{pmatrix} 364\\ 365 \end{pmatrix} \times \begin{pmatrix} 363\\ 365 \end{pmatrix} \times \dots$$

$$\times \left(\frac{365 - (n - 1)}{365}\right)$$

The probability that the *n*th person asked will provide a match is 1 minus this value:

*<sup>P</sup>* matching birthdays � � <sup>¼</sup>

$$\begin{aligned} 1 - \left(\frac{365}{365}\right) \times \left(\frac{364}{365}\right) \times \left(\frac{363}{365}\right) \times \dots \times \left(\frac{365 - (n - 1)}{365}\right) \\ = 1 - \frac{(365) \times (364) \times (363) \times \dots \times [365 - (n - 1)]}{365^\circ} = R\_E \end{aligned}$$

which shows that with 23 persons, the chances are 50.7%; with 55 persons, the chances are 98.6% or almost theoretically certain that at least two out of 55 people will have the same birthday. The table gives the theoretical probabilities of matching birthdays for a selected number of people *n* (**Table 1**).

Without using the probability theory, we can write a routine that uses the random number generator to compute the approximate chances for groups of *n* persons. Obviously, what is needed here is to choose *n* random integers from the set of integers {1, 2, 3, … , 365} and to check whether there is a match. When we repeat this experiment a large number of times, we can calculate afterward the probability of at least one match in any gathering of *n* persons. Note that if *n* ≥366, then *<sup>P</sup>* matching birthdays � � <sup>¼</sup> 1 by the famous pigeonhole principle.

Furthermore, the four figures (**Figures 8**–**11**) illustrate and prove the increasing convergence of Monte Carlo simulation and technique to the exact result *RE* ¼ 0*:*706316242719 … for *n* = 30 and for *N* = 50, 100, 500, and *N* ¼ *NC* ¼ 750, 000 iterations. Consequently, we have lim *<sup>N</sup>*!þ<sup>∞</sup> *Pr*ð Þ¼ *N* lim *<sup>N</sup>*!þ<sup>∞</sup> <sup>1</sup> � *RE*�*RA*ð Þ *<sup>N</sup> RE* � � � � � � n o <sup>¼</sup>

<sup>1</sup> � *RE*�*RE RE* � � � � � � <sup>¼</sup> <sup>1</sup> � <sup>0</sup> <sup>¼</sup> 1 which is equal to the probability of convergence of Monte Carlo technique as *N* ! þ∞.

Moreover, **Figure 12** shows undoubtedly and graphically the relation of all the parameters of the complex probability paradigm (*Chf*, *RA*, *Pr*, *MChf*, *RE*, *DOK*, *Pm=i*, *Pc*) to the Monte Carlo technique after applying *CPP* to this problem of matching birthday.


#### **Table 1.**

*Some theoretical probabilities of matching birthdays for n people where* 1≤*n*≤365*.*

#### **Figure 8.**

*The increasing convergence of the Monte Carlo method up to* N *= 50 iterations.*

#### *7.2.1 The cubes of complex probability*

In **Figure 13** and in the first cube, the simulation of *Chf* and *DOK* as functions of the iterations *N* and of each other is executed for the problem of matching birthday. If we project *Pc2* (*N*) = *DOK*(*N*) *Chf*(*N*)=1= *Pc*(*N*) on the plane *N* = 0 iterations, we will get the line in cyan. The starting point of this line is point J (*DOK* = 1, *Chf* = 0) when *N* = 0 iterations, and then the line gets to point (*DOK* = 0.5, *Chf* = 0.5) when *N* = 375,000 iterations and joins finally and again point J

and L correspond to the same points that exist in **Figure 12**.

*The increasing convergence of the Monte Carlo method up to* N *= 500 iterations.*

*The increasing convergence of the Monte Carlo method up to* N *= 100 iterations.*

*The Monte Carlo Techniques and the Complex Probability Paradigm*

*DOI: http://dx.doi.org/10.5772/intechopen.93048*

**Figure 9.**

**Figure 10.**

**21**

(*DOK* = 1, *Chf* = 0) when *N=NC* = 750,000 iterations. The graphs of *Chf*(*N*) (pink, green, blue) in different planes and *DOK*(*N*) (red) represent the other curves. We can notice that point K (*DOK* = 0.5, *Chf* = 0.5, *N =* 375,000 iterations) is the minimum of all these curves. We can notice also that point L has the coordinates (*DOK* = 1, *Chf* = 0, *N=NC* = 750,000 iterations). Additionally, the three points J, K, *The Monte Carlo Techniques and the Complex Probability Paradigm DOI: http://dx.doi.org/10.5772/intechopen.93048*

**Figure 9.** *The increasing convergence of the Monte Carlo method up to* N *= 100 iterations.*

**Figure 10.** *The increasing convergence of the Monte Carlo method up to* N *= 500 iterations.*

we will get the line in cyan. The starting point of this line is point J (*DOK* = 1, *Chf* = 0) when *N* = 0 iterations, and then the line gets to point (*DOK* = 0.5, *Chf* = 0.5) when *N* = 375,000 iterations and joins finally and again point J (*DOK* = 1, *Chf* = 0) when *N=NC* = 750,000 iterations. The graphs of *Chf*(*N*) (pink, green, blue) in different planes and *DOK*(*N*) (red) represent the other curves. We can notice that point K (*DOK* = 0.5, *Chf* = 0.5, *N =* 375,000 iterations) is the minimum of all these curves. We can notice also that point L has the coordinates (*DOK* = 1, *Chf* = 0, *N=NC* = 750,000 iterations). Additionally, the three points J, K, and L correspond to the same points that exist in **Figure 12**.

**Figure 11.** *The increasing convergence of the Monte Carlo method up to* N *= 750,000 iterations.*

**Figure 13.**

**Figure 14.**

**23**

Chf *and* DOK *in terms of each other and of* N *for the problem of matching birthday.*

*The Monte Carlo Techniques and the Complex Probability Paradigm*

*DOI: http://dx.doi.org/10.5772/intechopen.93048*

Pm*/*i *and* Pr *in terms of each other and of* N *for the problem of matching birthday.*

**Figure 12.** *The* CPP *parameters and the Monte Carlo techniques for the matching birthday problem.*

In **Figure 14** and in the second cube, we simulate the probability of convergence *Pr*(*N*) and its complementary real probability of divergence *Pm*(*N*)/*i* as functions of the iterations *N* for the problem of matching birthday. If we project *Pc*<sup>2</sup> (*N*) = *Pr*(*N*) + *Pm*(*N*)/*i* =1= *Pc*(*N*) on the plane *N* = 0 iterations, we will get the line in cyan. The starting point of this line is point (*Pr* = 0, *Pm*/*i* = 1), and the final point is

*The Monte Carlo Techniques and the Complex Probability Paradigm DOI: http://dx.doi.org/10.5772/intechopen.93048*

#### **Figure 13.**

Chf *and* DOK *in terms of each other and of* N *for the problem of matching birthday.*

**Figure 14.** Pm*/*i *and* Pr *in terms of each other and of* N *for the problem of matching birthday.*

*Pm* = 0.5*i* when *N* = 375,000 iterations. Additionally, the three points J, K, and L

In the current chapter, the extended and original Kolmogorov model of eight axioms (*EKA*) was connected and applied to the random and classical Monte Carlo techniques. Thus, we have bonded Monte Carlo algorithms to the novel *CPP* paradigm. Accordingly, the paradigm of "complex probability" was more expanded

Also, as it was proved and demonstrated in the original paradigm, when *N* = 0

nondeterministic phenomenon (*N* > 0), we have 0 < *MChf* ≤ 0.5, 0.5 ≤ *DOK* < 1, and � 0.5 ≤ *Chf* < 0, and it can be noticed that throughout this entire process, we have continually and incessantly *Pc*<sup>2</sup> <sup>=</sup> *DOK* � *Chf* <sup>=</sup> *DOK* <sup>+</sup> *MChf* =1= *Pc*, which means that the simulation which seemed to be random and nondeterministic in the set **R** is now deterministic and certain in the set **C** ¼ **R** þ**M**, and this after adding the contributions of **M** to the experiment happening in **R** and thus after removing and subtracting the chaotic factor from the degree of our knowledge. Additionally, the probabilities of convergence and divergence of the random Monte Carlo procedure that correspond to each iteration cycle *N* have been determined in the three sets of probabilities which are **C**, **M**, and **R** by *Pc*, *Pm*, and *Pr*, respectively. Subsequently, at each instance of *N*, the novel Monte Carlo techniques and *CPP* parameters *DOK*, *Chf*, *MChf*, *RE*, *RA*, *Pr*, *Pm*, *Pm=i*, *Pc*, and *Z* are perfectly and surely predicted in the set of complex probabilities **C** with *Pc* kept as equal to 1 continuously and forever. Also, referring to all these shown simulations and obtained graphs all over the entire chapter, we can visualize and quantify both the system chaos and stochastic influences and aspects (expressed by *Chf* and *MChf*) and the certain knowledge (expressed by *DOK* and *Pc*) of Monte Carlo algorithms. This is definitely very wonderful, fruitful, and fascinating and demonstrates once again the advantages of extending the five axioms of probability of Kolmogorov and thus the benefits and novelty of this original theory in applied mathematics and prognostics

(before the beginning of the random simulation) and when *N* = *N*<sup>C</sup> (after the convergence of Monte Carlo algorithm to the exact result), then the chaotic factor (*Chf* and *MChf*) is 0, and the degree of our knowledge (*DOK*) is 1 since the stochastic aspects and variations have either not commenced yet or they have terminated their job on the random phenomenon. During the course of the

correspond to the same points that exist in **Figure 12**.

*The Monte Carlo Techniques and the Complex Probability Paradigm*

beyond the scope of my 14 earlier studies on this topic.

that can be called verily: "the complex probability paradigm."

doubt the success of my innovative theory.

applied mathematics.

**25**

Moreover, it is important to mention here that one essential and very wellknown probability distribution was taken into consideration in the current chapter which is the uniform and discrete probability distribution as well as a specific generator of uniform random numbers, knowing that the original *CPP* model can be applied to any generator of uniform random numbers that exists in literature. This will yield certainly analogous results and conclusions and will confirm without any

As a prospective and future challenges and research, we intend to more develop

the novel conceived prognostic paradigm and to apply it to a diverse set of nondeterministic events like for other stochastic phenomena as in the classical theory of probability and in stochastic processes. Additionally, we will implement *CPP* to the first-order reliability method (FORM) in the field of prognostic in engineering and also to the problems of random walk which have huge consequences when applied to economics, to chemistry, to physics, and to pure and

**8. Perspectives and conclusion**

*DOI: http://dx.doi.org/10.5772/intechopen.93048*

**Figure 15.** *The vector of complex probability* Z *in terms of* N *for the problem of matching birthday.*

point (*Pr* = 1, *Pm*/*i* = 0). The graph of *Pr*(*N*) in the plane *Pr*(*N*) = *Pm*(*N*)/*i* is represented by the red curve. The starting point of this graph is point J (*Pr* = 0, *Pm*/*i* = 1, *N* = 0 iterations), and then it gets to point K (*Pr* = 0.5, *Pm*/*i* = 0.5, *N* = 375,000 iterations) and joins finally point L (*Pr* = 1, *Pm*/*i* = 0, *N* = *NC* = 750,000 iterations). The graph of *Pm*(*N*)/*i* in the plane *Pr*(*N*) + *Pm*(*N*)/*i* = 1 is represented by the blue curve. We can notice how much point K is important and which is the intersection of the blue and red graphs when *Pr*(*N*) = *Pm*(*N*)/*i* = 0.5 at *N* = 375,000 iterations. Additionally, the three points J, K, and L correspond to the same points that exist in **Figure 12**.

In **Figure 15** and in the third cube, we simulate the vector of complex probabilities *Z*(*N*) in **C** as a function of the real probability of convergence *Pr*(*N*) = Re(*Z*) in **R** and of its complementary imaginary probability of divergence *Pm*(*N*) = *i* Im(*Z*) in **M**, and as a function of the iterations *N* for the problem of matching birthday. The graph of *Pr*(*N*) in the plane *Pm*(*N*) = 0 is represented by the red curve, and the graph of *Pm*(*N*) in the plane *Pr*(*N*) = 0 is represented by the blue curve. The graph of the vector of complex probabilities *Z*(*N*) = *Pr*(*N*) + *Pm*(*N*) = Re(*Z*) + *i* Im(*Z*) in the plane *Pr*(*N*) = *iPm*(*N*) + 1 is represented by the green curve. The graph of *Z*(*N*) has point J (*Pr* = 0, *Pm* = *i*, *N* = 0 iterations) as the starting point and point L (*Pr* = 1, *Pm* = 0, *N* = *NC* = 750,000 iterations) as the end point. If we project *Z*(*N*) curve on the plane of complex probabilities whose equation is *N* = 0 iterations, we get the line in cyan which is *Pr*(0) = *iPm*(0) + 1. This projected line has point J (*Pr* = 0, *Pm* = *i*, *N* = 0 iterations) as the starting point and point (*Pr* = 1, *Pm* = 0, *N* = 0 iterations) as the end point. We can notice how much point K is important, and it corresponds to *Pr* = 0.5 and

*Pm* = 0.5*i* when *N* = 375,000 iterations. Additionally, the three points J, K, and L correspond to the same points that exist in **Figure 12**.

#### **8. Perspectives and conclusion**

In the current chapter, the extended and original Kolmogorov model of eight axioms (*EKA*) was connected and applied to the random and classical Monte Carlo techniques. Thus, we have bonded Monte Carlo algorithms to the novel *CPP* paradigm. Accordingly, the paradigm of "complex probability" was more expanded beyond the scope of my 14 earlier studies on this topic.

Also, as it was proved and demonstrated in the original paradigm, when *N* = 0 (before the beginning of the random simulation) and when *N* = *N*<sup>C</sup> (after the convergence of Monte Carlo algorithm to the exact result), then the chaotic factor (*Chf* and *MChf*) is 0, and the degree of our knowledge (*DOK*) is 1 since the stochastic aspects and variations have either not commenced yet or they have terminated their job on the random phenomenon. During the course of the nondeterministic phenomenon (*N* > 0), we have 0 < *MChf* ≤ 0.5, 0.5 ≤ *DOK* < 1, and � 0.5 ≤ *Chf* < 0, and it can be noticed that throughout this entire process, we have continually and incessantly *Pc*<sup>2</sup> <sup>=</sup> *DOK* � *Chf* <sup>=</sup> *DOK* <sup>+</sup> *MChf* =1= *Pc*, which means that the simulation which seemed to be random and nondeterministic in the set **R** is now deterministic and certain in the set **C** ¼ **R** þ**M**, and this after adding the contributions of **M** to the experiment happening in **R** and thus after removing and subtracting the chaotic factor from the degree of our knowledge. Additionally, the probabilities of convergence and divergence of the random Monte Carlo procedure that correspond to each iteration cycle *N* have been determined in the three sets of probabilities which are **C**, **M**, and **R** by *Pc*, *Pm*, and *Pr*, respectively. Subsequently, at each instance of *N*, the novel Monte Carlo techniques and *CPP* parameters *DOK*, *Chf*, *MChf*, *RE*, *RA*, *Pr*, *Pm*, *Pm=i*, *Pc*, and *Z* are perfectly and surely predicted in the set of complex probabilities **C** with *Pc* kept as equal to 1 continuously and forever. Also, referring to all these shown simulations and obtained graphs all over the entire chapter, we can visualize and quantify both the system chaos and stochastic influences and aspects (expressed by *Chf* and *MChf*) and the certain knowledge (expressed by *DOK* and *Pc*) of Monte Carlo algorithms. This is definitely very wonderful, fruitful, and fascinating and demonstrates once again the advantages of extending the five axioms of probability of Kolmogorov and thus the benefits and novelty of this original theory in applied mathematics and prognostics that can be called verily: "the complex probability paradigm."

Moreover, it is important to mention here that one essential and very wellknown probability distribution was taken into consideration in the current chapter which is the uniform and discrete probability distribution as well as a specific generator of uniform random numbers, knowing that the original *CPP* model can be applied to any generator of uniform random numbers that exists in literature. This will yield certainly analogous results and conclusions and will confirm without any doubt the success of my innovative theory.

As a prospective and future challenges and research, we intend to more develop the novel conceived prognostic paradigm and to apply it to a diverse set of nondeterministic events like for other stochastic phenomena as in the classical theory of probability and in stochastic processes. Additionally, we will implement *CPP* to the first-order reliability method (FORM) in the field of prognostic in engineering and also to the problems of random walk which have huge consequences when applied to economics, to chemistry, to physics, and to pure and applied mathematics.

### **Nomenclature**


**References**

**9**(4):289-304

[1] Abou Jaoude A, El-Tawil K, Kadry S. Prediction in complex dimension using Kolmogorov's set of axioms. Journal of Mathematics and Statistics, Science Publications. 2010;**6**(2):116-124

*DOI: http://dx.doi.org/10.5772/intechopen.93048*

*The Monte Carlo Techniques and the Complex Probability Paradigm*

prognostic for unburied petrochemical pipelines. Systems Science and Control Engineering, Taylor and Francis Publishers. 2017;**5**(1):178-214

[10] Abou Jaoude A. The paradigm of complex probability and Claude Shannon's information theory. Systems Science and Control Engineering, Taylor and Francis Publishers. 2017;**5**(1):

[11] Abou Jaoude A. The paradigm of complex probability and analytic nonlinear prognostic for unburied petrochemical pipelines. Systems Science and Control Engineering, Taylor and Francis Publishers. 2017;**5**(1):495-534

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[4] Abou Jaoude A. Complex probability theory and prognostic. Journal of Mathematics and Statistics, Science Publications. 2014;**10**(1):1-24

probability paradigm and analytic linear prognostic for vehicle suspension systems. American Journal of Engineering and Applied Sciences, Science Publications. 2015;**8**(1):147-175

[6] Abou Jaoude A. The paradigm of complex probability and the Brownian motion. Systems Science and Control Engineering, Taylor and Francis Publishers. 2015;**3**(1):478-503

[7] Abou Jaoude A. The paradigm of complex probability and Chebyshev's inequality. Systems Science and Control

[8] Abou Jaoude A. The paradigm of complex probability and analytic nonlinear prognostic for vehicle suspension systems. Systems Science and Control Engineering, Taylor and Francis Publishers. 2016;**4**(1):99-137

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**27**

Engineering, Taylor and Francis Publishers. 2016;**4**(1):99-137

reliability method. Journal of Mathematics and Statistics, Science Publications. 2013;**9**(4):310-324

### **Author details**

Abdo Abou Jaoude

Department of Mathematics and Statistics, Faculty of Natural and Applied Sciences, Notre Dame University - Louaize, Lebanon

\*Address all correspondence to: abdoaj@idm.net.lb

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

*The Monte Carlo Techniques and the Complex Probability Paradigm DOI: http://dx.doi.org/10.5772/intechopen.93048*

#### **References**

[1] Abou Jaoude A, El-Tawil K, Kadry S. Prediction in complex dimension using Kolmogorov's set of axioms. Journal of Mathematics and Statistics, Science Publications. 2010;**6**(2):116-124

[2] Abou Jaoude A. The complex statistics paradigm and the law of large numbers. Journal of Mathematics and Statistics, Science Publications. 2013; **9**(4):289-304

[3] Abou Jaoude A. The theory of complex probability and the first order reliability method. Journal of Mathematics and Statistics, Science Publications. 2013;**9**(4):310-324

[4] Abou Jaoude A. Complex probability theory and prognostic. Journal of Mathematics and Statistics, Science Publications. 2014;**10**(1):1-24

[5] Abou Jaoude A. The complex probability paradigm and analytic linear prognostic for vehicle suspension systems. American Journal of Engineering and Applied Sciences, Science Publications. 2015;**8**(1):147-175

[6] Abou Jaoude A. The paradigm of complex probability and the Brownian motion. Systems Science and Control Engineering, Taylor and Francis Publishers. 2015;**3**(1):478-503

[7] Abou Jaoude A. The paradigm of complex probability and Chebyshev's inequality. Systems Science and Control Engineering, Taylor and Francis Publishers. 2016;**4**(1):99-137

[8] Abou Jaoude A. The paradigm of complex probability and analytic nonlinear prognostic for vehicle suspension systems. Systems Science and Control Engineering, Taylor and Francis Publishers. 2016;**4**(1):99-137

[9] Abou Jaoude A. The paradigm of complex probability and analytic linear prognostic for unburied petrochemical pipelines. Systems Science and Control Engineering, Taylor and Francis Publishers. 2017;**5**(1):178-214

[10] Abou Jaoude A. The paradigm of complex probability and Claude Shannon's information theory. Systems Science and Control Engineering, Taylor and Francis Publishers. 2017;**5**(1): 380-425

[11] Abou Jaoude A. The paradigm of complex probability and analytic nonlinear prognostic for unburied petrochemical pipelines. Systems Science and Control Engineering, Taylor and Francis Publishers. 2017;**5**(1):495-534

[12] Abou Jaoude A. The paradigm of complex probability and Ludwig Boltzmann's entropy. Systems Science and Control Engineering, Taylor and Francis Publishers. 2018;**6**(1):108-149

[13] Abou Jaoude A. The paradigm of complex probability and Monte Carlo methods. Systems Science and Control Engineering, Taylor and Francis Publishers. 2019;**7**(1):407-451

[14] Abou Jaoude A. Analytic prognostic in the linear damage case applied to buried petrochemical pipelines and the complex probability paradigm. In: Fault Detection, Diagnosis and Prognosis. London, UK: IntechOpen; 2020. DOI: 10.5772/intechopen.90157

[15] Abou Jaoude A. The Computer Simulation of Monté Carlo Methods and Random Phenomena. United Kingdom: Cambridge Scholars Publishing; 2019

[16] Abou Jaoude A. The Analysis of Selected Algorithms for the Stochastic Paradigm. United Kingdom: Cambridge Scholars Publishing; 2019

[17] Abou Jaoude A. Applied mathematics: Numerical methods and algorithms for applied mathematicians [PhD thesis]. Spain: Bircham International University. 2004. Available from: http://www.bircham.edu

[18] Abou Jaoude A. Computer science: Computer simulation of Monté Carlo methods and random phenomena [PhD thesis]. Spain: Bircham International University. 2005. Available from: http:// www.bircham.edu

[19] Abou Jaoude A. Applied statistics and probability: Analysis and algorithms for the statistical and stochastic paradigm [PhD thesis]. Spain: Bircham International University. 2007. Available from: http://www.bircham.edu

[20] Metropolis N. The Beginning of the Monte Carlo Method. Los Alamos Science (1987 Special Issue dedicated to Stanislaw Ulam); 1987. pp. 125-130

[21] Eckhardt R. Stan Ulam, John von Neumann, and the Monte Carlo Method. Los Alamos Science, Special Issue (15); 1987. pp. 131-137

[22] Mazhdrakov M, Benov D, Valkanov N. The Monte Carlo Method. Engineering Applications. Cambridge: ACMO Academic Press; 2018. p. 250. ISBN: 978-619-90684-3-4

[23] Peragine M. The Universal Mind: The Evolution of Machine Intelligence and Human Psycholog*y*. San Diego, CA: Xiphias Press; 2013. [Retrieved: 17 December 2018]

[24] McKean HP. Propagation of chaos for a class of non-linear parabolic equations. In: Lecture Series in Differential Equations, Session 7. Arlington, VA: Catholic University; 1967. pp. 41-57. Bibcode: 1966PNAS...56.1907M. DOI: 10.1073/ pnas.56.6.1907. PMC: 220210. PMID: 16591437

[25] Herman K, Theodore HE. Estimation of particle transmission by random sampling. National Bureau of Standards: Applied Mathematics Series. 1951;**12**:27-30

[34] Gordon NJ, Salmond DJ, Smith AFM. Novel approach to

ISSN 0956-375X

JSTOR 1390750

1997;**33**(3):835-850

1991. p. 94

s0036139996307371

s004400050131

**29**

nonlinear/non-Gaussian Bayesian state estimation. Radar and Signal Processing, IEE Proceedings F. April 1993;**140**(2): 107-113. DOI: 10.1049/ip-f-2.1993.0015.

*DOI: http://dx.doi.org/10.5772/intechopen.93048*

*The Monte Carlo Techniques and the Complex Probability Paradigm*

[35] Kitagawa G. Monte Carlo filter and smoother for non-Gaussian nonlinear

Computational and Graphical Statistics. 1996;**5**(1):1-25. DOI: 10.2307/1390750.

[36] Carvalho H, Del Moral P, Monin A, Salut G. Optimal non-linear filtering in GPS/INS integration. IEEE Transactions on Aerospace and Electronic Systems.

[37] Del Moral P, Rigal G, Salut G. Nonlinear and Non-Gaussian Particle Filters Applied to Inertial Platform Repositioning. LAAS-CNRS, Toulouse,

Research Report No. 92207, STCAN/DIGILOG-LAAS/CNRS Convention STCAN No. A.91.77.013.

[38] Crisan D, Gaines J, Lyons T. Convergence of a branching particle method to the solution of the Zakai. SIAM Journal on Applied Mathematics. 1998;**58**(5):1568-1590. DOI: 10.1137/

[39] Crisan D, Lyons T. Nonlinear filtering and measure-valued processes. Probability Theory and Related Fields. 1997;**109**(2):217-244. DOI: 10.1007/

state space models. Journal of

[26] Turing AM. Computing machinery and intelligence. Mind. LIX. 1950;**238**: 433-460. DOI: 10.1093/mind/LIX.236.433

[27] Barricelli NA. Symbiogenetic evolution processes realized by artificial methods. Methodos. 1957:143-182

[28] Del Moral P. Feynman–Kac Formulae. Genealogical and Interacting Particle Approximations. Series: Probability and Applications. Berlin: Springer; 2004. p. 575

[29] Assaraf R, Caffarel M, Khelif A. Diffusion Monte Carlo methods with a fixed number of walkers. Physical Review E. 2000;**61**(4):4566-4575. Bibcode: 2000PhRvE...61.4566A. DOI: 10.1103/physreve.61.4566. Archived from the original (PDF) on 2014-11-07

[30] Caffarel M, Ceperley D, Kalos M. Comment on Feynman–Kac pathintegral calculation of the ground-state energies of atoms. Physical Review Letters. 1993;**71**(13):2159. Bibcode: 1993PhRvL...71.2159C. DOI: 10.1103/ physrevlett.71.2159. PMID: 10054598

[31] Hetherington JH. Observations on the statistical iteration of matrices. Physical Review A. 1984;**30**(2713): 2713-2719. Bibcode: 1984PhRvA...30.2713H. DOI: 10.1103/ PhysRevA.30.2713

[32] Fermi E, Richtmyer RD. Note on Census-Taking in Monte Carlo Calculations (PDF). LAM. 805 (A). Declassified Report Los Alamos Archive; 1948

[33] Rosenbluth MN, Rosenbluth AW. Monte-Carlo calculations of the average extension of macromolecular chains. The Journal of Chemical Physics. 1955; **23**(2):356-359. Bibcode: 1955JChPh ...23...356R. DOI: 10.1063/1.1741967

*The Monte Carlo Techniques and the Complex Probability Paradigm DOI: http://dx.doi.org/10.5772/intechopen.93048*

[34] Gordon NJ, Salmond DJ, Smith AFM. Novel approach to nonlinear/non-Gaussian Bayesian state estimation. Radar and Signal Processing, IEE Proceedings F. April 1993;**140**(2): 107-113. DOI: 10.1049/ip-f-2.1993.0015. ISSN 0956-375X

[35] Kitagawa G. Monte Carlo filter and smoother for non-Gaussian nonlinear state space models. Journal of Computational and Graphical Statistics. 1996;**5**(1):1-25. DOI: 10.2307/1390750. JSTOR 1390750

[36] Carvalho H, Del Moral P, Monin A, Salut G. Optimal non-linear filtering in GPS/INS integration. IEEE Transactions on Aerospace and Electronic Systems. 1997;**33**(3):835-850

[37] Del Moral P, Rigal G, Salut G. Nonlinear and Non-Gaussian Particle Filters Applied to Inertial Platform Repositioning. LAAS-CNRS, Toulouse, Research Report No. 92207, STCAN/DIGILOG-LAAS/CNRS Convention STCAN No. A.91.77.013. 1991. p. 94

[38] Crisan D, Gaines J, Lyons T. Convergence of a branching particle method to the solution of the Zakai. SIAM Journal on Applied Mathematics. 1998;**58**(5):1568-1590. DOI: 10.1137/ s0036139996307371

[39] Crisan D, Lyons T. Nonlinear filtering and measure-valued processes. Probability Theory and Related Fields. 1997;**109**(2):217-244. DOI: 10.1007/ s004400050131

**31**

**Chapter 2**

**Abstract**

**1. Introduction**

provides support to local energy companies.

ANFIS TVA Power Plants

In the present chapter, the evaluation of the Tennessee Valley Authority (TVA) Markov model transient behavior is derived and studied. It is focused on finding the models of the transient-state availability and unavailability of the four (TVA) models among using an adaptive neuro-fuzzy inference system (ANFIS). The developed ANFIS model for the TVA models is derived, and both availability and unavailability of the four TVA models are derived using the curve fitting technique, where each model of the transient availability of the three-state models of the TVA models is found. Each model is considered as a three-state model, and its equations obtained using the curve fitting technique are helping for the future availabilities and unavailabilities. The availability is a very important measure of performance for the availability of TVA power plants. The technique is used and applied on the four models in the present study to formulate and obtain the TVA models' results and are compared. In addition, the generation effects on the reliability investigation. The

generation study evaluates the improvement in reliability over a time.

**Keywords:** transient availability, TVA, Markov model, ANFIS, curve fitting

The Tennessee Valley Authority (TVA) is a business organization in the United States. TVA has been a key force for success in the Tennessee Valley since 1933 [1]. TVA supplies electricity to commercial customers and local energy supply companies. It serves approximately 10 million people in parts of seven southeastern states. TVA not only serves and invests its revenues in its electrical system but also provides flood control, navigation, and land management on the Tennessee River system and

Ren et al. [2] in their study went through the multi-microgrid system which is studied as a basic part of the intelligent network, and the radical system for several microgrids is the complete and standard system. The evaluation of the reliability of multi-microgrid systems is widely discussed to guarantee its reliability and steadystate operation. Balancing power between supply and demand is the key criterion for assessing the reliability of multiple microsystems, such as equipment shortages and inadequate distribution capacity in the network. In their research, few have developed a reliability model for partial failures and incomplete equipment repairs, and the transferability of distribution network (DN) is generally overlooked. In their study, they summarize the multi-microgrid (MMG) radial system as a

Availability Modeling

*Isa Qamber and Mohamed Al-Hamad*

Development

#### **Chapter 2**

## ANFIS TVA Power Plants Availability Modeling Development

*Isa Qamber and Mohamed Al-Hamad*

#### **Abstract**

In the present chapter, the evaluation of the Tennessee Valley Authority (TVA) Markov model transient behavior is derived and studied. It is focused on finding the models of the transient-state availability and unavailability of the four (TVA) models among using an adaptive neuro-fuzzy inference system (ANFIS). The developed ANFIS model for the TVA models is derived, and both availability and unavailability of the four TVA models are derived using the curve fitting technique, where each model of the transient availability of the three-state models of the TVA models is found. Each model is considered as a three-state model, and its equations obtained using the curve fitting technique are helping for the future availabilities and unavailabilities. The availability is a very important measure of performance for the availability of TVA power plants. The technique is used and applied on the four models in the present study to formulate and obtain the TVA models' results and are compared. In addition, the generation effects on the reliability investigation. The generation study evaluates the improvement in reliability over a time.

**Keywords:** transient availability, TVA, Markov model, ANFIS, curve fitting

#### **1. Introduction**

The Tennessee Valley Authority (TVA) is a business organization in the United States. TVA has been a key force for success in the Tennessee Valley since 1933 [1]. TVA supplies electricity to commercial customers and local energy supply companies. It serves approximately 10 million people in parts of seven southeastern states. TVA not only serves and invests its revenues in its electrical system but also provides flood control, navigation, and land management on the Tennessee River system and provides support to local energy companies.

Ren et al. [2] in their study went through the multi-microgrid system which is studied as a basic part of the intelligent network, and the radical system for several microgrids is the complete and standard system. The evaluation of the reliability of multi-microgrid systems is widely discussed to guarantee its reliability and steadystate operation. Balancing power between supply and demand is the key criterion for assessing the reliability of multiple microsystems, such as equipment shortages and inadequate distribution capacity in the network. In their research, few have developed a reliability model for partial failures and incomplete equipment repairs, and the transferability of distribution network (DN) is generally overlooked. In their study, they summarize the multi-microgrid (MMG) radial system as a

performance sharing system and present a uniform modeling approach based on the Bayesian network for performance and reliability to fill research gaps. The operation of MMG radial systems is analyzed to create an abstract model to simplify the problem. The operational program is provided for the BN reliability assessment method. In addition, system modeling and BN parameter modeling are introduced. Finally, the MMG radial system composed of nine micronetworks (MNs) were examined and the variability of the system reliability indicator is analyzed in network-connected mode and in island mode.

Pham et al. [3] in their study highlight on the microgrid which mainly consists of distribution generators and energy storage systems, which are used to supply local loads. Distributed generators are mainly renewable energy sources. The aggregated system with numerous battery energy storage devices should be used to improve the reliability of the power supply from these renewable energy sources in the microgrid as a collective storage system battery power. The complete battery energy storage system is used to control the balance of source charging power so that microgrid can operate with high stability and reliability to supply electricity to a variety of customers. To demonstrate the importance of the complete battery energy storage system in microgrid, the reliability of its operation is examined in their study. Microgrid offers a systematic way to assess the reliability performance under various dynamic operational issues. An analytical method based on Markov models has been developed to assess the reliability of the entire energy storage system of the overall battery. In addition to the time-dependent failure rate, the voltage-dependent failure rate, and the power loss failure rate, important components, such as bilateral DC-to-DC converters, DC-to-AC converters, switching and protection devices, battery modules, and a battery charger/ controller, are also designed and included in the reliability assessment. Depending on the dynamic operating problems of the microarrays with the full battery energy storage system and the photovoltaic (PV) generation systems, the overall battery energy storage system will be affected differently. The dynamic random operating conditions of the microgrid analyzed include the change in the load power, the intermittent and unstable operation of the PV sources, the modes of operation and distribution of the microarrays, and the cost/off-grid conditions. The complete battery energy storage system is used to control the balance of source charging power. The results of the simulation tests are presented and discussed to confirm that the operational reliability of the entire battery energy storage system in the microgrid strongly depends on its different dynamic operating strategies as well as activated voltage constraints.

Min et al. [4] in their paper highlight on the plan to switch to renewable energy, which recently has been a key element in the energy policy of the electricity system in South Korea. The renewable energy has raised questions about the reliability and flexibility of the electricity system. The researchers provide a research framework to assess the new policy in South Korea from different dimensions using three consecutive simulation models. The first-generation model in their study, which they derived from the best electricity generation mix, provides the overall cost of production and the environmental impact of the long-term capacity expansion plan. In addition, the researchers deal with other two simulation models to assess the reliability and flexibility of electrical systems, respectively. Also, they introduced the way to measure the results predictability of applying the framework in the new policy which shows that the policy does not guarantee a target level of reliability and increases costs and emissions. Achieving target system reliability requires additional reliability, and system flexibility is very sensitive to the type of capacity added.

Čepin, in his paper [5], published the modern electrical systems which introduce an increasing number of more dispersed and smaller energy sources, which are slowly replacing larger and more compact energy sources. The aim of the article is to examine the relocation of nuclear power plants to wind power plants to compare

**33**

*ANFIS TVA Power Plants Availability Modeling Development*

each scenario in terms of reliability of the electricity network. The author highlights on the load diagram and the variability of power plants. The load diagram power depends on environmental parameters over time. In addition, the article's updated method measures the power of the installation as a function of time instead of its nominal power. The first model, in the study, includes a dozen power plants including a nuclear power plant. This power plant will then be replaced by three wind turbines whose total power will be five times that of the nuclear power plant. Reliability is compared to pressure drop using actual weather data for a calendar year. The results show that a decrease in the reliability of the power system leads to a

To determine the dynamic systems of a nuclear power plant, it is necessary to consider the effects of dynamic interactions, such as components, software, and operating processes. But at present, Lee et al. [6] in their study concentrate that there is no simple and easy-to-use tool to assess the availability of these dynamic systems. The method, for example the Markov chains, has a precise solution, but it is difficult to model the system. Using standard error trees, the reliability of a system with dynamic characteristics cannot be accurately estimated because error trees measure the reliability of a system configuration. The dynamic reliability graph with general gates (DRGGGs) allows intuitive modeling similar to actual system configuration, which can reduce the human error that occurs when modeling the target system. As the current dynamic reliability graph with general gates cannot assess the dynamic system in terms of reliability without repair, a new evaluation method enabling the availability of the dynamic repair system to be calculated is proposed through this study. The proposed method extends dynamic reliability graph with general gates by adding the repair condition to dynamic doors. A comparison of the method proposed by the Markov chains in terms of a simple validation model shows that the measured value converges toward the solution. Sabouhi et al. [7] in their paper deal with the proper functioning of a power tool which depends on its subsystem and its components. Due to the current financial constraints in the energy sector, power plant operators face a wide range of challenges when dealing with the maintenance schedule and asset management practices. Knowing the roles and judgment of the components on the overall performance of the plant will help to plan for smooth, safe, and economical operation. To determine the technical and economic decisions for the maintenance of power plant equipment, this study focuses on modeling the reliability of combined cycle power plants (CCPPs). Reliability models are first developed for gas turbine power plants (GTPPs) and steam turbine power plants (STPPs), which provide the data needed to assess the overall reliability of the CCPP from a system perspective. Reliability indices with the reliability of the abovementioned types of supply devices are recommended to identify the critical components of a plant, that is, those which have the most significant impact on the reliability and availability objectives of the system. By identifying essential system components, it is possible to determine effective maintenance strategies for power tool components so that the available

*DOI: http://dx.doi.org/10.5772/intechopen.93422*

decrease in the reliability of the power system.

resources are well designed and technically allocated.

Uncertain sources of intermittent generation and load demand in addition to the transmission line access is not a threat to the security of electricity grids. In Sharifzadeh et al.'s [8] study, to address these uncertainties, an optimal stochastic energy flow is recommended when examining security constraints. A site generation method is also presented to assess the uncertainty of wind generations and load requests when evaluating their connections. In the proposed model, uncertainty is addressed through a combination of common decisions to be the best decision. The efficiency of the proposed model is demonstrated in the known 24 IEEE bus test system. The greater efficiency of the proposed model is shown numerically

#### *ANFIS TVA Power Plants Availability Modeling Development DOI: http://dx.doi.org/10.5772/intechopen.93422*

*Forecasting in Mathematics - Recent Advances, New Perspectives and Applications*

network-connected mode and in island mode.

performance sharing system and present a uniform modeling approach based on the Bayesian network for performance and reliability to fill research gaps. The operation of MMG radial systems is analyzed to create an abstract model to simplify the problem. The operational program is provided for the BN reliability assessment method. In addition, system modeling and BN parameter modeling are introduced. Finally, the MMG radial system composed of nine micronetworks (MNs) were examined and the variability of the system reliability indicator is analyzed in

Pham et al. [3] in their study highlight on the microgrid which mainly consists of distribution generators and energy storage systems, which are used to supply local loads. Distributed generators are mainly renewable energy sources. The aggregated system with numerous battery energy storage devices should be used to improve the reliability of the power supply from these renewable energy sources in the microgrid as a collective storage system battery power. The complete battery energy storage system is used to control the balance of source charging power so that microgrid can operate with high stability and reliability to supply electricity to a variety of customers. To demonstrate the importance of the complete battery energy storage system in microgrid, the reliability of its operation is examined in their study. Microgrid offers a systematic way to assess the reliability performance under various dynamic operational issues. An analytical method based on Markov models has been developed to assess the reliability of the entire energy storage system of the overall battery. In addition to the time-dependent failure rate, the voltage-dependent failure rate, and the power loss failure rate, important components, such as bilateral DC-to-DC converters, DC-to-AC converters, switching and protection devices, battery modules, and a battery charger/ controller, are also designed and included in the reliability assessment. Depending on the dynamic operating problems of the microarrays with the full battery energy storage system and the photovoltaic (PV) generation systems, the overall battery energy storage system will be affected differently. The dynamic random operating conditions of the microgrid analyzed include the change in the load power, the intermittent and unstable operation of the PV sources, the modes of operation and distribution of the microarrays, and the cost/off-grid conditions. The complete battery energy storage system is used to control the balance of source charging power. The results of the simulation tests are presented and discussed to confirm that the operational reliability of the entire battery energy storage system in the microgrid strongly depends on its different dynamic operating strategies as well as activated voltage constraints.

Min et al. [4] in their paper highlight on the plan to switch to renewable energy, which recently has been a key element in the energy policy of the electricity system in South Korea. The renewable energy has raised questions about the reliability and flexibility of the electricity system. The researchers provide a research framework to assess the new policy in South Korea from different dimensions using three consecutive simulation models. The first-generation model in their study, which they derived from the best electricity generation mix, provides the overall cost of production and the environmental impact of the long-term capacity expansion plan. In addition, the researchers deal with other two simulation models to assess the reliability and flexibility of electrical systems, respectively. Also, they introduced the way to measure the results predictability of applying the framework in the new policy which shows that the policy does not guarantee a target level of reliability and increases costs and emissions. Achieving target system reliability requires additional reliability, and

Čepin, in his paper [5], published the modern electrical systems which introduce

an increasing number of more dispersed and smaller energy sources, which are slowly replacing larger and more compact energy sources. The aim of the article is to examine the relocation of nuclear power plants to wind power plants to compare

system flexibility is very sensitive to the type of capacity added.

**32**

each scenario in terms of reliability of the electricity network. The author highlights on the load diagram and the variability of power plants. The load diagram power depends on environmental parameters over time. In addition, the article's updated method measures the power of the installation as a function of time instead of its nominal power. The first model, in the study, includes a dozen power plants including a nuclear power plant. This power plant will then be replaced by three wind turbines whose total power will be five times that of the nuclear power plant. Reliability is compared to pressure drop using actual weather data for a calendar year. The results show that a decrease in the reliability of the power system leads to a decrease in the reliability of the power system.

To determine the dynamic systems of a nuclear power plant, it is necessary to consider the effects of dynamic interactions, such as components, software, and operating processes. But at present, Lee et al. [6] in their study concentrate that there is no simple and easy-to-use tool to assess the availability of these dynamic systems. The method, for example the Markov chains, has a precise solution, but it is difficult to model the system. Using standard error trees, the reliability of a system with dynamic characteristics cannot be accurately estimated because error trees measure the reliability of a system configuration. The dynamic reliability graph with general gates (DRGGGs) allows intuitive modeling similar to actual system configuration, which can reduce the human error that occurs when modeling the target system. As the current dynamic reliability graph with general gates cannot assess the dynamic system in terms of reliability without repair, a new evaluation method enabling the availability of the dynamic repair system to be calculated is proposed through this study. The proposed method extends dynamic reliability graph with general gates by adding the repair condition to dynamic doors. A comparison of the method proposed by the Markov chains in terms of a simple validation model shows that the measured value converges toward the solution.

Sabouhi et al. [7] in their paper deal with the proper functioning of a power tool which depends on its subsystem and its components. Due to the current financial constraints in the energy sector, power plant operators face a wide range of challenges when dealing with the maintenance schedule and asset management practices. Knowing the roles and judgment of the components on the overall performance of the plant will help to plan for smooth, safe, and economical operation. To determine the technical and economic decisions for the maintenance of power plant equipment, this study focuses on modeling the reliability of combined cycle power plants (CCPPs). Reliability models are first developed for gas turbine power plants (GTPPs) and steam turbine power plants (STPPs), which provide the data needed to assess the overall reliability of the CCPP from a system perspective. Reliability indices with the reliability of the abovementioned types of supply devices are recommended to identify the critical components of a plant, that is, those which have the most significant impact on the reliability and availability objectives of the system. By identifying essential system components, it is possible to determine effective maintenance strategies for power tool components so that the available resources are well designed and technically allocated.

Uncertain sources of intermittent generation and load demand in addition to the transmission line access is not a threat to the security of electricity grids. In Sharifzadeh et al.'s [8] study, to address these uncertainties, an optimal stochastic energy flow is recommended when examining security constraints. A site generation method is also presented to assess the uncertainty of wind generations and load requests when evaluating their connections. In the proposed model, uncertainty is addressed through a combination of common decisions to be the best decision. The efficiency of the proposed model is demonstrated in the known 24 IEEE bus test system. The greater efficiency of the proposed model is shown numerically

compared to four other definitive and stochastic methods for determining the reserve obtained and the "subsequent" conditions. In addition, the impact of the number of cases on the performance of the proposed model is assessed using a sensitivity analysis. Slower changes in the conditions created have also been shown to reflect correlations and can better capture the uncertain load behavior.

In recent years, there has been growing interest in developing maintenance activities for nuclear power plants in the form of risk-based models as studied by Mohammadhasani and Pirouzmand [9]. Maintenance design plays a key role in security likelihood assessment applications. The importance of these actions, especially when considering the effects of component degradation, is very clear for resolving controversial goals such as achieving the highest level of opportunity and reducing implementation costs of these actions, given the limitations of using standard safety assessment methods to test complex maintenance policies, as well as the difficulty of formulating the impact of component maintenance and degradation strategies using analysis of fault trees. Mohammadhasani and Pirouzmand [9] in their article present Markov's multi-time continuous approach to modeling three-component test and repair policies focusing on changing technical specifications such as test times. First, Markov models will be developed for three different test policies considering the effects of degradation. These models are then applied to analyze the availability of essential components of an emergency cooling system for the core of a VVER-1000/ V446 nuclear power plant as a case study. In the first policy, the other components are not tested further. The test method is carried out in accordance with the test plan as originally recorded. In the second policy, additional testing should take place after the repair of the defective part and, in the third policy, the remaining redundant components should be subjected to extensive testing after the first detection.

Managing instant access to content servers (source servers) in a network in knowledge-centric networks has received little attention in Banerjee et al.'s [10] study. Banerjee et al. [10] considered a case in their study where content repositories in an information-centric network are temporarily unavailable, perhaps for reasons such as device malfunction, interruption of power outage, or denial of service attacks. Unlike the traditional host-centric Internet, users can still be served on informationcentric networks, because content is stored on network nodes. The authors [10] begin their study by observing whether caches continue to operate using their native cache and expulsion policies after content repositories are unavailable, and over time the diversity of network content decreases. Therefore, the authors [10] recommend freezing network caches as soon as the custodians are not available. Next, the authors [10] provide a routing and search algorithm, content analysis under server availability, which uses breadcrumbs to efficiently find content stored on network nodes and to satisfy user requests when guards are not available. The authors [10] perform in-depth simulations of the real Internet topology in the Icarus simulator and show that content analysis on server accessibility can easily detect content stored on the network. The content of the server availability survey exceeds the shortest content demand on average by 56% and meets up to 98.5% of the total server demand.

#### **2. Power plant model**

The four TVA power plant models [11] are differing by nature of different numerical values for transition rates and are shown in **Table 1**. These transition rates are shown for the four TVA models represented by **Figure 1**, which are relevant to power plants operated by the TVA. Each TVA model is formed as illustrated in **Figure 2**, which is called Markov model. The general model for each of the TVA model can be represented by the following differential equations:

**35**

**Figure 1.**

*TVA power plants' representation.*

*ANFIS TVA Power Plants Availability Modeling Development*

( ) ( ) ( ) ( ) ( ) *up*

( ) ( ) ( ) ( ) ( ) *derated*

( ) ( ) ( ) ( ) ( ) *down up derated down*

*dP t bP t eP t d f P t dt*

The three differential equations are solved using the MATLAB Simulink 7.10 package to find the transient probabilities of each model of the TVA, where the

Derated-state: the system operates at less than full capacity due to generators

And the initial probabilities for each model are P1 (0) = 1, P2 (0) = 0, and P3 (0) = 0. The results are obtained for the four TVA models. The MATLAB Simulink 7.10 package is used to obtain the transient-state probabilities for the four TVA models and

 0.0003 0.0010 0.0225 0.0350 0.0008 0.0004 0.0006 0.0050 0.0400 0.1000 0.0004 0.0004 0.0005 0.0002 0.0240 0.0430 0.0001 0.0001 0.0010 0.0006 0.0200 0.1000 0.0002 0.0020

**a b c d e f**

Down-state: the system has no power at all due to forced outage rate.

**Model Transition rates (per hour)**

*dP t aP t c e P t f P t dt*

*up derated down*

=− + + + (1)

= −+ + (2)

= + −+ (3)

*a b P t cP t dP t*

*up derated down*

*DOI: http://dx.doi.org/10.5772/intechopen.93422*

*dP t*

*dt*

three states of each model are defined as follows:

outages.

**Table 1.**

*The four TVA models' transition rate values [11].*

Up-state: the system operates at full capacity.

*ANFIS TVA Power Plants Availability Modeling Development DOI: http://dx.doi.org/10.5772/intechopen.93422*

$$\frac{dP\_{up}\left(t\right)}{dt} = -\left(a+b\right)P\_{up}\left(t\right) + c\left.P\_{dead}\left(t\right)\right| + d\left.P\_{down}\left(t\right)\right|\tag{1}$$

$$\frac{dP\_{demand}\left(t\right)}{dt} = a\,P\_{up}\left(t\right) - \left(c+e\right)P\_{dead}\left(t\right) + f\,P\_{down}\left(t\right) \tag{2}$$

$$\frac{dP\_{down}(t)}{dt} = b \, P\_{up}(t) + e \, P\_{druated}(t) - (d+f)P\_{down}(t) \tag{3}$$

The three differential equations are solved using the MATLAB Simulink 7.10 package to find the transient probabilities of each model of the TVA, where the three states of each model are defined as follows:

Up-state: the system operates at full capacity.

Derated-state: the system operates at less than full capacity due to generators outages.

Down-state: the system has no power at all due to forced outage rate.

And the initial probabilities for each model are P1 (0) = 1, P2 (0) = 0, and P3 (0) = 0. The results are obtained for the four TVA models. The MATLAB Simulink 7.10 package is used to obtain the transient-state probabilities for the four TVA models and


#### **Table 1.**

*Forecasting in Mathematics - Recent Advances, New Perspectives and Applications*

compared to four other definitive and stochastic methods for determining the reserve obtained and the "subsequent" conditions. In addition, the impact of the number of cases on the performance of the proposed model is assessed using a sensitivity analysis. Slower changes in the conditions created have also been shown

In recent years, there has been growing interest in developing maintenance activities for nuclear power plants in the form of risk-based models as studied by Mohammadhasani and Pirouzmand [9]. Maintenance design plays a key role in security likelihood assessment applications. The importance of these actions, especially when considering the effects of component degradation, is very clear for resolving controversial goals such as achieving the highest level of opportunity and reducing implementation costs of these actions, given the limitations of using standard safety assessment methods to test complex maintenance policies, as well as the difficulty of formulating the impact of component maintenance and degradation strategies using analysis of fault trees. Mohammadhasani and Pirouzmand [9] in their article present Markov's multi-time continuous approach to modeling three-component test and repair policies focusing on changing technical specifications such as test times. First, Markov models will be developed for three different test policies considering the effects of degradation. These models are then applied to analyze the availability of essential components of an emergency cooling system for the core of a VVER-1000/ V446 nuclear power plant as a case study. In the first policy, the other components are not tested further. The test method is carried out in accordance with the test plan as originally recorded. In the second policy, additional testing should take place after the repair of the defective part and, in the third policy, the remaining redundant components should be subjected to extensive testing after the first detection. Managing instant access to content servers (source servers) in a network in knowledge-centric networks has received little attention in Banerjee et al.'s [10] study. Banerjee et al. [10] considered a case in their study where content repositories in an information-centric network are temporarily unavailable, perhaps for reasons such as device malfunction, interruption of power outage, or denial of service attacks. Unlike the traditional host-centric Internet, users can still be served on informationcentric networks, because content is stored on network nodes. The authors [10] begin their study by observing whether caches continue to operate using their native cache and expulsion policies after content repositories are unavailable, and over time the diversity of network content decreases. Therefore, the authors [10] recommend freezing network caches as soon as the custodians are not available. Next, the authors [10] provide a routing and search algorithm, content analysis under server availability, which uses breadcrumbs to efficiently find content stored on network nodes and to satisfy user requests when guards are not available. The authors [10] perform in-depth simulations of the real Internet topology in the Icarus simulator and show that content analysis on server accessibility can easily detect content stored on the network. The content of the server availability survey exceeds the shortest content demand on average by 56% and meets up to 98.5% of the total server demand.

The four TVA power plant models [11] are differing by nature of different numerical values for transition rates and are shown in **Table 1**. These transition rates are shown for the four TVA models represented by **Figure 1**, which are relevant to power plants operated by the TVA. Each TVA model is formed as illustrated in **Figure 2**, which is called Markov model. The general model for each of the TVA

model can be represented by the following differential equations:

to reflect correlations and can better capture the uncertain load behavior.

**34**

**2. Power plant model**

*The four TVA models' transition rate values [11].*

**Figure 1.** *TVA power plants' representation.*

is reproduced by the fourth order Runge-Kutta method. Out of the obtained results, the availabilities and unavailabilities are obtained. The availability is the summation of the up-state and derated-state, where the unavailability is the down-state.

**Table 2** summarizes the six transition rates of the four TVA models. **Figure 3a–d** illustrate the four TVA models with their transition rate values.

### **Figure 2.**

*TVA three-state Markov model.*

**37**

**Figure 3.**

*ANFIS TVA Power Plants Availability Modeling Development*

Al-Hamad and Qamber in their study [12] targeted a numerical evaluation of the nonlinear behavior of the Markov model and discussed it. Their study aims to obtain the transient probability between the states of four models recognized by the Tennessee Valley Authority (TVA). Three approaches are being studied to obtain the three transient probabilities after modeling. These techniques are Laplace transforms, curve fitting, and neuro-fuzzy. The MATLAB Simulink 7.10 package is used to obtain steady-state probabilities for the four VAT models while reproducing these solutions with Laplace transforms. For each model, a three-state model is considered, where its equations can be obtained using curve fitting and neuro-fuzzy methods. All the methods are used and applied in Al-Hamad and Qamber's study [12] and are used to design and obtain the TVA models. Al-Hamad and Qamber's study [12] proposes three technical approaches. These techniques increased the performance of the models in terms of execution time. These techniques are Laplace transformations, curve fitting, and neuro-fuzzy. The main objective of their study [12] is to temporarily reshape the transient-state probabilities for the four TVA models and find the best TVA models, even if any researcher has worked

*Transition rate values for (a) TVA-1 model, (b) TVA-2 model, (c) TVA-3 model, and (d) TVA-4 model.*

*DOI: http://dx.doi.org/10.5772/intechopen.93422*

**Table 2.** *Transition rates of each TVA model.*

**Figure 3.**

*Forecasting in Mathematics - Recent Advances, New Perspectives and Applications*

illustrate the four TVA models with their transition rate values.

is reproduced by the fourth order Runge-Kutta method. Out of the obtained results, the availabilities and unavailabilities are obtained. The availability is the summation

**Table 2** summarizes the six transition rates of the four TVA models. **Figure 3a–d**

of the up-state and derated-state, where the unavailability is the down-state.

**36**

**Table 2.**

**Figure 2.**

*TVA three-state Markov model.*

*Transition rates of each TVA model.*

*Transition rate values for (a) TVA-1 model, (b) TVA-2 model, (c) TVA-3 model, and (d) TVA-4 model.*

Al-Hamad and Qamber in their study [12] targeted a numerical evaluation of the nonlinear behavior of the Markov model and discussed it. Their study aims to obtain the transient probability between the states of four models recognized by the Tennessee Valley Authority (TVA). Three approaches are being studied to obtain the three transient probabilities after modeling. These techniques are Laplace transforms, curve fitting, and neuro-fuzzy. The MATLAB Simulink 7.10 package is used to obtain steady-state probabilities for the four VAT models while reproducing these solutions with Laplace transforms. For each model, a three-state model is considered, where its equations can be obtained using curve fitting and neuro-fuzzy methods. All the methods are used and applied in Al-Hamad and Qamber's study [12] and are used to design and obtain the TVA models. Al-Hamad and Qamber's study [12] proposes three technical approaches. These techniques increased the performance of the models in terms of execution time. These techniques are Laplace transformations, curve fitting, and neuro-fuzzy. The main objective of their study [12] is to temporarily reshape the transient-state probabilities for the four TVA models and find the best TVA models, even if any researcher has worked

on modeling using curve fitting and other methods. TVA data are very useful for modeling a three-state model system. In their work the neuro-fuzzy was studied and used. It is found that it is the best and most useful to model and calculate the transient probabilities. In addition, the performance of the TVA models is clear and satisfying. As the way of calculating the transient probabilities throughout the operation period, each TVA model is likely to be trained at some point to develop fuzzy IF-THEN rules that help to determine the input and output variables for the functions of the TVA model. In Al-Hamad and Qamber's study [12], the combination of fuzzy logic and the neural network (becoming neuro-fuzzy) is a powerful means of designing intelligent systems and obtaining precise results. In their study [12], curved fitting modulation methods and neuro-fuzzy methods were adopted and applied to the four TVA models to calculate the probability of nonlinear state of the models. The performance of the two approaches was also evaluated to find the right and the accurate approach for the four TVA models. The results obtained from Al-Hamad and Qamber's study [9] were found to be much closer to the results of Laplace transforms, in particular, the results obtained using neuro-fuzzy. The study [12] shows a successful development of a reliable relationship between the probabilities of movement and time. Results obtained with both curve fitting and neuro-fuzzy were compared and studied. Comparing exposures to neuro-fuzzy offers better accuracy—as already mentioned—in predicting the probability of a moving state carried out in their study [12].

#### **3. Applications and results**

The neuro-fuzzy has a combination of advantages of the neural networks and fuzzy-logic, where the aim of the neuro-fuzzy is to combine collectively the benefits of both approaches. The neural networks have two main benefits. These two can learn nonlinear mapping of numerical data and the performing of parallel computation. It is very hard to understand the meaning of weights and the incorporation of prior knowledge into the system, which is usually impossible. Fuzzy logic uses human understanding of linguistic terms to form the knowledge of the system. This makes a close interaction between the system and human operator possible. In addition, neuro-fuzzy systems allow the incorporation of both numerical and linguistic data into the system, where it is also capable of extracting fuzzy knowledge from numerical data.

The ANFIS system applies the artificial neural network to find a suitable fuzzy inference system (FIS) structure and parameters. The fuzzy system with its structure identifies the considered fuzzy rules to obtain the targeted results. In addition, the considered architecture of the ANFIS structure has five layers as shown in **Figure 4**, which is the developed model in the study.

The developed general model for the neuro-fuzzy system is shown in **Figure 4**. **Figure 4** shows the developed five-layer connection for six inputs and two outputs. These five layers represent the neuro-fuzzy model, which is used to represent the TVA models and helps to obtain the availabilities and unavailabilities of each TVA model.

The proposed model using the neuro-fuzzy has six inputs as mentioned earlier and two outputs model as illustrated in **Figure 4**. In addition to the fourth order Runge-Kutta method, ANFIS technique applied in the present chapter to model the three-state probabilities of the four TVA models (fourth layer of **Figure 4**) which deals to the fifth layer helps to find the availability and unavailability of the model. The availability is reaching the steady-state availability of each TVA

**39**

models are identical.

*ANFIS TVA developed model.*

and PDown.

**Figure 4.**

of time.

*ANFIS TVA Power Plants Availability Modeling Development*

model, where the steady-state availability is defined as the steady-state probability of the system which is IN service. The steady-state probability that the TVA model is OUT of service is calculated as out of service due to failures. Any system can be interpreted as the long-run fraction of time spent in the failure state. The ANFIS and the curve fitting techniques are compared based on the availability and unavailability results. Through the obtained results, it is found that the results from both techniques are very close to each other. ANFIS and the curve fitting

With reference to the TVA models, the fuzzy inference system (FIS) is applied. Six inputs are shown in layer (1), which are the transition rates between the three states model (**Figure 2**). The outputs of the model (layer (5)) results are the availability and unavailability of the TVA model. The second layer shows the three-states model. This process of converting an output fuzzy set for a solution variable into a single model is represented by the layer (3) which is TVA model. The results of the TVA model are shown in layer (4), which are the three probabilities PUp, PDerated,

The results obtained using the curve fitting for the four TVA models are shown in **Figures 5–8** as availability and unavailability. The curve fittings models' equations are obtained for the four TVAs' availability and unavailability as a function

The availabilities as a function of time for the four TVA models are shown in **Figures 5a**–**8a**, respectively. The availability of each TVA model decreased gradually with time and then finally reached steady state (stable). The unavailability of each TVA model is illustrated in **Figures 5b**–**8b**, respectively. The unavailability of each

TVA model increased gradually till it reached the steady state (stability). When the generation reached the stability, the availabilities are 98.72%, 98.56%, 97.99%, and 95.26%, respectively, for the corresponding TVA models. Under the same conditions, the unavailabilities are 1.28%, 1.44%, 2.01%, and

4.74%, respectively, for the TVA models.

*DOI: http://dx.doi.org/10.5772/intechopen.93422*

*ANFIS TVA Power Plants Availability Modeling Development DOI: http://dx.doi.org/10.5772/intechopen.93422*

**Figure 4.** *ANFIS TVA developed model.*

*Forecasting in Mathematics - Recent Advances, New Perspectives and Applications*

moving state carried out in their study [12].

**Figure 4**, which is the developed model in the study.

**3. Applications and results**

numerical data.

TVA model.

on modeling using curve fitting and other methods. TVA data are very useful for modeling a three-state model system. In their work the neuro-fuzzy was studied and used. It is found that it is the best and most useful to model and calculate the transient probabilities. In addition, the performance of the TVA models is clear and satisfying. As the way of calculating the transient probabilities throughout the operation period, each TVA model is likely to be trained at some point to develop fuzzy IF-THEN rules that help to determine the input and output variables for the functions of the TVA model. In Al-Hamad and Qamber's study [12], the combination of fuzzy logic and the neural network (becoming neuro-fuzzy) is a powerful means of designing intelligent systems and obtaining precise results. In their study [12], curved fitting modulation methods and neuro-fuzzy methods were adopted and applied to the four TVA models to calculate the probability of nonlinear state of the models. The performance of the two approaches was also evaluated to find the right and the accurate approach for the four TVA models. The results obtained from Al-Hamad and Qamber's study [9] were found to be much closer to the results of Laplace transforms, in particular, the results obtained using neuro-fuzzy. The study [12] shows a successful development of a reliable relationship between the probabilities of movement and time. Results obtained with both curve fitting and neuro-fuzzy were compared and studied. Comparing exposures to neuro-fuzzy offers better accuracy—as already mentioned—in predicting the probability of a

The neuro-fuzzy has a combination of advantages of the neural networks and fuzzy-logic, where the aim of the neuro-fuzzy is to combine collectively the benefits of both approaches. The neural networks have two main benefits. These two can learn nonlinear mapping of numerical data and the performing of parallel computation. It is very hard to understand the meaning of weights and the incorporation of prior knowledge into the system, which is usually impossible. Fuzzy logic uses human understanding of linguistic terms to form the knowledge of the system. This makes a close interaction between the system and human operator possible. In addition, neuro-fuzzy systems allow the incorporation of both numerical and linguistic data into the system, where it is also capable of extracting fuzzy knowledge from

The ANFIS system applies the artificial neural network to find a suitable fuzzy inference system (FIS) structure and parameters. The fuzzy system with its structure identifies the considered fuzzy rules to obtain the targeted results. In addition, the considered architecture of the ANFIS structure has five layers as shown in

The developed general model for the neuro-fuzzy system is shown in **Figure 4**. **Figure 4** shows the developed five-layer connection for six inputs and two outputs. These five layers represent the neuro-fuzzy model, which is used to represent the TVA models and helps to obtain the availabilities and unavailabilities of each

The proposed model using the neuro-fuzzy has six inputs as mentioned earlier and two outputs model as illustrated in **Figure 4**. In addition to the fourth order Runge-Kutta method, ANFIS technique applied in the present chapter to model the three-state probabilities of the four TVA models (fourth layer of **Figure 4**) which deals to the fifth layer helps to find the availability and unavailability of the model. The availability is reaching the steady-state availability of each TVA

**38**

model, where the steady-state availability is defined as the steady-state probability of the system which is IN service. The steady-state probability that the TVA model is OUT of service is calculated as out of service due to failures. Any system can be interpreted as the long-run fraction of time spent in the failure state. The ANFIS and the curve fitting techniques are compared based on the availability and unavailability results. Through the obtained results, it is found that the results from both techniques are very close to each other. ANFIS and the curve fitting models are identical.

With reference to the TVA models, the fuzzy inference system (FIS) is applied. Six inputs are shown in layer (1), which are the transition rates between the three states model (**Figure 2**). The outputs of the model (layer (5)) results are the availability and unavailability of the TVA model. The second layer shows the three-states model. This process of converting an output fuzzy set for a solution variable into a single model is represented by the layer (3) which is TVA model. The results of the TVA model are shown in layer (4), which are the three probabilities PUp, PDerated, and PDown.

The results obtained using the curve fitting for the four TVA models are shown in **Figures 5–8** as availability and unavailability. The curve fittings models' equations are obtained for the four TVAs' availability and unavailability as a function of time.

The availabilities as a function of time for the four TVA models are shown in **Figures 5a**–**8a**, respectively. The availability of each TVA model decreased gradually with time and then finally reached steady state (stable). The unavailability of each TVA model is illustrated in **Figures 5b**–**8b**, respectively. The unavailability of each TVA model increased gradually till it reached the steady state (stability).

When the generation reached the stability, the availabilities are 98.72%, 98.56%, 97.99%, and 95.26%, respectively, for the corresponding TVA models. Under the same conditions, the unavailabilities are 1.28%, 1.44%, 2.01%, and 4.74%, respectively, for the TVA models.

**Figure 5.** *(a) Availability model for TVA1 and (b) unavailability model for TVA1.*

**41**

**Figure 8.**

**Figure 7.**

*ANFIS TVA Power Plants Availability Modeling Development*

*(a) Availability model for TVA3 and (b) unavailability model for TVA3.*

*(a) Availability model for TVA4 and (b) unavailability model for TVA4.*

*DOI: http://dx.doi.org/10.5772/intechopen.93422*

**Figure 6.** *(a) Availability model for TVA2 and (b) unavailability model for TVA2.*

*ANFIS TVA Power Plants Availability Modeling Development DOI: http://dx.doi.org/10.5772/intechopen.93422*

*Forecasting in Mathematics - Recent Advances, New Perspectives and Applications*

**40**

**Figure 6.**

**Figure 5.**

*(a) Availability model for TVA2 and (b) unavailability model for TVA2.*

*(a) Availability model for TVA1 and (b) unavailability model for TVA1.*

**Figure 7.** *(a) Availability model for TVA3 and (b) unavailability model for TVA3.*

**Figure 8.** *(a) Availability model for TVA4 and (b) unavailability model for TVA4.*

The curve fitting technique MyCurveFit (online curve fitting) is applied to find the model as shown in **Figures 5–8** as mentioned earlier. The results for the four TVA models are summarized in **Table 3**, showing the availabilities of each model, where **Table 4** shows the unavailabilities of the same models. The obtained availability equation using the curve fitting technique MyCurveFit is shown in Eq. (4) for the four TVA models.

$$Availability\left(time\right) = d + \frac{\left(a - d\right)}{a + \left[\frac{Time}{c}\right]^b} \tag{4}$$

The obtained unavailability equation using the curve fitting technique MyCurveFit is shown in Eq. (5) for the four TVA models.

$$\text{Un} - \text{Avaailability} \left( \text{Time} \right) = d + \frac{\left( a - d \right)}{\mathbf{1} + \left[ \frac{\text{Time}}{c} \right]^b} \tag{5}$$

Strong access to a complex system, such as a gas turbine, is linked to a part's reliability and maintenance policy. This policy affects not only the repair time of parts but also the reliability of parts, which affects the contamination and accessibility of the system. In any study, different methods are used for assessing the reliability and availability of gas turbines installed in a power plant. The investigated methods are based on concepts of system reliability, such as the development of the functional tree structure. The application of the failure mode and the analysis of the power


**43**

the system.

**Table 4.**

*Unavailabilities of the four TVA models.*

**4. Conclusion**

finally becoming stable.

application study.

*ANFIS TVA Power Plants Availability Modeling Development*

**Unavailability model of TVA Coefficients** TVA1 a 0.000369658874527318

TVA2 a 0.000583437614778953

TVA3 a 0.000728795178208125

TVA4 a 0.000499149643278974

b 1.47698177420345 c 30.6343647596028 d 0.0130413214169725

b 1.64413595811526 c 17.7140736131654 d 0.0146075810962714

b 1.52929468363619 c 29.3840092626888 d 0.0205824453759151

b 1.26083143262919 c 35.1382710744811 d 0.0491382311673703

supply helps identify the critical components in improving the reliability of the system. The system and assessment of maintenance based on a historical fault database. This study focuses on the transition rates' changes (failure and repair rates) between the state which the model is passing through as observed and presented in this study. Implement trust-based maintenance concepts to implement the complex system maintenance policies of the generation system aimed at minimizing unexpected failure in critical components. The accessibility analysis shows different results for each TVA model, showing differences in the installation and operation of

Various failure and repair transition rates have been taken in consideration in the present study. In the present study, the transition rates show the performance of the TVA models based on their variation and nature of each model. Different cases as obtained graphs show the availability and unavailability of the TVA models, where their models are produced by the curve fitting. The availability and unavailability values are obtained through a period of times. The unavailability increases until it is

The reliability of the availability and unavailability generation study seems important to obtain the means for the designer to apprehend the reliability for each design of TVA model. This means that the experience feedback is necessary. In the present study, the main part is to obtain the general TVA model transient availabilities and unavailabilities which has been modeled. This leads to the final target of the

*DOI: http://dx.doi.org/10.5772/intechopen.93422*

#### **Table 3.**

*Availabilities of the four TVA models.*


#### *ANFIS TVA Power Plants Availability Modeling Development DOI: http://dx.doi.org/10.5772/intechopen.93422*

#### **Table 4.**

*Forecasting in Mathematics - Recent Advances, New Perspectives and Applications*

for the four TVA models.

The curve fitting technique MyCurveFit (online curve fitting) is applied to find the model as shown in **Figures 5–8** as mentioned earlier. The results for the four TVA models are summarized in **Table 3**, showing the availabilities of each model, where **Table 4** shows the unavailabilities of the same models. The obtained availability equation using the curve fitting technique MyCurveFit is shown in Eq. (4)

( ) ( )

*a d Availability time d Time*

The obtained unavailability equation using the curve fitting technique

*a d Un Availability Time d Time*

**Availability model of TVA Coefficients** TVA1 a 1.00014234742424

TVA2 a 1.00022688188532

TVA3 a 0.999265231533308

TVA4 a 0.999505792270978

<sup>−</sup> − = <sup>+</sup>

Strong access to a complex system, such as a gas turbine, is linked to a part's reliability and maintenance policy. This policy affects not only the repair time of parts but also the reliability of parts, which affects the contamination and accessibility of the system. In any study, different methods are used for assessing the reliability and availability of gas turbines installed in a power plant. The investigated methods are based on concepts of system reliability, such as the development of the functional tree structure. The application of the failure mode and the analysis of the power

MyCurveFit is shown in Eq. (5) for the four TVA models.

1

( ) ( )

1

b 1.646673111807 c 32.8722769925419 d 0.98712380612843

b 1.89339095988375 c 20.6089357501677 d 0.985496773685337

b 1.53764424728804 c 29.218900622036 d 0.97949890714863

b 1.26046003465127 c 35.1534160245152 d 0.950859080757418

<sup>−</sup> = +

*b*

*b*

*c*

 <sup>+</sup>  (4)

(5)

*c*

 <sup>+</sup> 

**42**

**Table 3.**

*Availabilities of the four TVA models.*

*Unavailabilities of the four TVA models.*

supply helps identify the critical components in improving the reliability of the system. The system and assessment of maintenance based on a historical fault database. This study focuses on the transition rates' changes (failure and repair rates) between the state which the model is passing through as observed and presented in this study. Implement trust-based maintenance concepts to implement the complex system maintenance policies of the generation system aimed at minimizing unexpected failure in critical components. The accessibility analysis shows different results for each TVA model, showing differences in the installation and operation of the system.

#### **4. Conclusion**

Various failure and repair transition rates have been taken in consideration in the present study. In the present study, the transition rates show the performance of the TVA models based on their variation and nature of each model. Different cases as obtained graphs show the availability and unavailability of the TVA models, where their models are produced by the curve fitting. The availability and unavailability values are obtained through a period of times. The unavailability increases until it is finally becoming stable.

The reliability of the availability and unavailability generation study seems important to obtain the means for the designer to apprehend the reliability for each design of TVA model. This means that the experience feedback is necessary. In the present study, the main part is to obtain the general TVA model transient availabilities and unavailabilities which has been modeled. This leads to the final target of the application study.

#### **Acknowledgements**

The authors would like to thank Ms. Aysha Isa Qamber for the great help during the preparation of this book chapter.

### **Conflict of interest**

The authors declare no conflict of interest.

### **Author details**

Isa Qamber1 \* and Mohamed Al-Hamad<sup>2</sup>

1 Former University of Bahrain, Bahrain Society of Engineers Member, Isa Town, Kingdom of Bahrain

2 GCC Interconnection Authority, Dammam, Kingdom of Saudi Arabia

\*Address all correspondence to: i.s.qamber@gmail.com

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

**45**

*ANFIS TVA Power Plants Availability Modeling Development*

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[1] Tennessee Valley Authority. Available from: https://www.tva.com/

About-TVA

**References**

cie.2019.106145

ijepes.2019.105786

*ANFIS TVA Power Plants Availability Modeling Development DOI: http://dx.doi.org/10.5772/intechopen.93422*

#### **References**

*Forecasting in Mathematics - Recent Advances, New Perspectives and Applications*

The authors would like to thank Ms. Aysha Isa Qamber for the great help during

**44**

**Author details**

**Acknowledgements**

**Conflict of interest**

the preparation of this book chapter.

The authors declare no conflict of interest.

Kingdom of Bahrain

\* and Mohamed Al-Hamad<sup>2</sup>

\*Address all correspondence to: i.s.qamber@gmail.com

provided the original work is properly cited.

1 Former University of Bahrain, Bahrain Society of Engineers Member, Isa Town,

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

2 GCC Interconnection Authority, Dammam, Kingdom of Saudi Arabia

Isa Qamber1

[1] Tennessee Valley Authority. Available from: https://www.tva.com/ About-TVA

[2] Yi R, Cui B, Feng Q, Yang D, Fan D, Sun B, et al. A reliability evaluation method for radial multi-microgrid systems considering distribution network transmission capacity. Computers & Industrial Engineering Journal. 2020;**139**:1-10. DOI: 10.1016/j. cie.2019.106145

[3] Pham TT, Kuo T, Bui DM. Reliability evaluation of an aggregate battery energy storage system in microgrids under dynamic operation. International Journal of Electrical Power and Energy Systems. 2020;**118**:1-22. DOI: 10.1016/j. ijepes.2019.105786

[4] Min D, Ryu J, Choi DG. Effects of the move towards renewables on the power system reliability and flexibility in South Korea. Energy Reports. 2020;**6**:406-417. DOI: 10.1016/j.egyr.2020.02.007

[5] Čepin M. Evaluation of the power system reliability if a nuclear power plant is replaced with wind power plants. Reliability Engineering and System Safety. 2019;**185**:455-464. DOI: 10.1016/j.ress.2019.01.010

[6] Lee EC, Shin SK, Seong PH. Evaluation of availability of nuclear power plant dynamic systems using extended dynamic reliability graph with general gates (DRGGG). Nuclear Engineering and Technology Journal. 2018;**51**:444-452. DOI: 10.1016/j. net.2018.10.009

[7] Sabouhi H, Abbaspour A, Fotuhi-Firuzabad M, Dehghanian P. Reliability modeling and availability analysis of combined cycle power plants. International Journal of Electrical Power & Energy Systems. 2016;**79**:108-119. DOI: 10.1016/j. ijepes.2016.01.007

[8] Sharifzadeh H, Amjady N, Zareipour H. Multi-period stochastic security-constrained OPF considering the uncertainty sources of wind power, load demand and equipment unavailability. Electric Power Systems Research. 2017;**146**:33-42. DOI: 10.1016/j.epsr.2017.01.011

[9] Mohammadhasani F, Pirouzmand A. Multi-state unavailability analysis of safety system redundant components with aging effect under surveillance testing. Progress in Nuclear Energy Journal. 2020;**126**:1-15. DOI: 10.1016/j. pnucene.2020.103415

[10] Banerjee A, Banerjee B, Seetharam A, Tellambura C. Content search and routing under custodian unavailability in information-centric networks. Computer Networks Journal. 2018;**141**:92-101. DOI: 10.1016/j. comnet.2018.05.015

[11] Biggerstaff BE, Jackson TM. The Markov process as a means of determining generating-unit state probabilities for use in spinning reserve applications. IEEE Transactions on Power Apparatus and Systems. 1969;**88**:423-430. DOI: 10.1109/ TPAS.1969.292464

[12] Al-Hamad MY, Qamber IS. TVA generating unit modeling using MATLAB. Journal of the Association of Arab Universities for Basic and Applied Sciences. 2015;**17**:73-81. DOI: 10.1016/j. jaubas.2014.05.003

**Chapter 3**

and *PM*2.5

**Abstract**

neural network

**1. Introduction**

**47**

*and Elvira Kovač-Andrić*

performance of other forecasting models.

A Layered Recurrent Neural

Pollutants Missing Data and

Prediction of *NO*2, *O*3, *PM*10,

To fulfill the national air quality standards, many countries have created emissions monitoring strategies on air quality. Nowadays, policymakers and air quality executives depend on scientific computation and prediction models to monitor that cause air pollution, especially in industrial cities. Air pollution is considered one of the primary problems that could cause many human health problems such as asthma, damage to lungs, and even death. In this study, we present investigated development forecasting models for air pollutant attributes including Particulate Matters (PM2.5, PM10), ground-level Ozone (O3), and Nitrogen Oxides (NO2). The dataset used was collected from Dubrovnik city, which is located in the east of Croatia. The collected data has missing values. Therefore, we suggested the use of a Layered Recurrent Neural Network (L-RNN) to impute the missing value(s) of air pollutant attributes then build forecasting models. We adopted four regression models to forecast air pollutant attributes, which are: Multiple Linear Regression (MLR), Decision Tree Regression (DTR), Artificial Neural Network (ANN) and L-RNN. The obtained results show that the proposed method enhances the overall

**Keywords:** imputing missing data, air pollutants, prediction, layered recurrent

Air quality monitoring and management have drawn much attention in recent years and attracted great attention from the public. Air pollution poses serious problems and infection for living organisms and environmental risks [1]. Harmful emission of industrial waste on air is one of the common environmental influences that disturb the air quality specifications and the national economy [2]. Significant publications have shown that air pollution has harmful effects on human health [3]. Air pollution affects the living organisms by producing impacts on cardiac, vascular, pulmonary, and neurological systems [4]. For example, air pollution in the City

Network for Imputing Air

*Hamza Turabieh, Alaa Sheta, Malik Braik*

#### **Chapter 3**

## A Layered Recurrent Neural Network for Imputing Air Pollutants Missing Data and Prediction of *NO*2, *O*3, *PM*10, and *PM*2.5

*Hamza Turabieh, Alaa Sheta, Malik Braik and Elvira Kovač-Andrić*

#### **Abstract**

To fulfill the national air quality standards, many countries have created emissions monitoring strategies on air quality. Nowadays, policymakers and air quality executives depend on scientific computation and prediction models to monitor that cause air pollution, especially in industrial cities. Air pollution is considered one of the primary problems that could cause many human health problems such as asthma, damage to lungs, and even death. In this study, we present investigated development forecasting models for air pollutant attributes including Particulate Matters (PM2.5, PM10), ground-level Ozone (O3), and Nitrogen Oxides (NO2). The dataset used was collected from Dubrovnik city, which is located in the east of Croatia. The collected data has missing values. Therefore, we suggested the use of a Layered Recurrent Neural Network (L-RNN) to impute the missing value(s) of air pollutant attributes then build forecasting models. We adopted four regression models to forecast air pollutant attributes, which are: Multiple Linear Regression (MLR), Decision Tree Regression (DTR), Artificial Neural Network (ANN) and L-RNN. The obtained results show that the proposed method enhances the overall performance of other forecasting models.

**Keywords:** imputing missing data, air pollutants, prediction, layered recurrent neural network

#### **1. Introduction**

Air quality monitoring and management have drawn much attention in recent years and attracted great attention from the public. Air pollution poses serious problems and infection for living organisms and environmental risks [1]. Harmful emission of industrial waste on air is one of the common environmental influences that disturb the air quality specifications and the national economy [2]. Significant publications have shown that air pollution has harmful effects on human health [3]. Air pollution affects the living organisms by producing impacts on cardiac, vascular, pulmonary, and neurological systems [4]. For example, air pollution in the City of New York causes the death of more than 3000 people and causes hospitalization of 200 persons [5]. It was found that many of these reported incidences were caused by the exposure to PM2.5 and other pollutant attributes [6]. In 2010, it was estimated that ambient particulate matter (PM) caused 3.2 million premature deaths [7]. Moreover, several-analysis and research papers highlight that there is an exponential relationship between PM values and cardiovascular disease, and significant relation between NO2 concentrations and cardiovascular disease [8, 9].

Air pollution arises from many sources such as vehicle fumes, agricultural, industrial, and natural sources like volcanoes [10]. Common air pollutants are classified into two groups: trace gases such as carbon monoxide (CO), nitrogen dioxide (NO2), ground-level ozone (O3), and sulfur dioxide (SO2) or particulate matter (PM2.5) or (PM10) in aerodynamic diameter [11]. Tropospheric ground-level ozone (O3) is a secondary factor that can damage human health and ecosystem [12–41]. O3 concentration is one of the most serious oxidant factors that are harmful to human skin and lung tissues when inhaled [15, 16]. Several side effects impair pulmonary function and cause respiratory symptoms such as headache, weight loss, cough, shortness of breath, hoarseness, and pain while breathing [17]. Moreover, several epidemiological research studies focus on the relation between O3 pollution and mortality [18].

#### **1.1 Challenges**

Air pollution monitoring and control is a major global challenge [19, 20]. To develop and train air quality prediction models, meteorological data for the investigated area should be collected and used. This data mostly consists of physical parameters that include temperature, dew point, wind direction, wind speed, cloud cover, cloud layer(s), ceiling height, visibility, current weather, amount of precipitation, and many more [21, 42]. These attributes greatly influence the concentration of pollutants in the area of interest.

learning methods to impute missing data [25]. Imputing missing values based on

*A Layered Recurrent Neural Network for Imputing Air Pollutants Missing Data…*

The main goal of this study is to propose a hybrid model that can predict the daily average of the concentration of air pollutants based on missing data imputation. The proposed model is a machine learning approach that can enhance the performance of monitoring systems of air pollution inside cities. Layered Recurrent Neural Network (L-RNN) [26] was successfully used to solve a variety of stateof-the-art applications such as detection of heart failure [27] and time-series data classification [28]. L-RNNs for missing data were explored earlier to handle the missing data problem [25, 29]. In this research, we first explore the use of L-RNN for imputing the missing data collected from Dubrovnik city that is in the east of Croatia. In the second step, we develop a series of models for predicting NO2, *O*3, *PM*10, and *PM*2*:*<sup>5</sup> using the machine learning model (i.e., MLR, DTR, ANN, and

The rest of this chapter is organized as follows: In Section 2, the related works of air pollution as well as the literature of missing data is presented. Section 3 describes the proposed methodology. Section 4 presents predictive models using machine learning concepts. Section 5 presents the data collection process. The evaluation criteria used in this chapter are presented in Section 6. Section 7 presents the experimental setup used in this paper. Finally, a conclusion of the work is presented

One of the most common problems in the process of developing prediction models is the Data Cleaning/Exploratory Analysis. This phase becomes a challenge

average is not accurate compared to the other one.

*The process of collecting data using air pollutants sensors.*

*DOI: http://dx.doi.org/10.5772/intechopen.93678*

L-RNN).

**Figure 1.**

in Section 8.

**49**

**2. Related works**

**2.1 Imputation vs. removing data**

Recently, cities are exposed to air pollutants either indoors or outdoors [22]. Several monitoring stations (i.e., sensors) are used to monitor the air quality by collecting data from different locations inside cities. These stations are used to collect data for gases or particulate matter in an accurate manner [23]. These sensors can be categorized as wired or wireless sensors. Wired sensors need great efforts for deployment and maintenance. Wired sensors can be easily breakdown due to several reasons (e.g., environment close to a volcano, where the hot gases and steams can damage a wired network easily [24, 43]). Wireless sensors still in an early stage. However, they show a great performance compared to wired sensors either in deployment or maintenance. Both types of sensors send the collected data to a central station for further processing. However, sometimes the process of collecting data suffers from different problems such as power failure, sensor fault, man-made error in measurements, and many others. **Figure 1** depicts the process of collecting data from different sensors. For example, if the gas sensor (i.e., O3) does not work accurately, the collected data will not be complete and accurate. As a result, the air quality of the prediction model may not be accurate if the percentage of missing data is high.

Missing data cause serious problems for developing prediction models. The presence of missing data could severely reduce the quality of air quality prediction models. To solve this problem, we may either remove the missing data or imputing it. Removing the missing data may reduce the application performance [25], while imputing missing data may enhance the overall performance and without losing the collected data. Many methods exist to impute the missing data. Researchers either applied simple methods such as average value or complex methods such as machine *A Layered Recurrent Neural Network for Imputing Air Pollutants Missing Data… DOI: http://dx.doi.org/10.5772/intechopen.93678*

**Figure 1.**

of New York causes the death of more than 3000 people and causes hospitalization of 200 persons [5]. It was found that many of these reported incidences were caused by the exposure to PM2.5 and other pollutant attributes [6]. In 2010, it was estimated that ambient particulate matter (PM) caused 3.2 million premature deaths [7]. Moreover, several-analysis and research papers highlight that there is an exponential relationship between PM values and cardiovascular disease, and significant

Air pollution arises from many sources such as vehicle fumes, agricultural, industrial, and natural sources like volcanoes [10]. Common air pollutants are classified into two groups: trace gases such as carbon monoxide (CO), nitrogen dioxide (NO2), ground-level ozone (O3), and sulfur dioxide (SO2) or particulate matter (PM2.5) or (PM10) in aerodynamic diameter [11]. Tropospheric ground-level ozone (O3) is a secondary factor that can damage human health and ecosystem [12–41]. O3 concentration is one of the most serious oxidant factors that are harmful to human skin and lung tissues when inhaled [15, 16]. Several side effects impair pulmonary function and cause respiratory symptoms such as headache, weight loss, cough, shortness of breath, hoarseness, and pain while breathing [17]. Moreover, several epidemiological research studies focus on the relation between O3 pollution

Air pollution monitoring and control is a major global challenge [19, 20]. To develop and train air quality prediction models, meteorological data for the investigated area should be collected and used. This data mostly consists of physical parameters that include temperature, dew point, wind direction, wind speed, cloud cover, cloud layer(s), ceiling height, visibility, current weather, amount of precipitation, and many more [21, 42]. These attributes greatly influence the concentra-

Recently, cities are exposed to air pollutants either indoors or outdoors [22]. Several monitoring stations (i.e., sensors) are used to monitor the air quality by collecting data from different locations inside cities. These stations are used to collect data for gases or particulate matter in an accurate manner [23]. These sensors can be categorized as wired or wireless sensors. Wired sensors need great efforts for deployment and maintenance. Wired sensors can be easily breakdown due to several reasons (e.g., environment close to a volcano, where the hot gases and steams can damage a wired network easily [24, 43]). Wireless sensors still in an early stage. However, they show a great performance compared to wired sensors either in deployment or maintenance. Both types of sensors send the collected data to a central station for further processing. However, sometimes the process of collecting data suffers from different problems such as power failure, sensor fault, man-made error in measurements, and many others. **Figure 1** depicts the process of collecting data from different sensors. For example, if the gas sensor (i.e., O3) does not work accurately, the collected data will not be complete and accurate. As a result, the air quality of the prediction model may not be accurate if the percentage

Missing data cause serious problems for developing prediction models. The presence of missing data could severely reduce the quality of air quality prediction models. To solve this problem, we may either remove the missing data or imputing it. Removing the missing data may reduce the application performance [25], while imputing missing data may enhance the overall performance and without losing the collected data. Many methods exist to impute the missing data. Researchers either applied simple methods such as average value or complex methods such as machine

relation between NO2 concentrations and cardiovascular disease [8, 9].

*Forecasting in Mathematics - Recent Advances, New Perspectives and Applications*

and mortality [18].

of missing data is high.

**48**

tion of pollutants in the area of interest.

**1.1 Challenges**

*The process of collecting data using air pollutants sensors.*

learning methods to impute missing data [25]. Imputing missing values based on average is not accurate compared to the other one.

The main goal of this study is to propose a hybrid model that can predict the daily average of the concentration of air pollutants based on missing data imputation. The proposed model is a machine learning approach that can enhance the performance of monitoring systems of air pollution inside cities. Layered Recurrent Neural Network (L-RNN) [26] was successfully used to solve a variety of stateof-the-art applications such as detection of heart failure [27] and time-series data classification [28]. L-RNNs for missing data were explored earlier to handle the missing data problem [25, 29]. In this research, we first explore the use of L-RNN for imputing the missing data collected from Dubrovnik city that is in the east of Croatia. In the second step, we develop a series of models for predicting NO2, *O*3, *PM*10, and *PM*2*:*<sup>5</sup> using the machine learning model (i.e., MLR, DTR, ANN, and L-RNN).

The rest of this chapter is organized as follows: In Section 2, the related works of air pollution as well as the literature of missing data is presented. Section 3 describes the proposed methodology. Section 4 presents predictive models using machine learning concepts. Section 5 presents the data collection process. The evaluation criteria used in this chapter are presented in Section 6. Section 7 presents the experimental setup used in this paper. Finally, a conclusion of the work is presented in Section 8.

#### **2. Related works**

#### **2.1 Imputation vs. removing data**

One of the most common problems in the process of developing prediction models is the Data Cleaning/Exploratory Analysis. This phase becomes a challenge when missing values are in presence. In general, there is no fundamental method to deal with missing data. Missing data problem occurs if no value(s) is assigned while collecting data. In general, the missing data are presented by different symbols such as ?, N/A, or . There are two methods adopted in the literature to handle missing data. They are:

needed. Air monitoring consists of several steps; 1) Monitoring sites based on wired or wireless sensors, 2) collecting data that should be accurate and complete, 3) data analysis using predictive models based on machine learning to predict and analyze the collected data, and, 4) the final step is making decisions to reduce the concentrations of air pollution. This process should be performed correctly to ensure that

*A Layered Recurrent Neural Network for Imputing Air Pollutants Missing Data…*

Different types of machine learning methods have been used to predict the concentrations of air pollutant indicators by many researchers. For example, Perez and Gramsch [37] applied a feed-forward neural network to predict the concentration of PM2.5 and PM10 in Santiago, Chile. The obtained accurate results show that the proposed approach enhances the prediction of PM2.5 and PM10. Lana et al. [38] employed regression models to predict several air pollutants such as CO, NO, NO2, O3 and PM10) for the city of Madrid (Spain). The obtained results explore the importance of reducing air pollutants in the city of Madrid. Kamińska [39] employed an ensemble learning method based on random forests to model the relationship between the concentrations of air pollutants and nine variables describing meteorological conditions, temporal conditions, and traffic flow. The collected data was for 2 years 2015 and 2016 for WrocÅ'aw city. The data consists of hourly values of wind speed, wind direction, temperature, air pressure and relative humidity, temporal variables, and traffic flow. The obtained results show that the season plays a vital role in the overall performance. Kamińska [40] proposed a probabilistic forecasting method to predict the concentrations of NO2. The dataset represents the hourly values of the concentration of NO2 wind speed, and traffic flow for the main intersection in Wrocław city. The obtained results show that wind

Shang et al. [41] employed a novel prediction method that hybridized the regression tree (CART) and ensemble extreme learning machine (EELM) methods to predict the hourly concentration of PM2.5 air pollutant. The training dataset used in this research obtained from the meteorological data of Yancheng urban area, while the testing data (i.e., the air pollutant concentration) obtained from the City Monitoring Centre. The obtained results demonstrate the effectiveness of the proposed method to predict PM2.5. A hybrid framework based on three different machine learning methods (i.e., genetic algorithm [GA], random forests [RFs], and backpropagation neural networks [BPNN]) proposed by Dotse et al. [44]. The proposed hybrid approach is used to predict daily PM10 in Brunei Darussalam. Sun and Sun [45] proposed a hybrid model to predict PM2.5 in Baoding city in China, where a combination of three machine learning methods (i.e., principal component analysis [PCA], least squares support vector machine [LSSVM], and cuckoo search [CS]). The obtained results show that the PCA algorithm works as a feature selection algorithm that reduces the dimensionality of the input dataset while CS shows promising results to predict PM2.5. The main shortfall of this work that is applicable

A dynamic fuzzy synthetic evaluation model for predicting the concentration of three air pollutants (i.e., of PM2.5, PM10 and SO2) in two cities from China have been proposed by Xu et al. [46]. The obtained results show that the proposed model can be employed to build a robust monitoring air quality system for early warning. A novel hybrid model based on extreme learning machine (ELM) is employed to predict the concentration level of PM10 for Beijing and Harbin cities in China by Luo et al. [47]. Aznarte [48] proposed an ELM approach that is optimized by cuckoo search (CS) to enhance the overall performance of ELM. A probabilistic forecasting approach is applied to predict NO2 in Madrid city from Spain. Wang et al. [49] proposed a novel hybrid machine learning approach based on a decomposition method and extreme learning machine (ELM) that is optimized by differential

the concentration of air pollution is under control.

*DOI: http://dx.doi.org/10.5772/intechopen.93678*

speed plays a vital factor in the concentration of NO2.

for short-term PM2.5 forecasting.

**51**

#### *2.1.1 Deletion*

Several researchers remove the missing data from the collected dataset if the percentage of the missing data is less than 5%. However, if the percentage of missing data is greater than 5%, the dataset should be examined carefully [30]. Many approaches have been investigated by researchers to solve the missing data problem. For example, the data list wise deletion method removes the missing data or incomplete data from the collected dataset. This method works fine if the percentage of missing data is very small and does not affect the overall accuracy [31]. The pairwise data deletion method keeps the missing data and tries to reduce the loss that occurs in the list wise deletion method. However, deleting missing values is an acceptable approach for some applications. Mary and Arockiam [32] investigated the missing data as a case study of air pollution. They proposed an ST-correlated proximate approach to impute the incomplete dataset for the air pollution system. The authors compared the obtained results of the proposed approach with different statistical methods. Sta [33] investigated the process of collecting data for modern urban cities. The author proposed a framework to cluster the collected data into three clusters: complete, ambiguous, and missing data. The author imputed the missing data and enhanced the overall performance of the proposed system. Xiaodong et al. [34] proposed a Hot Deck imputation approach that imputes the incomplete records (missing data) using the similarity between complete and incomplete data.

#### *2.1.2 Imputation*

In statistics, imputation is defined as the process of substituting missing data with swapped values. Unit imputation is used when we replace a single data point while the replacement of a component of a data point, is called, item imputation. Imputation is considered a successful solution to avoid difficulties associated with list wise deletion of missing values. Suhani et al. [35] proposed a machine learning approach based on the fuzzy kNN technique to impute the missing data for a selected case from the medical field. The authors ignore the missing data whose entropy value is less than a predetermined value and recover the incomplete data that are higher than the predetermined value based on a fuzzy kNN algorithm. Chen et al. [36] applied a machine learning approach based on a convolutional neural network to impute the missing data for a real medical dataset. The authors improve the overall performance after imputing missing data. Turabieh et al. [25] proposed a dynamic model based on deep learning neural networks for missing data imputation. The authors showed that the proposed model improves the overall performance of medical applications after imputing missing data.

#### **2.2 Air pollution prediction**

Air pollution is a serious problem that negatively affects human health, environment, and climate. Governments and organizations published several initiatives to reduce the concentrations of air pollutants, but high levels of concentrations of air pollutants still exist. As a result, monitoring the concentrations of air pollution is

#### *A Layered Recurrent Neural Network for Imputing Air Pollutants Missing Data… DOI: http://dx.doi.org/10.5772/intechopen.93678*

needed. Air monitoring consists of several steps; 1) Monitoring sites based on wired or wireless sensors, 2) collecting data that should be accurate and complete, 3) data analysis using predictive models based on machine learning to predict and analyze the collected data, and, 4) the final step is making decisions to reduce the concentrations of air pollution. This process should be performed correctly to ensure that the concentration of air pollution is under control.

Different types of machine learning methods have been used to predict the concentrations of air pollutant indicators by many researchers. For example, Perez and Gramsch [37] applied a feed-forward neural network to predict the concentration of PM2.5 and PM10 in Santiago, Chile. The obtained accurate results show that the proposed approach enhances the prediction of PM2.5 and PM10. Lana et al. [38] employed regression models to predict several air pollutants such as CO, NO, NO2, O3 and PM10) for the city of Madrid (Spain). The obtained results explore the importance of reducing air pollutants in the city of Madrid. Kamińska [39] employed an ensemble learning method based on random forests to model the relationship between the concentrations of air pollutants and nine variables describing meteorological conditions, temporal conditions, and traffic flow. The collected data was for 2 years 2015 and 2016 for WrocÅ'aw city. The data consists of hourly values of wind speed, wind direction, temperature, air pressure and relative humidity, temporal variables, and traffic flow. The obtained results show that the season plays a vital role in the overall performance. Kamińska [40] proposed a probabilistic forecasting method to predict the concentrations of NO2. The dataset represents the hourly values of the concentration of NO2 wind speed, and traffic flow for the main intersection in Wrocław city. The obtained results show that wind speed plays a vital factor in the concentration of NO2.

Shang et al. [41] employed a novel prediction method that hybridized the regression tree (CART) and ensemble extreme learning machine (EELM) methods to predict the hourly concentration of PM2.5 air pollutant. The training dataset used in this research obtained from the meteorological data of Yancheng urban area, while the testing data (i.e., the air pollutant concentration) obtained from the City Monitoring Centre. The obtained results demonstrate the effectiveness of the proposed method to predict PM2.5. A hybrid framework based on three different machine learning methods (i.e., genetic algorithm [GA], random forests [RFs], and backpropagation neural networks [BPNN]) proposed by Dotse et al. [44]. The proposed hybrid approach is used to predict daily PM10 in Brunei Darussalam. Sun and Sun [45] proposed a hybrid model to predict PM2.5 in Baoding city in China, where a combination of three machine learning methods (i.e., principal component analysis [PCA], least squares support vector machine [LSSVM], and cuckoo search [CS]). The obtained results show that the PCA algorithm works as a feature selection algorithm that reduces the dimensionality of the input dataset while CS shows promising results to predict PM2.5. The main shortfall of this work that is applicable for short-term PM2.5 forecasting.

A dynamic fuzzy synthetic evaluation model for predicting the concentration of three air pollutants (i.e., of PM2.5, PM10 and SO2) in two cities from China have been proposed by Xu et al. [46]. The obtained results show that the proposed model can be employed to build a robust monitoring air quality system for early warning. A novel hybrid model based on extreme learning machine (ELM) is employed to predict the concentration level of PM10 for Beijing and Harbin cities in China by Luo et al. [47]. Aznarte [48] proposed an ELM approach that is optimized by cuckoo search (CS) to enhance the overall performance of ELM. A probabilistic forecasting approach is applied to predict NO2 in Madrid city from Spain. Wang et al. [49] proposed a novel hybrid machine learning approach based on a decomposition method and extreme learning machine (ELM) that is optimized by differential

when missing values are in presence. In general, there is no fundamental method to deal with missing data. Missing data problem occurs if no value(s) is assigned while collecting data. In general, the missing data are presented by different symbols such as ?, N/A, or . There are two methods adopted in the literature to handle missing

*Forecasting in Mathematics - Recent Advances, New Perspectives and Applications*

Several researchers remove the missing data from the collected dataset if the percentage of the missing data is less than 5%. However, if the percentage of missing data is greater than 5%, the dataset should be examined carefully [30]. Many approaches have been investigated by researchers to solve the missing data problem. For example, the data list wise deletion method removes the missing data or incomplete data from the collected dataset. This method works fine if the percentage of missing data is very small and does not affect the overall accuracy [31]. The pairwise data deletion method keeps the missing data and tries to reduce the loss that occurs in the list wise deletion method. However, deleting missing values is an acceptable approach for some applications. Mary and Arockiam [32] investigated the missing data as a case study of air pollution. They proposed an ST-correlated proximate approach to impute the incomplete dataset for the air pollution system. The authors compared the obtained results of the proposed approach with different statistical methods. Sta [33] investigated the process of collecting data for modern urban cities. The author proposed a framework to cluster the collected data into three clusters: complete, ambiguous, and missing data. The author imputed the missing data and enhanced the overall performance of the proposed system. Xiaodong et al. [34] proposed a Hot Deck imputation approach that imputes the incomplete records (missing data) using the similarity between complete and

In statistics, imputation is defined as the process of substituting missing data with swapped values. Unit imputation is used when we replace a single data point while the replacement of a component of a data point, is called, item imputation. Imputation is considered a successful solution to avoid difficulties associated with list wise deletion of missing values. Suhani et al. [35] proposed a machine learning approach based on the fuzzy kNN technique to impute the missing data for a selected case from the medical field. The authors ignore the missing data whose entropy value is less than a predetermined value and recover the incomplete data that are higher than the predetermined value based on a fuzzy kNN algorithm. Chen et al. [36] applied a machine learning approach based on a convolutional neural network to impute the missing data for a real medical dataset. The authors improve the overall performance after imputing missing data. Turabieh et al. [25] proposed a dynamic model based on deep learning neural networks for missing data imputation. The authors showed that the proposed model improves the overall

Air pollution is a serious problem that negatively affects human health, environment, and climate. Governments and organizations published several initiatives to reduce the concentrations of air pollutants, but high levels of concentrations of air pollutants still exist. As a result, monitoring the concentrations of air pollution is

performance of medical applications after imputing missing data.

data. They are:

*2.1.1 Deletion*

incomplete data.

*2.1.2 Imputation*

**2.2 Air pollution prediction**

**50**

evolution (DE) to predict air pollutants in Beijing and Shanghai cities from China. Kumar and Goyal [50] applied Multiple Linear Regression (MLR) and Principal Component Regression (PCR) methods to predict several air pollutants in Delhi city from India. A MultiLayer Perceptron (MLP) neural network is adopted to predict PM10 in Delhi city from India by Aly et al. [51]. The authors also applied two algorithms (i.e., Naïve Bayes [NB] and Support Vector Machine [SVM]) and the performance of MLP outperforms NB and SVM. Vibha and Satyendra [52] applied seven models of neural networks using Levenberg–Marquardt (LM) to predict the daily PM10 in two cities from India.

#### **3. Methodology**

The main purpose of this research is to evolve different machine learning methods to predict daily average air pollutant concentrations such as O3, PM2.5, PM10, and NO2 values given data with many missing values. The process consists of two phases: 1) imputing missing data based L-RNN model, and 2) development of predictive models using several machine learning algorithms which include LR, DTR, ANN, and L-RNN. Our proposed approach starts by collecting data from sensors. If the collected data suffer from missing data, an imputation process will start based on the L-RNN hat predicts the concentration of air pollutants. This process will be repeated until the collected data have no missing value(s). Once the collected dataset is complete, a machine learning model is selected to predict the daily average of air pollutant concentrations. The selected model is evaluated based on two evaluation criteria that are Root Mean Square Error (RMSE), and coefficient of determination (*R*<sup>2</sup> ). The proposed approach is depicted in **Figure 2**. The following subsections demonstrate the proposed approach.

connection between output layer and hidden layer(s). **Figure 3** demonstrates the connection feedback. In simple, during the training process, the output of the recurrent neural network is added to the output of the hidden layer. The result of summation is employed as an argument of the transfer function to gain the output in the succeeding iteration. Eq.(1) demonstrates the output of the L-RNN, where *u k*ð Þ presents the input values for hidden layer, *v k*ð Þ presents the input values for output layer. *Wu*,*<sup>i</sup>* and *Wv*,*<sup>j</sup>* presents the weights between *u* and *v*, respectively. The final output *y k*ð Þ is obtained from Eq.(2), where *f*ðÞ is a transfer function. In this work, we employed back-propagation through time in the training phase for the

*A Layered Recurrent Neural Network for Imputing Air Pollutants Missing Data…*

*Wu*,*<sup>i</sup>*ð Þ*<sup>k</sup> u k*ð ÞþX*<sup>m</sup>*

*j*¼0

*wv*,*<sup>j</sup>*ð Þ*k v k*ð Þ (1)

<sup>1</sup> <sup>þ</sup> *exp* ð�ð Þ *v k*ð Þ (2)

proposed L-RNN structure.

*Layer recurrent neural network (L-RNN).*

*DOI: http://dx.doi.org/10.5772/intechopen.93678*

**Figure 3.**

**3.2 Data imputation using L-RNN**

concentration value of O3, NO2, PM2.5, PM10).

missing value(s).

**53**

*v k*ð Þ¼ <sup>þ</sup> <sup>1</sup> <sup>X</sup>*<sup>n</sup>*

*i*¼0

*y k*ð Þ¼ *f vk* ð Þ ð Þ *where* : *f vk* ð Þ¼ ð Þ <sup>1</sup>

To implement the data imputation process, we clustered the data into two groups 1) complete dataset [without a missing value(s)] and 2) incomplete dataset [with a missing value(s)]. A holdout method is used to train and test the L-RNN. The complete dataset is divided into three datasets: training dataset (70%), testing dataset (15%), and validation dataset (15%). While the incomplete dataset is used to simulate the trained L-RNN model to impute the missing value(s). This process will be repeated dynamically while receiving any records with a

The computational complexity of the model depends on the structure of the L-RNN and the number of missing data in the received record. The computational complexity will increase exponentially if the number of missing values increases. **Figure 4** illustrates the process used to impute the missing value(s) (i.e., the

#### **3.1 L-RNN**

A layered recurrent neural network is known as a neural network that has local feedback, which is particularly suited to predict the daily air pollutant attributes since it incorporates a time delay while training process through a feedback

**Figure 2.** *The proposed approach.*

*A Layered Recurrent Neural Network for Imputing Air Pollutants Missing Data… DOI: http://dx.doi.org/10.5772/intechopen.93678*

#### **Figure 3.** *Layer recurrent neural network (L-RNN).*

evolution (DE) to predict air pollutants in Beijing and Shanghai cities from China. Kumar and Goyal [50] applied Multiple Linear Regression (MLR) and Principal Component Regression (PCR) methods to predict several air pollutants in Delhi city from India. A MultiLayer Perceptron (MLP) neural network is adopted to predict PM10 in Delhi city from India by Aly et al. [51]. The authors also applied two algorithms (i.e., Naïve Bayes [NB] and Support Vector Machine [SVM]) and the performance of MLP outperforms NB and SVM. Vibha and Satyendra [52] applied seven models of neural networks using Levenberg–Marquardt (LM) to predict the

*Forecasting in Mathematics - Recent Advances, New Perspectives and Applications*

The main purpose of this research is to evolve different machine learning methods to predict daily average air pollutant concentrations such as O3, PM2.5, PM10, and NO2 values given data with many missing values. The process consists of two phases: 1) imputing missing data based L-RNN model, and 2) development of predictive models using several machine learning algorithms which include LR, DTR, ANN, and L-RNN. Our proposed approach starts by collecting data from sensors. If the collected data suffer from missing data, an imputation process will start based on the L-RNN hat predicts the concentration of air pollutants. This process will be repeated until the collected data have no missing value(s). Once the collected dataset is complete, a machine learning model is selected to predict the daily average of air pollutant concentrations. The selected model is evaluated based on two evaluation criteria that are Root Mean Square Error (RMSE), and coefficient

). The proposed approach is depicted in **Figure 2**. The follow-

A layered recurrent neural network is known as a neural network that has local feedback, which is particularly suited to predict the daily air pollutant attributes since it incorporates a time delay while training process through a feedback

daily PM10 in two cities from India.

**3. Methodology**

of determination (*R*<sup>2</sup>

**3.1 L-RNN**

**Figure 2.**

**52**

*The proposed approach.*

ing subsections demonstrate the proposed approach.

connection between output layer and hidden layer(s). **Figure 3** demonstrates the connection feedback. In simple, during the training process, the output of the recurrent neural network is added to the output of the hidden layer. The result of summation is employed as an argument of the transfer function to gain the output in the succeeding iteration. Eq.(1) demonstrates the output of the L-RNN, where *u k*ð Þ presents the input values for hidden layer, *v k*ð Þ presents the input values for output layer. *Wu*,*<sup>i</sup>* and *Wv*,*<sup>j</sup>* presents the weights between *u* and *v*, respectively. The final output *y k*ð Þ is obtained from Eq.(2), where *f*ðÞ is a transfer function. In this work, we employed back-propagation through time in the training phase for the proposed L-RNN structure.

$$v(k+1) = \sum\_{i=0}^{n} W\_{u,i}(k)u(k) + \sum\_{j=0}^{m} w\_{v\_j}(k)v(k) \tag{1}$$

$$y(k) = f(v(k)) \quad where \; f(v(k)) = \frac{1}{1 + \exp\left(-(v(k))\right)} \tag{2}$$

#### **3.2 Data imputation using L-RNN**

To implement the data imputation process, we clustered the data into two groups 1) complete dataset [without a missing value(s)] and 2) incomplete dataset [with a missing value(s)]. A holdout method is used to train and test the L-RNN. The complete dataset is divided into three datasets: training dataset (70%), testing dataset (15%), and validation dataset (15%). While the incomplete dataset is used to simulate the trained L-RNN model to impute the missing value(s). This process will be repeated dynamically while receiving any records with a missing value(s).

The computational complexity of the model depends on the structure of the L-RNN and the number of missing data in the received record. The computational complexity will increase exponentially if the number of missing values increases. **Figure 4** illustrates the process used to impute the missing value(s) (i.e., the concentration value of O3, NO2, PM2.5, PM10).

where

*y*1 *y*2 *: : :*

, *ε* ¼

square method as shown in Eq.(6).

**4.2 Decision tree regression**

called the optimum tree.

**4.3 Artificial neural network**

**55**

*ε*1 *ε*2 *: : : εn*

, *β* ¼

independent input variable in the *mth* sample, and *ε<sup>i</sup>* is the *i*

*β*1 *β*2 *: : : βn*

*A Layered Recurrent Neural Network for Imputing Air Pollutants Missing Data…*

, *and X* ¼

where *n* represents the number of samples, *xm*,*<sup>i</sup>* represents the value of *i*

*<sup>β</sup>* <sup>¼</sup> *<sup>X</sup>TX* � ��<sup>1</sup>

can predict the dependent output variable based on the independent input

Therefore, when the parameter vector *β* is known, the generated MLR model

The DTR is employed in this chapter to predict the air pollutant attributes due to its ability to handle complex data and takes less training execution time compared to other prediction models. In simple, DTR uses if-then conditions to predict the appropriate output value(s) [55]. The DTR has three steps to predict the output

accuracy, selecting splits, when to stop splitting, and selecting the optimal tree.

variable, which usually measured with node impurity measure which provides an indication of the relative homogeneity of cases in the terminal nodes.

• **Step 3**: Determining when to stop the splitting which is related to the minimum number of nodes. Which means to select the best rightly-sized tree, which is

Artificial Neural Network (ANN) has been used widely in many forecasting applications due to its ability to handle complex data. Without having any information about the mathematical model that represents the relation between input and output variables, ANN can learn the learn hidden knowledge between input

Feedforward Neural Network (FFNN), Recurrent Neural Network (RNN), and Convolutional Neural Network (CNN) [56]. In this chapter, we adopted two types of neural networks based on a feed-forward network using the propagation

and output variables. In general, there are many kinds of ANNs such as

• **Step 1**: Determining the parameter settings for DTR such as: predicting

• **Step 2**: Selecting the splits to predict values of the continuous dependent

*mth* sample. The coefficient vector *β* can be calculated based on the standard least-

1, *x*1,1, … , *x*1,*<sup>k</sup>* 1, *x*2,1, … , *x*2,*<sup>k</sup> : : :* 1, *xn*,1, … , *xn*,*<sup>k</sup>*

*n*�ð Þ *k*þ1

*th* residual error in the

*XTY* (6)

(5)

*th*

*DOI: http://dx.doi.org/10.5772/intechopen.93678*

*yn*

*y* ¼

variable (s).

value(s) as follows:

#### **4. Predictive models using machine learning**

Several machine learning methods can be used to predict air quality. However, we have limited our research paper into four methods: MLR, DTR, ANN, and L-RNN. To avoid the over-fitting problem in the training process, we employed the k-fold cross-validation method with k-fold = 5. The following subsection explores each learning method in more detail.

#### **4.1 Multiple linear regression**

Linear regression (LR) is one of the most well-known algorithms in statistics and machine learning. The main idea of LR is to find the relationship between input and output numerical variables. There are several types of LR such as Simple linear regression, multiple linear regression, logistic regression, ordinal regression, Multinomial regression, and Discriminant Analysis. LR has been employed successfully in many areas as a machine learning algorithm [53, 54]. MLR is a classical statistical method that tries to find a relationship between complex input–output variables. In simple, MLR tries to find an approximation linear function between independent input variables and dependent output variable without loss of generality. Eq.(3) explores the regression line in MLR.

$$\boldsymbol{y} = \boldsymbol{\beta}\_0 + \boldsymbol{\beta}\_1 \boldsymbol{\kappa}\_1 + \dots + \boldsymbol{\beta}\_i \boldsymbol{\kappa}\_i + \dots + \boldsymbol{\beta}\_k \boldsymbol{\kappa}\_k + \varepsilon \tag{3}$$

where *y* is dependent output variable, *xi* is the *i th* independent input variable, *β<sup>i</sup>* is polynomial coefficients of *xi*, *k* is the number of independent input variables, and *ε*is the possible variation form. Eq.(4) presents a compact version of Eq.(3).

$$
\varphi = X\beta + \varepsilon \tag{4}
$$

*A Layered Recurrent Neural Network for Imputing Air Pollutants Missing Data… DOI: http://dx.doi.org/10.5772/intechopen.93678*

where

$$\mathbf{y} = \begin{bmatrix} \boldsymbol{\mathcal{y}\_1} \\ \boldsymbol{\mathcal{y}\_2} \\ \vdots \\ \boldsymbol{\mathcal{x}} \\ \vdots \\ \boldsymbol{\mathcal{x}} \\ \boldsymbol{\mathcal{y}}\_n \end{bmatrix}, \boldsymbol{\varepsilon} = \begin{bmatrix} \boldsymbol{c}\_1 \\ \boldsymbol{c}\_2 \\ \vdots \\ \boldsymbol{c}\_n \\ \vdots \\ \boldsymbol{c}\_n \end{bmatrix}, \boldsymbol{\beta} = \begin{bmatrix} \boldsymbol{\beta}\_1 \\ \boldsymbol{\beta}\_2 \\ \vdots \\ \boldsymbol{\cdot} \\ \vdots \\ \boldsymbol{\cdot} \\ \boldsymbol{\beta}\_n \end{bmatrix}, \text{and } \quad \boldsymbol{X} = \begin{bmatrix} \mathbf{1}, \mathbf{x}\_{1,1}, \dots, \mathbf{x}\_{1,k} \\ \vdots \\ \mathbf{1}, \mathbf{x}\_{2,1}, \dots, \mathbf{x}\_{2,k} \\ \vdots \\ \mathbf{1}, \mathbf{x}\_{n,1}, \dots, \mathbf{x}\_{n,k} \end{bmatrix} \tag{5}$$

where *n* represents the number of samples, *xm*,*<sup>i</sup>* represents the value of *i th* independent input variable in the *mth* sample, and *ε<sup>i</sup>* is the *i th* residual error in the *mth* sample. The coefficient vector *β* can be calculated based on the standard leastsquare method as shown in Eq.(6).

$$\boldsymbol{\beta} = \left(\mathbf{X}^T\mathbf{X}\right)^{-1}\mathbf{X}^T\mathbf{Y} \tag{6}$$

Therefore, when the parameter vector *β* is known, the generated MLR model can predict the dependent output variable based on the independent input variable (s).

#### **4.2 Decision tree regression**

**4. Predictive models using machine learning**

each learning method in more detail.

**4.1 Multiple linear regression**

in MLR.

**Figure 4.**

*Impute missing data approach.*

of Eq.(3).

**54**

Several machine learning methods can be used to predict air quality. However,

Linear regression (LR) is one of the most well-known algorithms in statistics and machine learning. The main idea of LR is to find the relationship between input and output numerical variables. There are several types of LR such as Simple linear regression, multiple linear regression, logistic regression, ordinal regression, Multinomial regression, and Discriminant Analysis. LR has been employed successfully in many areas as a machine learning algorithm [53, 54]. MLR is a classical statistical method that tries to find a relationship between complex input–output variables. In simple, MLR tries to find an approximation

linear function between independent input variables and dependent

where *y* is dependent output variable, *xi* is the *i*

output variable without loss of generality. Eq.(3) explores the regression line

variable, *β<sup>i</sup>* is polynomial coefficients of *xi*, *k* is the number of independent input variables, and *ε*is the possible variation form. Eq.(4) presents a compact version

*y* ¼ *β*<sup>0</sup> þ *β*1*x*<sup>1</sup> þ … þ *βixi* þ … þ *βkxk* þ *ε* (3)

*th* independent input

*y* ¼ *Xβ* þ *ε* (4)

we have limited our research paper into four methods: MLR, DTR, ANN, and L-RNN. To avoid the over-fitting problem in the training process, we employed the k-fold cross-validation method with k-fold = 5. The following subsection explores

*Forecasting in Mathematics - Recent Advances, New Perspectives and Applications*

The DTR is employed in this chapter to predict the air pollutant attributes due to its ability to handle complex data and takes less training execution time compared to other prediction models. In simple, DTR uses if-then conditions to predict the appropriate output value(s) [55]. The DTR has three steps to predict the output value(s) as follows:


#### **4.3 Artificial neural network**

Artificial Neural Network (ANN) has been used widely in many forecasting applications due to its ability to handle complex data. Without having any information about the mathematical model that represents the relation between input and output variables, ANN can learn the learn hidden knowledge between input and output variables. In general, there are many kinds of ANNs such as Feedforward Neural Network (FFNN), Recurrent Neural Network (RNN), and Convolutional Neural Network (CNN) [56]. In this chapter, we adopted two types of neural networks based on a feed-forward network using the propagation

training method, which is: the standard neural network (ANN) and Layered Recurrent Neural Network (LRNN).

#### **5. Data collection**

The data set used in this research is collected from Dubrovnik city that is located in the east of Croatia. Dubrovnik city has a Mediterranean climate and has over 2600 hours of sunshine per year, which is considered the sunniest place in Croatia. In this dataset, the concentration of O3 has been monitored with a commercial Teledyne API 400E UV photometric O3 analyzer. While the concentration of NO2 has been monitored with Teledyne API 200E chemiluminescent NO2 analyzer. O3 and NO2 concentrations were measured every minute and the output signals were stored in a datalogger. The collected data are validated and averaged. The concentration of PM10, and PM2.5 have been monitored with the GRIMM model EDM 180. Samples of PM particles were collected by gravimetric methods throughout the day to obtain 24–hour averages of concentrations. All instruments are regularly maintained and calibrated. Meteorological data were obtained from the Meteorological and Hydrological Services of Croatia. The dataset is collected during the 2015 and 2016.

**Table 1** shows the number of records in each dataset used in this paper. For example, the O3 dataset has 699 total records, where 200 records (28*:*80%) are incomplete. **Figure 5b** demonstrates the missing data pattern for each dataset, where the x-axis presents the 24-hours (i.e., input variables), while the y-axis presents the observations during the 2 years. **Figure 5a** shows that there is a missing data in the second year for NO2 dataset, where NO2 sensors do not work. Since the missing data are higher than 5%, we examined the collected data carefully to maintain the performance of air quality prediction systems. As a result, imputing missing data are needed.

#### **6. Evaluation criteria**

In this research, we employed two different evaluation criteria: Root Mean Square Error (RMSE), and coefficient of determination (*R*<sup>2</sup> ), defined below.

$$RMSE = \sqrt{\frac{1}{s} \sum\_{i=1}^{s} \left( \mathcal{y}\_{i\,predicted} - \mathcal{y}\_{i\, observed} \right)^{2}} \tag{7}$$

where *yi observed* and *yi predicted* denote the actual and predicted values of air pollution concentrations, respectively, *s* represents the number of instances and ^*yi observed* stands for the average of the actual values of the air pollution concentrations. Eqs.(7) and (8) show the evaluation process for each criteria. The minimum value of RMSE means better forecasting, while the maximum value of *R*<sup>2</sup> means

*Actual () and predicted (.) values for No2 using all regression methods for NO2 dataset.*

*A Layered Recurrent Neural Network for Imputing Air Pollutants Missing Data…*

*DOI: http://dx.doi.org/10.5772/intechopen.93678*

In this work, two different types of experiments were performed to develop a prediction model for pollutant parameters with missing data. They are: (i) removing missing or incomplete records, and (ii) imputing the missing data. Four regression models were employed in this work (i.e., MLR, DT, ANN, and LRNN). All experiments were performed using MATLAB-R2019b environment. The following

The first experiments that we employed in this chapter are based on removing

all the missing data (i.e., records). **Table 2** shows the obtained results of four different regression models. The LRNN model outperforms other models in three

better forecasting.

**Figure 5.**

**57**

**7. Experimental results**

subsections discussed the obtained results.

**7.1 Results without imputing missing data**

$$R^2 = \mathbf{1} - \frac{\sum\_{i=1}^{\epsilon} \left(\mathbf{y}\_{i\,predicted} - \mathbf{y}\_{i\,observed}\right)^2}{\sum\_{i=1}^{\epsilon} \left(\mathbf{y}\_{i\,observed} - \mathbf{\hat{y}}\_{i\,observed}\right)^2} \tag{8}$$


**Table 1.**

*Number of samples in each dataset.*

*A Layered Recurrent Neural Network for Imputing Air Pollutants Missing Data… DOI: http://dx.doi.org/10.5772/intechopen.93678*

**Figure 5.** *Actual () and predicted (.) values for No2 using all regression methods for NO2 dataset.*

where *yi observed* and *yi predicted* denote the actual and predicted values of air pollution concentrations, respectively, *s* represents the number of instances and ^*yi observed* stands for the average of the actual values of the air pollution concentrations.

Eqs.(7) and (8) show the evaluation process for each criteria. The minimum value of RMSE means better forecasting, while the maximum value of *R*<sup>2</sup> means better forecasting.

#### **7. Experimental results**

In this work, two different types of experiments were performed to develop a prediction model for pollutant parameters with missing data. They are: (i) removing missing or incomplete records, and (ii) imputing the missing data. Four regression models were employed in this work (i.e., MLR, DT, ANN, and LRNN). All experiments were performed using MATLAB-R2019b environment. The following subsections discussed the obtained results.

#### **7.1 Results without imputing missing data**

The first experiments that we employed in this chapter are based on removing all the missing data (i.e., records). **Table 2** shows the obtained results of four different regression models. The LRNN model outperforms other models in three

training method, which is: the standard neural network (ANN) and Layered

*Forecasting in Mathematics - Recent Advances, New Perspectives and Applications*

to obtain 24–hour averages of concentrations. All instruments are regularly maintained and calibrated. Meteorological data were obtained from the Meteorological and Hydrological Services of Croatia. The dataset is collected during the 2015

**Table 1** shows the number of records in each dataset used in this paper. For example, the O3 dataset has 699 total records, where 200 records (28*:*80%) are incomplete. **Figure 5b** demonstrates the missing data pattern for each dataset, where the x-axis presents the 24-hours (i.e., input variables), while the y-axis presents the observations during the 2 years. **Figure 5a** shows that there is a missing data in the second year for NO2 dataset, where NO2 sensors do not work. Since the missing data are higher than 5%, we examined the collected data carefully to maintain the performance of air quality prediction systems. As a result, imputing

In this research, we employed two different evaluation criteria: Root Mean

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

*<sup>i</sup>*¼<sup>1</sup> *yi predicted* � *yi observed* � �<sup>2</sup>

*<sup>i</sup>*¼<sup>1</sup> *yi observed* � ^*yi observed*

**Dataset PM2.5 PM10 O3 NO2** InComplete 179 179 200 270 Complete 551 551 499 461 Percentage of missing data % 16.96 17.08 20.26 28.80 Total number of records 730 731 699 731

*<sup>i</sup>*¼<sup>1</sup> *yi predicted* � *yi observed* � �<sup>2</sup>

), defined below.

� �<sup>2</sup> (8)

(7)

Square Error (RMSE), and coefficient of determination (*R*<sup>2</sup>

1 *s* X*<sup>s</sup>*

P*<sup>s</sup>*

P*<sup>s</sup>*

r

*RMSE* ¼

*<sup>R</sup>*<sup>2</sup> <sup>¼</sup> <sup>1</sup> �

The data set used in this research is collected from Dubrovnik city that is located in the east of Croatia. Dubrovnik city has a Mediterranean climate and has over 2600 hours of sunshine per year, which is considered the sunniest place in Croatia. In this dataset, the concentration of O3 has been monitored with a commercial Teledyne API 400E UV photometric O3 analyzer. While the concentration of NO2 has been monitored with Teledyne API 200E chemiluminescent NO2 analyzer. O3 and NO2 concentrations were measured every minute and the output signals were stored in a datalogger. The collected data are validated and averaged. The concentration of PM10, and PM2.5 have been monitored with the GRIMM model EDM 180. Samples of PM particles were collected by gravimetric methods throughout the day

Recurrent Neural Network (LRNN).

**5. Data collection**

and 2016.

**Table 1.**

**56**

*Number of samples in each dataset.*

missing data are needed.

**6. Evaluation criteria**


**Table 2.**

*Results without imputing missing data.*

datasets (i.e., NO2, PM10, and PM2.5) based on RMSE and *R*<sup>2</sup> values. While ANN outperforms other models in O3 dataset based on RMSE. The performance of the MLR method is the worst overall datasets.

#### **7.2 Imputing data using LRNN**

For imputing missing data, we employed L-RNN as a dynamic prediction model based on the current states of the collected data. In general, there are two different training algorithms for L-RNN: real-time recurrent learning, where a fixed set of weights recursively applied while training process and back-propagation through time, where the L-RNN structure altered between feed-forward and feedback structures. In this work, we used back-propagation through a time training process.

The parameters setting used, in this case, are shown in **Table 3**. A holdout method is employed to train the L-RNN based on the complete dataset, where 70% for training, 15% for validation, and 15% for testing. The reason for employing the holdout method is to reduce the complexity and execution time for the proposed imputing model. After imputing missing data, we employed a k-fold across-validation method in the training process for four machine learning methods (i.e., MLR, DTR, L-RNN, and ANN) with k-fold = 5 to evaluate the complete dataset.

#### **7.3 Results after imputing missing data**

#### *7.3.1 MLR models*

In this part, we employed MLR as a prediction model after imputing missing data. In Eq.(9), Eq.(10), Eq.(11), and Eq.(12) we show the MLR results for PM2.5, PM10, O3, and NO2, respectively. **Table 4** shows the obtained results of MLR

*PM*2*:*<sup>5</sup> ¼ 4*:*4447 � 0*:*3753 � *NO*<sup>2</sup> þ 0*:*61551 � *PM*<sup>10</sup> � 0*:*025394 � *O*<sup>3</sup> (9)

*PM*<sup>10</sup> ¼ �3*:*8704 þ 0*:*6183 � *NO*<sup>2</sup> þ 0*:*033796 � *O*<sup>3</sup> þ 1*:*2886 � *PM*2*:*<sup>5</sup> (10)

*O*<sup>3</sup> ¼ 95*:*039 � 0*:*22434 � *NO*<sup>2</sup> þ 0*:*96905 � *PM*<sup>10</sup> � 1*:*493 � *PM*2*:*<sup>5</sup> (11)

*NO*<sup>2</sup> ¼ 93*:*9595 þ 0*:*0017985 � *O*<sup>3</sup> þ 0*:*15471 � *PM*<sup>10</sup> � 0*:*17977 � *PM*2*:*<sup>5</sup> (12)

In this work, the minimum leave size used is 4, and the maximum number of splits is 6. The main reason for using this setting is to simplify the generated tree.

method. The performance of of MLR is acceptable over all datasets.

*Obtained tree for all datasets after imputing missing data.*

**Dataset MLR results**

*A Layered Recurrent Neural Network for Imputing Air Pollutants Missing Data…*

NO2 1.61 0.79 O3 1.78 0.82 PM10 3.84 0.46 PM2.5 1.76 0.62

**RMSE** *R***<sup>2</sup>**

*7.3.2 DT models*

**59**

**Figure 6.**

**Table 4.**

*MLR results after imputing missing data.*

*DOI: http://dx.doi.org/10.5772/intechopen.93678*


#### **Table 3.**

*Parameter settings for the L-RNN model during imputing missing data.*


*A Layered Recurrent Neural Network for Imputing Air Pollutants Missing Data… DOI: http://dx.doi.org/10.5772/intechopen.93678*

**Table 4.**

datasets (i.e., NO2, PM10, and PM2.5) based on RMSE and *R*<sup>2</sup> values. While ANN outperforms other models in O3 dataset based on RMSE. The performance of the

**Regression model NO2 O3 PM10 PM2.5**

*Forecasting in Mathematics - Recent Advances, New Perspectives and Applications*

MLR 1.79 0.10 22.11 0.05 3.68 0.83 2.61 0.81 DT 1.73 0.16 20.44 0.19 3.68 0.83 2.57 0.82 LRNN **0.26 0.85** 10.06 0.61 **0.30 0.88 2.39 0.90** ANN 1.02 0.74 **8.39 0.76** 3.85 0.67 2.92 0.85

**RMSE** *R***<sup>2</sup> RMSE** *R***<sup>2</sup> RMSE** *R***<sup>2</sup> RMSE** *R***<sup>2</sup>**

For imputing missing data, we employed L-RNN as a dynamic prediction model based on the current states of the collected data. In general, there are two different training algorithms for L-RNN: real-time recurrent learning, where a fixed set of weights recursively applied while training process and back-propagation through time, where the L-RNN structure altered between feed-forward and feedback structures. In this work, we used back-propagation through a time

The parameters setting used, in this case, are shown in **Table 3**. A holdout method is employed to train the L-RNN based on the complete dataset, where 70% for training, 15% for validation, and 15% for testing. The reason for employing the holdout method is to reduce the complexity and execution time for the proposed imputing model. After imputing missing data, we employed a k-fold across-validation method in the training process for four machine learning methods (i.e., MLR, DTR, L-RNN, and ANN) with k-fold = 5 to evaluate the complete

In this part, we employed MLR as a prediction model after imputing missing data. In Eq.(9), Eq.(10), Eq.(11), and Eq.(12) we show the MLR results for PM2.5,

Number of neurons in the input layer Number of input data Number of neurons in the hidden layer Number of input data /2

**Parameters Values** Number of iterations 1000

Number of neurons in the output layer 1

*Parameter settings for the L-RNN model during imputing missing data.*

MLR method is the worst overall datasets.

**7.3 Results after imputing missing data**

**7.2 Imputing data using LRNN**

*Results without imputing missing data.*

training process.

**Table 2.**

dataset.

**Table 3.**

**58**

*7.3.1 MLR models*

*MLR results after imputing missing data.*

**Figure 6.**

*Obtained tree for all datasets after imputing missing data.*

PM10, O3, and NO2, respectively. **Table 4** shows the obtained results of MLR method. The performance of of MLR is acceptable over all datasets.

$$\text{PM}\_{2.5} = 4.4447 - 0.3753 \times \text{NO}\_2 + 0.61551 \times \text{PM}\_{10} - 0.025394 \times \text{O}\_3 \tag{9}$$

$$PM\_{10} = -3.8704 + 0.6183 \times NO\_2 + 0.033796 \times O\_3 + 1.2886 \times PM\_{2.5} \tag{10}$$

$$\dot{Q}\_3 = 95.039 - 0.22434 \times \text{NO}\_2 + 0.96905 \times \text{PM}\_{10} - 1.493 \times \text{PM}\_{2.5} \tag{11}$$

$$\text{NO}\_2 = \text{93.9595} + 0.0017985 \times O\_3 + 0.15471 \times \text{PM}\_{10} - 0.17977 \times \text{PM}\_{2.5} \tag{12}$$

*7.3.2 DT models*

In this work, the minimum leave size used is 4, and the maximum number of splits is 6. The main reason for using this setting is to simplify the generated tree.

#### *Forecasting in Mathematics - Recent Advances, New Perspectives and Applications*


#### **Table 5.**

*DT results after imputing missing data.*

#### **Figure 7.** *ANN block diagram structure.*


#### **Table 6.**

*ANN results after imputing missing data.*


ANN over all datasets. The performance of ANN is excellent compared to MLR

**Dataset LRNN results**

*A Layered Recurrent Neural Network for Imputing Air Pollutants Missing Data…*

NO2 0.22 0.93 O3 2.76 0.80 PM10 0.02 0.98 PM2.5 0.02 0.93

**RMSE** *R***<sup>2</sup>**

In this chapter, we employed the LRNN as a regression model to predict the daily average of air pollutant attributes. **Table 7** shows the parameters setting for LRNN as a regression method. These settings have been selected carefully to fit our data based on a set ore preliminary experiments. **Table 8** shows the obtained results of LRNN. The performance of LRNN is outstanding based on the convergence curves as shown in **Figure 8**. LRNN method can converge within 1000 epochs.

and DT.

**61**

**Figure 8.**

**Table 8.**

*LRNN results after imputing missing data.*

*DOI: http://dx.doi.org/10.5772/intechopen.93678*

*7.3.4 LRNN models*

*Convergence curves for LRNN over all datasets.*

#### **Table 7.**

*Parameters setting for LRNN as a regression method.*

The obtained models of DT for each dataset were shown in **Figure 6**. **Table 5** explores the obtained results of DT over all datasets.

#### *7.3.3 ANN models*

**Figure 7** shows the ANN structure used in this chapter, where we have three inputs and a single output. **Table 6** shows the obtained results of


*A Layered Recurrent Neural Network for Imputing Air Pollutants Missing Data… DOI: http://dx.doi.org/10.5772/intechopen.93678*

**Table 8.**

*LRNN results after imputing missing data.*

**Figure 8.** *Convergence curves for LRNN over all datasets.*

ANN over all datasets. The performance of ANN is excellent compared to MLR and DT.

#### *7.3.4 LRNN models*

In this chapter, we employed the LRNN as a regression model to predict the daily average of air pollutant attributes. **Table 7** shows the parameters setting for LRNN as a regression method. These settings have been selected carefully to fit our data based on a set ore preliminary experiments. **Table 8** shows the obtained results of LRNN. The performance of LRNN is outstanding based on the convergence curves as shown in **Figure 8**. LRNN method can converge within 1000 epochs.

The obtained models of DT for each dataset were shown in **Figure 6**. **Table 5**

**LRNN** Number of epoch 1000

**Dataset- ANN results**

**Dataset DT results**

*Forecasting in Mathematics - Recent Advances, New Perspectives and Applications*

NO2 2.36 0.65 O3 7.32 0.54 PM10 3.21 0.89 PM2.5 3.14 0.85

NO2 0.05 0.96 O3 3.25 0.78 PM10 0.33 0.82 PM2.5 0.16 0.97

**RMSE** *R***<sup>2</sup>**

**RMSE** *R***<sup>2</sup>**

**Parameters Value**

Layer delays 1:2 Hidden sizes 10 Training function Back propagation

**Figure 7** shows the ANN structure used in this chapter, where we have three inputs and a single output. **Table 6** shows the obtained results of

explores the obtained results of DT over all datasets.

*Parameters setting for LRNN as a regression method.*

*7.3.3 ANN models*

**Figure 7.**

**Table 5.**

**Table 6.**

**Table 7.**

**60**

*ANN block diagram structure.*

*DT results after imputing missing data.*

*ANN results after imputing missing data.*


For example, the *R*<sup>2</sup> value of ANN over O3 dataset before imputing missing data was 0.76, and after imputing missing data becomes 0.78, while the RMSE is improved 39%. So, we can conclude that imputing missing data will improve the air quality measurement systems without losing any record of

*A Layered Recurrent Neural Network for Imputing Air Pollutants Missing Data…*

Data collection from remote sensors suffers from missing data which reduces

the overall performance of air quality monitoring systems. Monitoring air pollution is not an easy task, where several measurements are used to evaluate air quality. In this study, four measurements are used to predict air pollution concentrations (i.e., O3, NO2, PM2.5, and PM10). We imputed the missing data using the Layered recurrent neural network (L-RNN). The performance of four different machine learning models (i.e., LR, DTR, ANN, and L-RNN) was investigated to predict the average daily air pollution concentrations. The performance of the proposed method presented an improvement in the performance of the air quality monitoring system. In future work, we plan to study different methods based on machine learning concepts to enhance the prediction of air pollutant systems. Moreover, we will investigate the general design of the Internet

of Things (IoT) applications to improve the performance of the air quality

The authors would like to acknowledgement Croatian Meteorological and

collected data.

monitoring system.

**Acknowledgements**

**63**

Hydrological Service for their support.

**8. Conclusion and future work**

*DOI: http://dx.doi.org/10.5772/intechopen.93678*

#### *Forecasting in Mathematics - Recent Advances, New Perspectives and Applications*

#### **Table 9.**

*Results before and after imputing missing data.*

Moreover, the obtained results of LRNN compared to the other previous methods are promising.

#### **7.4 Analysis of the results**

**Table 9** shows the obtained results before and after imputing missing data. The performance of the LRNN model outperforms other models in three datasets (i.e., NO2, PM10, and PM2.5) based on RMSE and *R*<sup>2</sup> values. While ANN outperforms other models in O3 dataset based on RMSE. The performance of ANN over O3 outperforms other methods. While the performance of MLR is the worst one.

From the obtained results, it can be seen that the performance of the LRNN model has an outstanding performance, where *R*<sup>2</sup> equals 0.90 in three datasets. However, these obtained results are not perfect since 16*:*96% of the data is removed for PM2.5, and 28*:*80% of the data is removed from NO2. Removing the missing data will neglect several records and the dataset may lose important information. **Figure 5** shows the actual and predicted values for NO2 dataset using all regression methods after imputing missing data.

For more analysis, comparing the obtained results that are reported in **Table 9**, we can notice that the performance of MLR over PM10 after imputing the missing data is reduced 19%, while the performance of DT, LRNN, and ANN is improved after imputing missing data for PM10 dataset. In general, the performance of the regression models is improved compared to the results reported in **Table 2**.

*A Layered Recurrent Neural Network for Imputing Air Pollutants Missing Data… DOI: http://dx.doi.org/10.5772/intechopen.93678*

For example, the *R*<sup>2</sup> value of ANN over O3 dataset before imputing missing data was 0.76, and after imputing missing data becomes 0.78, while the RMSE is improved 39%. So, we can conclude that imputing missing data will improve the air quality measurement systems without losing any record of collected data.

#### **8. Conclusion and future work**

Data collection from remote sensors suffers from missing data which reduces the overall performance of air quality monitoring systems. Monitoring air pollution is not an easy task, where several measurements are used to evaluate air quality. In this study, four measurements are used to predict air pollution concentrations (i.e., O3, NO2, PM2.5, and PM10). We imputed the missing data using the Layered recurrent neural network (L-RNN). The performance of four different machine learning models (i.e., LR, DTR, ANN, and L-RNN) was investigated to predict the average daily air pollution concentrations. The performance of the proposed method presented an improvement in the performance of the air quality monitoring system. In future work, we plan to study different methods based on machine learning concepts to enhance the prediction of air pollutant systems. Moreover, we will investigate the general design of the Internet of Things (IoT) applications to improve the performance of the air quality monitoring system.

#### **Acknowledgements**

Moreover, the obtained results of LRNN compared to the other previous methods

**Dataset Regression model After imputing Without imputing**

*Forecasting in Mathematics - Recent Advances, New Perspectives and Applications*

NO2 MLR 1.61 0.79 1.79 0.1

O3 MLR 1.78 0.82 22.11 0.05

PM10 MLR 3.84 0.46 3.68 0.83

PM2.5 MLR 1.76 0.62 2.61 0.81

**RMSE** *R***<sup>2</sup> RMSE R2**

DT 2.36 0.65 1.73 0.16 LRNN 0.22 0.93 0.26 0.85 ANN **0.05 0.96** 1.02 0.74

DT 7.32 0.54 20.44 0.19 LRNN **2.76 0.80** 10.06 0.61 ANN 3.25 0.78 8.39 0.76

DT 3.21 0.89 3.68 0.83 LRNN **0.02 0.98** 0.3 0.88 ANN 0.33 0.82 3.85 0.67

DT 3.14 0.85 2.57 0.82 LRNN **0.02 0.93** 2.39 0.9 ANN 0.16 0.97 2.92 0.85

**Table 9** shows the obtained results before and after imputing missing data.

From the obtained results, it can be seen that the performance of the LRNN model has an outstanding performance, where *R*<sup>2</sup> equals 0.90 in three datasets. However, these obtained results are not perfect since 16*:*96% of the data is removed for PM2.5, and 28*:*80% of the data is removed from NO2. Removing the missing data will neglect several records and the dataset may lose important information. **Figure 5** shows the actual and predicted values for NO2 dataset using all regression

For more analysis, comparing the obtained results that are reported in **Table 9**,

we can notice that the performance of MLR over PM10 after imputing the missing data is reduced 19%, while the performance of DT, LRNN, and ANN is improved after imputing missing data for PM10 dataset. In general, the performance of the regression models is improved compared to the results reported in **Table 2**.

The performance of the LRNN model outperforms other models in three datasets (i.e., NO2, PM10, and PM2.5) based on RMSE and *R*<sup>2</sup> values. While ANN outperforms other models in O3 dataset based on RMSE. The performance of ANN over O3 outperforms other methods. While the performance of MLR is the

are promising.

**Table 9.**

worst one.

**62**

**7.4 Analysis of the results**

*All significance values are in bold.*

*Results before and after imputing missing data.*

methods after imputing missing data.

The authors would like to acknowledgement Croatian Meteorological and Hydrological Service for their support.

**References**

550-558

459-473

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#### **Author details**

Hamza Turabieh<sup>1</sup> \*, Alaa Sheta2 , Malik Braik<sup>3</sup> and Elvira Kovač-Andrić<sup>4</sup>

1 Department of Information Technology, College of Computers and Information Technology, Taif University, Taif, Saudi Arabia

2 Computer Science Department, Southern Connecticut State University, New Haven, United States of America

3 Department of Computer Science, Al-Balqa Applied University, Salt, Jordan

4 Department of Chemistry, Josip Juraj Strossmayer University of Osijek, Osijek, Croatia

\*Address all correspondence to: turabieh@gmail.com

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

*A Layered Recurrent Neural Network for Imputing Air Pollutants Missing Data… DOI: http://dx.doi.org/10.5772/intechopen.93678*

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**Author details**

Hamza Turabieh<sup>1</sup>

Croatia

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\*, Alaa Sheta2

Technology, Taif University, Taif, Saudi Arabia

\*Address all correspondence to: turabieh@gmail.com

New Haven, United States of America

provided the original work is properly cited.

, Malik Braik<sup>3</sup> and Elvira Kovač-Andrić<sup>4</sup>

1 Department of Information Technology, College of Computers and Information

2 Computer Science Department, Southern Connecticut State University,

*Forecasting in Mathematics - Recent Advances, New Perspectives and Applications*

3 Department of Computer Science, Al-Balqa Applied University, Salt, Jordan

4 Department of Chemistry, Josip Juraj Strossmayer University of Osijek, Osijek,

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

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[49] Wang D, Wei S, Luo H, Yue C, Grunder O. A novel hybrid model for air quality index forecasting based on twophase decomposition technique and modified extreme learning machine. Science of the Total Environment. 2017; **580**:719-733

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[51] Akhtar A, Masood S, Gupta C, Masood A. Prediction and analysis of pollution levels in Delhi using multilayer perceptron. In: Satapathy SC, Bhateja V, Raju KS, Janakiramaiah B, editors. Data Engineering and Intelligent Computing. Singapore: Springer Singapore; 2018. pp. 563-572

[52] Yadav V, Nath S. Identification of relevant stochastic input variables for prediction of daily pm10 using artificial neural networks. In: Ray K, Sharma TK, Rawat S, Saini RK, Bandyopadhyay A, editors. Soft Computing: Theories and

Applications. Singapore: Springer Singapore; 2019. pp. 23-31

[53] Singh P. Linear Regression. Berkeley, CA: Apress; 2019. pp. 43-64 **Chapter 4**

**Abstract**

**1. Introduction**

**69**

Wind Power Forecasting

expected outcomes with the help of probabilistic measures.

generation technologies are solar, geothermal and tidal energy.

*Sumit Saroha, Sanjeev Kumar Aggarwal and Preeti Rana*

The wind power generation depends on wind speed and its derivatives like: wind speed and direction. With consideration of stochastic nature of wind power, this work addresses three main issues: first, it discusses the state of art of energy forecasting with emphasis on wind power forecasting. It provides an overview of different variables on which wind power generation depends and explains various key features regarding the design framework of forecasting models. Second, it performs an assessment, detailed comparison and evaluation of the forecasting performance of various types of models; and third, evaluates the uncertainty of

**Keywords:** forecasting, neural networks, probability, time series, wind power

Electricity sector especially in supply industry over the last various years across the world has underwent through numerous structural and systematic changes due to two main reasons: orientation of industry towards privatizations (reforms) and movement of electricity generation towards clean and pollution free renewable energy sources [1]. In this changing environment forecasting electricity becomes one of the most important exercises in managing the power systems. Forecasting plays a significant role in operation planning, scheduling and real time balancing of power system. Mainly, there are three forecasting issues in present day power systems namely electricity load, price and the renewable energy sources. Among the recently emerged renewable sources of energy (solar energy), the wind power industry has witnessed tremendous growth and has taken a leading role [2, 3]. Besides this, the electricity based on renewable energy sources perceived as an alternate source of energy and their penetration within the power system is rising at a very fast rate [4]. Among new sources of renewable energy, the wind energy has seen tremendous growth over recent years; in various countries, it is a true alternative to fossil fuels. Furthermore, wind power generation capacity varies constantly, stochastic, intermittent in nature and associated with generation of other ramp events. In spite of that, it is freely available & pollution free source of energy; so, it has gained an extensive interest and one of the most established renewable energy alternatives to the conventional energy resources. On approaching towards the end of 2016, 486.8 GW would be worldwide installed wind nameplate capacity due to growth rate of 12.5%. As per estimate, wind power towards the end of 2021 will approach to 817 GW with growth rate of 10.4%. These wind capacity installations are mainly utilized in electric power systems based on large grid and their interconnections [5, 6]. Now-a-days another fast growing eco-friendly electrical

[54] Wang S, Huang GH, He L. Development of a clusterwise-linearregression-based forecasting system for characterizing dnapl dissolution behaviors in porous media. Science of the Total Environment. 2012;**433**:141-150

[55] Swetapadma A, Yadav A. A novel decision tree regression-based fault distance estimation scheme for transmission lines. IEEE Transactions on Power Delivery. 2017;**32**(1):234-245

[56] Qin H, Gong R, Liu X, Bai X, Song J, Sebe N. Binary neural networks: A survey. Pattern Recognition. 2020;**105**:107281

## **Chapter 4** Wind Power Forecasting

*Sumit Saroha, Sanjeev Kumar Aggarwal and Preeti Rana*

#### **Abstract**

Journal of Environmental Management.

*Forecasting in Mathematics - Recent Advances, New Perspectives and Applications*

Applications. Singapore: Springer Singapore; 2019. pp. 23-31

[53] Singh P. Linear Regression. Berkeley, CA: Apress; 2019. pp. 43-64

[54] Wang S, Huang GH, He L. Development of a clusterwise-linearregression-based forecasting system for characterizing dnapl dissolution

behaviors in porous media. Science of the Total Environment. 2012;**433**:141-150

[55] Swetapadma A, Yadav A. A novel decision tree regression-based fault distance estimation scheme for

transmission lines. IEEE Transactions on Power Delivery. 2017;**32**(1):234-245

[56] Qin H, Gong R, Liu X, Bai X, Song J, Sebe N. Binary neural networks: A survey. Pattern Recognition. 2020;**105**:107281

[46] Xu Y, Du P, Wang J. Research and application of a hybrid model based on dynamic fuzzy synthetic evaluation for establishing air quality forecasting and early warning system: A case study in China. Environmental Pollution. 2017;

[47] Luo H, Wang D, Yue C, Liu Y, Guo H. Research and application of a novel hybrid decomposition-ensemble

learning paradigm with error correction for daily pm10 forecasting. Atmospheric

[48] Aznarte JL. Probabilistic forecasting for extreme no2 pollution episodes. Environmental Pollution. 2017;**229**:

[49] Wang D, Wei S, Luo H, Yue C, Grunder O. A novel hybrid model for air quality index forecasting based on twophase decomposition technique and modified extreme learning machine. Science of the Total Environment. 2017;

[50] Kumar A, Goyal P. Forecasting of air quality in Delhi using principal component regression technique. Atmospheric Pollution Research. 2011;**2**

[51] Akhtar A, Masood S, Gupta C, Masood A. Prediction and analysis of pollution levels in Delhi using multilayer perceptron. In: Satapathy SC, Bhateja V, Raju KS, Janakiramaiah B, editors. Data Engineering and Intelligent Computing. Singapore: Springer Singapore; 2018.

[52] Yadav V, Nath S. Identification of relevant stochastic input variables for prediction of daily pm10 using artificial neural networks. In: Ray K, Sharma TK, Rawat S, Saini RK, Bandyopadhyay A, editors. Soft Computing: Theories and

Research. 2018;**201**:34-45

2017;**188**:144-152

**223**:435-448

321-328

**580**:719-733

(4):436-444

pp. 563-572

**68**

The wind power generation depends on wind speed and its derivatives like: wind speed and direction. With consideration of stochastic nature of wind power, this work addresses three main issues: first, it discusses the state of art of energy forecasting with emphasis on wind power forecasting. It provides an overview of different variables on which wind power generation depends and explains various key features regarding the design framework of forecasting models. Second, it performs an assessment, detailed comparison and evaluation of the forecasting performance of various types of models; and third, evaluates the uncertainty of expected outcomes with the help of probabilistic measures.

**Keywords:** forecasting, neural networks, probability, time series, wind power

#### **1. Introduction**

Electricity sector especially in supply industry over the last various years across the world has underwent through numerous structural and systematic changes due to two main reasons: orientation of industry towards privatizations (reforms) and movement of electricity generation towards clean and pollution free renewable energy sources [1]. In this changing environment forecasting electricity becomes one of the most important exercises in managing the power systems. Forecasting plays a significant role in operation planning, scheduling and real time balancing of power system. Mainly, there are three forecasting issues in present day power systems namely electricity load, price and the renewable energy sources. Among the recently emerged renewable sources of energy (solar energy), the wind power industry has witnessed tremendous growth and has taken a leading role [2, 3].

Besides this, the electricity based on renewable energy sources perceived as an alternate source of energy and their penetration within the power system is rising at a very fast rate [4]. Among new sources of renewable energy, the wind energy has seen tremendous growth over recent years; in various countries, it is a true alternative to fossil fuels. Furthermore, wind power generation capacity varies constantly, stochastic, intermittent in nature and associated with generation of other ramp events. In spite of that, it is freely available & pollution free source of energy; so, it has gained an extensive interest and one of the most established renewable energy alternatives to the conventional energy resources. On approaching towards the end of 2016, 486.8 GW would be worldwide installed wind nameplate capacity due to growth rate of 12.5%. As per estimate, wind power towards the end of 2021 will approach to 817 GW with growth rate of 10.4%. These wind capacity installations are mainly utilized in electric power systems based on large grid and their interconnections [5, 6]. Now-a-days another fast growing eco-friendly electrical generation technologies are solar, geothermal and tidal energy.

The uncertainty associated with wind power originates from uncertainties in its derivatives such as: wind speed & direction forecasts. In coordination with fast deployment of wind farms establishes a demand for efficient forecasting methods related to wind power production. The high is forecast reliability, low will be reserve maintenance cost of the system, which will result technical and commercial implications for proper management and working of power systems. Wind power forecasting (WPF) depicts how much wind power is to be expected at particular instant of time in the days to come. WPF is one of the most critical aspects in wind power integration and operation [6–8]. As per time horizons, the WPF has been done on the basis of long, medium and short term.

**2. Methods of energy forecasting**

*DOI: http://dx.doi.org/10.5772/intechopen.94550*

**2.1 Deterministic or point forecasting**

**Single Step Ahead Forecasting.**

which these time series are produced.

advance as given below in Eq. (3, 4).

*Single Step*

*Second Step*

*KthStep*

**71**

**Multi Step Ahead Forecasting.**

using probability theory.

*Wind Power Forecasting*

With

The predicted values can be provided to end-users either in a deterministic or in probabilistic format, with the former, a specific value for energy production at a particular time step (15-minutes or one hour) is forecasted; whereas, in later, range of possible output is forecasted on the behalf of deterministic forecasted values

It is the estimation of any quantity today for the next coming day with utmost possible precision and reliability. We have at our disposal the past values of this quantity, the data of one or several time series along with other several factors on

By the Eq. (1), e, is the prediction error or noise present between present forecasting value and n previous observations. WP is the wind power, T is the target, for multiple step the target matrix is increased with respect to each step in

> *WP*11*WP*<sup>12</sup> ��� *WP*<sup>16</sup> *WP*21*WP*<sup>22</sup> ��� *WP*<sup>26</sup>

*WPN*1*WPN*<sup>2</sup> ��� *WPN*<sup>6</sup>

*WP*11*WP*<sup>12</sup> ��� *WP*<sup>16</sup> *WP*21*WP*<sup>22</sup> ��� *WP*<sup>26</sup>

*WPN*1*WPN*<sup>2</sup> ��� *WPN*<sup>6</sup>

The multiple steps ahead or multiple lead hour prediction is forecasting a pattern of values for given time series. It is an approach that works step-by-step by using current prediction for deterministic next stage prediction. In case of multi-step ahead prediction various anomalies like error accumulation and complexity of data prevails when prediction period is long. It all occurs due to propagation of bias and variances form previous prediction of future prediction. Because of this large forecasting horizon & error present in forecasting this method is suffered from the low performance & higher inaccuracy that is because of use of approximated values rather than actual values. The main reason for this higher inaccuracy is that the error is multiplied in every step-ahead prediction. So, the selection of input parameter function to fit the

time series can be a challenging task for the power system researchers.

*WP*11*WP*<sup>12</sup> ��� *WP*<sup>16</sup> *WP*21*WP*<sup>22</sup> ��� *WP*<sup>26</sup>

*WPN*1*WPN*<sup>2</sup> ��� *WPN*<sup>6</sup>

*WPt*þ<sup>1</sup> ¼ *f WPt* ð , … … ,*WPt*�*d*þ<sup>1</sup>Þ þ *e* (1)

*t*∈*f d*f g , … … , *N* � 1 (2)

*TK TK*þ<sup>1</sup>

*TN*þ*<sup>K</sup>*

*T*1 *T*2

(3)

(4)

(5)

*TN*

*TN*þ<sup>1</sup>

*T*2 *T*3

The availability of wind power is largely influenced by the prevailing weather conditions, seasonal variations and time spam variation and therefore, it is characterized by strong fluctuations, uncertainty and intermittency. These characteristics of wind power create a great attention towards it. Consequently, power generation from wind cannot be matched easily to the electricity demand like power generated with conventional plants. The penetration (share of wind power to meet demand) level of wind power introduces new challenges for the power system, some of them include:

**Integration with Grid:** The management of intermittence of wind generation is the key issue related to its integration with grid. The transmission utility is only responsible for the balancing of demand and supply at grid level. Therefore, it is necessary to schedule the supply in advance in order to meet the load profile. The load is corresponding to the total demand of electricity consumption over a definite area. The load forecast is usually given by the load forecasting models. The Mean Absolute Percentage Error (MAPE) of load is in the order of 0.87–1.34% [9] for the day ahead or week ahead predictions. Still continuous effort has been made by various researchers and practitioners for improving the performance of load forecasting models and techniques. i.e. it is reached in advance stage of research.

**Integration with Electricity Markets:** Generally, the electricity market is build by two mechanisms. The first one is spot energy market or so called Day Ahead market, where the bulk energy necessary to cover the load profile for the next coming day is traded on the generation cost. An auction process followed by bidding permits the settlement of electricity price and generation for the various bidding hours. The second mechanism is ancillary service market or so called intraday market, where differences between planned production and actual load are traded (due to the power plant failure or due to intermittence of wind power generation). The ancillary service market is very important for a stable operation of the power grid and span across various time frames. Therefore, it is additionally important for consumers as well as suppliers to know the future electricity price, so that they can make strategies. Like load forecasting the electricity price is in its advance stage of research and error rate (MAPE) reported is 3.96–4.92% [10].

Therefore, the accurate forecasts of wind power generation is an essential factor for a successful integration of large amounts of wind power into the electricity supply system, aiming at precise information on timing and magnitude of power generation from these variable sources.

Among requirements of wind power forecasting over three different forecasting horizons, there are different framework for the forecasting which includes single step ahead, multiple lead hours ahead and probabilistic forecasting. Typically multiple step and probabilistic forecasting is more complicated because in multiple, the error is multiples at every lead hour prediction; whereas, in probabilistic several statistical factors contribute additional complexity and additional complicacy. Moreover, it also affects the profits of a utility directly.

#### **2. Methods of energy forecasting**

#### **2.1 Deterministic or point forecasting**

The predicted values can be provided to end-users either in a deterministic or in probabilistic format, with the former, a specific value for energy production at a particular time step (15-minutes or one hour) is forecasted; whereas, in later, range of possible output is forecasted on the behalf of deterministic forecasted values using probability theory.

#### **Single Step Ahead Forecasting.**

It is the estimation of any quantity today for the next coming day with utmost possible precision and reliability. We have at our disposal the past values of this quantity, the data of one or several time series along with other several factors on which these time series are produced.

$$\text{WPP}\_{t+1} = f(\text{WP}\_t, \dots, \text{WP}\_{t-d+1}) + e \tag{1}$$

With

The uncertainty associated with wind power originates from uncertainties in its derivatives such as: wind speed & direction forecasts. In coordination with fast deployment of wind farms establishes a demand for efficient forecasting methods related to wind power production. The high is forecast reliability, low will be reserve maintenance cost of the system, which will result technical and commercial implications for proper management and working of power systems. Wind power forecasting (WPF) depicts how much wind power is to be expected at particular instant of time in the days to come. WPF is one of the most critical aspects in wind power integration and operation [6–8]. As per time horizons, the WPF has been

*Forecasting in Mathematics - Recent Advances, New Perspectives and Applications*

The availability of wind power is largely influenced by the prevailing weather conditions, seasonal variations and time spam variation and therefore, it is characterized by strong fluctuations, uncertainty and intermittency. These characteristics of wind power create a great attention towards it. Consequently, power generation from wind cannot be matched easily to the electricity demand like power generated with conventional plants. The penetration (share of wind power to meet demand) level of wind power introduces new challenges for the power system, some of them

**Integration with Grid:** The management of intermittence of wind generation is the key issue related to its integration with grid. The transmission utility is only responsible for the balancing of demand and supply at grid level. Therefore, it is necessary to schedule the supply in advance in order to meet the load profile. The load is corresponding to the total demand of electricity consumption over a definite area. The load forecast is usually given by the load forecasting models. The Mean Absolute Percentage Error (MAPE) of load is in the order of 0.87–1.34% [9] for the day ahead or week ahead predictions. Still continuous effort has been made by various researchers and practitioners for improving the performance of load forecasting models and techniques. i.e. it is reached in advance stage of

**Integration with Electricity Markets:** Generally, the electricity market is build by two mechanisms. The first one is spot energy market or so called Day Ahead market, where the bulk energy necessary to cover the load profile for the next coming day is traded on the generation cost. An auction process followed by bidding permits the settlement of electricity price and generation for the various bidding hours. The second mechanism is ancillary service market or so called intraday market, where differences between planned production and actual load are traded (due to the power plant failure or due to intermittence of wind power generation). The ancillary service market is very important for a stable operation of the power grid and span across various time frames. Therefore, it is additionally important for consumers as well as suppliers to know the future electricity price, so that they can make strategies. Like load forecasting the electricity price is in its advance stage of research and error rate (MAPE) reported is 3.96–4.92% [10].

Therefore, the accurate forecasts of wind power generation is an essential factor

Among requirements of wind power forecasting over three different forecasting horizons, there are different framework for the forecasting which includes single step ahead, multiple lead hours ahead and probabilistic forecasting. Typically multiple step and probabilistic forecasting is more complicated because in multiple, the error is multiples at every lead hour prediction; whereas, in probabilistic several statistical factors contribute additional complexity and additional complicacy.

for a successful integration of large amounts of wind power into the electricity supply system, aiming at precise information on timing and magnitude of power

generation from these variable sources.

Moreover, it also affects the profits of a utility directly.

done on the basis of long, medium and short term.

include:

research.

**70**

$$t \in f\{d, \dots, \dots, N-1\} \tag{2}$$

By the Eq. (1), e, is the prediction error or noise present between present forecasting value and n previous observations. WP is the wind power, T is the target, for multiple step the target matrix is increased with respect to each step in advance as given below in Eq. (3, 4).

$$\begin{aligned} \text{Single } Step \begin{bmatrix} \text{WP1}\,\text{WP1}\,\text{2} & -- & \text{WP1}\_{6} \\ \text{WP2}\,\text{1}\,\text{WP2}\,\text{2} & -- & \text{WP2}\_{6} \\\\ \text{WP}\_{N1}\,\text{WP}\_{N2} & -- & \text{WP}\_{N6} \end{bmatrix} \begin{bmatrix} T\_{1} \\ T\_{2} \\ T\_{N} \end{bmatrix} \\ \text{Second } \text{Step} \begin{bmatrix} \text{WP1}\,\text{1}\,\text{WP}\_{12} & -- & - & \text{WP1}\_{6} \\ \text{WP}\_{21}\,\text{WP}\_{22} & -- & - & \text{WP2} \\\\ \text{WP}\_{N1}\,\text{WP}\_{N2} & -- & - & \text{WP}\_{N6} \end{bmatrix} \begin{bmatrix} T\_{2} \\ T\_{3} \\ T\_{N+1} \end{bmatrix} \end{aligned} \tag{4}$$

#### **Multi Step Ahead Forecasting.**

The multiple steps ahead or multiple lead hour prediction is forecasting a pattern of values for given time series. It is an approach that works step-by-step by using current prediction for deterministic next stage prediction. In case of multi-step ahead prediction various anomalies like error accumulation and complexity of data prevails when prediction period is long. It all occurs due to propagation of bias and variances form previous prediction of future prediction. Because of this large forecasting horizon & error present in forecasting this method is suffered from the low performance & higher inaccuracy that is because of use of approximated values rather than actual values. The main reason for this higher inaccuracy is that the error is multiplied in every step-ahead prediction. So, the selection of input parameter function to fit the time series can be a challenging task for the power system researchers.

$$K^{th}Step\begin{bmatrix}\textit{WPP}\_{11}\textit{WPP}\_{12}---\textit{WPP}\_{16} \\ \textit{WPP}\_{21}\textit{WPP}\_{22}---\textit{WPP}\_{26} \\\\ \textit{WPP}\_{N1}\textit{WPP}\_{N2}---\textit{WPP}\_{N6}\end{bmatrix}\begin{bmatrix}T\_K\\T\_{K+1}\\\\T\_{N+K}\end{bmatrix}\tag{5}$$

#### **2.2 Probabilistic or interval forecasting**

The probabilistic forecast systems are designed to estimate the uncertainty of a forecast and used to produce the application of probabilistic forecasting. The verification is an essential part of probabilistic forecast systems. The correct and accurate use of probability forecasts means that, given a large sample, on average and event will occur at the same frequency as the forecast probability [11].

#### **3. State of art for wind power forecast**

As far as literature is concerned, number of forecasting methods have been designed and analyzed over last few decades. Based on information in research papers, author has examined various developments in the field of wind power generation & its derivatives prediction such as speed or direction. The major emphasis is led on facilitation of a number of issues concerned with techniques involved in WPF, focuses on complexity reduction in forecasting issues with higher accuracy in forecasting for different time span. This research mainly focuses on motivating power system researchers to design highly efficient and accurate models whether online/offline considering various issues related to wind power which in twin result in reliable operation of power system models by utilizing energy resources economically. On carrying out comparative study and analysis of accuracy in forecasting models, hybrid models outperformed all other models.

The generation of wind power is highly influenced by nature and seasons. So, it has been a tedious task to design a sound prediction model by taking in account above two factors. But, AI and machine learning have come with an advantage for developing new models due to their higher efficiency and accuracy. After a deep insight of various research papers authors have observed that the NN is the most prevailing approach for wind power and its derivatives estimation. It has also been observed that, hybrid models have been found to be more accurate model and for getting more accuracy, the training data should be updated regularly with small time span. Although for real time operation of power system, researchers have to move towards online models. There are three main steps involved in WPF (i) Input Selection, (ii) Data Pre-processing, & (iii) Forecasting models (tool) used.

*3.1.1 Physical or numerical weather prediction (NWP) models*

wind speed of surrounding area of wind mill.

*3.1.2 Statistical models*

**73**

1. Atmospheric Characteristics

*Wind Power Forecasting*

2. Topographic Characteristics

3. Wind Power Characteristics

4. Behavior Indices

5. Other Stochastic Uncertainty

**Table 1.**

6. Geographical Conditions

*Factors affecting wind power generation.*

These are very common model in which wind is a function of exogenous vari-

Here, *ρ* denotes the air density; v denotes the wind velocity through an intercepting area *A* of wind turbine. Actually, this equation follows the different physical variables corresponding to wind turbine. The purpose of NWP models is to predict the

In statistical models, wind remains a function that works using past captured values. These models are trained by providing data patterns that are measured statistically. They are based on historical data patterns generated by wind power and hence, they are not based on computation of any form of mathematical expression. These models outperform other short term forecasting horizon over prediction accuracy and these models are easy to implement & validate. They employed the statistics like: Cross Correlation (CC), Auto Correlation Function (ACF) and Partial Auto Correlation Function (PACF) for input selection on the basis of standard deviation, variance, mean and slope of input curve etc. The **Figure 1** shows ACF and PACF of hourly Wind Power time series based on these two parameters input

.

*f*(wind Speed); (d-m, t), m = 1,2,3,4,7,8, 168, 365

*f*(wind power; (d-m,tn), m = 1,2,3,4,7,8, 168, 365 and n = 0,1,2,3,4

*f*(wind direction; (d-m,t-n), m = 1,2,3, 168, 365 and n = 0,1,2,3,4

ables and forecasting tool input is the output of NWP models. These physical models forecasting process depends on entire input corresponding to wind power derivatives and are deterministic one. Their implementation process is very complex to perform, take high computation time to carry out forecasting process and depends on physical variables concerned with wind farm location. The equation which is used to convert wind speed into power is as follows as: *Wp = 0.5.ρ.A.v<sup>3</sup>*

**Class Input variable Input data**

(10) Turbine position, (11) Turbine size, (12) Hub height, (13) Tower height, (14) Elevation, (15) Degree in

(16) Wind speed, (17) Wind direction, (18) Radiation transmission, (19) Sine & Cosine of wind direction, (20)

(22) Hydrological cycle, (23) cloud-radiation interaction, (24) spatial behavior, (25) Temporal behavior, (26)

(27) Ocean-land interactions, (28) Regime switching, (29) Exchanges of momentum, (30) Load distribution among parallel turbines, (3) 1Thunders, (32) Storms,

(35) Orography, (36) Surface roughness, (37) Obstacles, (38) Geographical height, (39) Mean sea level pressure, (40) Air temperature, (41) Soil wetness, (42) Atmosphere covering, (43) Snow covering, (44) Moisture with land surface, (45) Complex terrain, (46)

(8) Radiations Effect, (9) Density

Air density, (21) Local wind profile

(33) Risk index, (34) Guest wind speed

Latitude

*DOI: http://dx.doi.org/10.5772/intechopen.94550*

Spatial resolution

Terrain roughness

(1) Temperature (2) Pressure, (3) Humidity (4) Rainfall, (5) Cloud formation, (6) Cloud cover, (7) Turbulance,

#### **3.1 Input parameters & their selection methods**

The higher uncertainty in wind nature is result of uncertainties in its derivatives that affect systems of reliability. If forecast reliability is higher than operational cost of wind power system is lowered, in turn benefitting wind farm owners as they will have more substantial saving as well as have better efficiency of the system [12]. Apart from all this, wind power prediction is still a tedious task because wind flow is an unpredictable natural phenomenon and wind speed time series possesses various characteristics like: high volatility, high complexity, non linearity and nonstationary due to prevent physical conditions of place [13, 14]. After an extensive study of various research papers more than 46 exogenous variables have been observed as given in **Table 1**.

The input variables selection is main task because the accurate prediction by a forecasting model is highly influenced by proper input variables and their past results in the field of wind speed & power prediction and estimation. Furthermore, the selection of input variables for a prediction model mainly depends on exogenous and without exogenous variables. The various input selection techniques are as discussed.


#### **Table 1.**

**2.2 Probabilistic or interval forecasting**

**3. State of art for wind power forecast**

The probabilistic forecast systems are designed to estimate the uncertainty of a forecast and used to produce the application of probabilistic forecasting. The verification is an essential part of probabilistic forecast systems. The correct and accurate use of probability forecasts means that, given a large sample, on average and

As far as literature is concerned, number of forecasting methods have been designed and analyzed over last few decades. Based on information in research papers, author has examined various developments in the field of wind power generation & its derivatives prediction such as speed or direction. The major emphasis is led on facilitation of a number of issues concerned with techniques involved in WPF, focuses on complexity reduction in forecasting issues with higher accuracy in forecasting for different time span. This research mainly focuses on motivating power system researchers to design highly efficient and accurate models whether online/offline considering various issues related to wind power which in twin result in reliable operation of power system models by utilizing energy resources economically. On carrying out comparative study and analysis of accuracy in forecasting models, hybrid models outperformed all other models.

The generation of wind power is highly influenced by nature and seasons. So, it has been a tedious task to design a sound prediction model by taking in account above two factors. But, AI and machine learning have come with an advantage for developing new models due to their higher efficiency and accuracy. After a deep insight of various research papers authors have observed that the NN is the most prevailing approach for wind power and its derivatives estimation. It has also been observed that, hybrid models have been found to be more accurate model and for getting more accuracy, the training data should be updated regularly with small time span. Although for real time operation of power system, researchers have to move towards online models. There are three main steps involved in WPF (i) Input

The higher uncertainty in wind nature is result of uncertainties in its derivatives that affect systems of reliability. If forecast reliability is higher than operational cost of wind power system is lowered, in turn benefitting wind farm owners as they will have more substantial saving as well as have better efficiency of the system [12]. Apart from all this, wind power prediction is still a tedious task because wind flow is an unpredictable natural phenomenon and wind speed time series possesses various

The input variables selection is main task because the accurate prediction by a forecasting model is highly influenced by proper input variables and their past results in the field of wind speed & power prediction and estimation. Furthermore, the selection of input variables for a prediction model mainly depends on exogenous and without exogenous variables. The various input selection techniques are as

Selection, (ii) Data Pre-processing, & (iii) Forecasting models (tool) used.

characteristics like: high volatility, high complexity, non linearity and nonstationary due to prevent physical conditions of place [13, 14]. After an extensive study of various research papers more than 46 exogenous variables have been

**3.1 Input parameters & their selection methods**

observed as given in **Table 1**.

discussed.

**72**

event will occur at the same frequency as the forecast probability [11].

*Forecasting in Mathematics - Recent Advances, New Perspectives and Applications*

*Factors affecting wind power generation.*

#### *3.1.1 Physical or numerical weather prediction (NWP) models*

These are very common model in which wind is a function of exogenous variables and forecasting tool input is the output of NWP models. These physical models forecasting process depends on entire input corresponding to wind power derivatives and are deterministic one. Their implementation process is very complex to perform, take high computation time to carry out forecasting process and depends on physical variables concerned with wind farm location. The equation which is used to convert wind speed into power is as follows as: *Wp = 0.5.ρ.A.v<sup>3</sup>* . Here, *ρ* denotes the air density; v denotes the wind velocity through an intercepting area *A* of wind turbine. Actually, this equation follows the different physical variables corresponding to wind turbine. The purpose of NWP models is to predict the wind speed of surrounding area of wind mill.

#### *3.1.2 Statistical models*

In statistical models, wind remains a function that works using past captured values. These models are trained by providing data patterns that are measured statistically. They are based on historical data patterns generated by wind power and hence, they are not based on computation of any form of mathematical expression. These models outperform other short term forecasting horizon over prediction accuracy and these models are easy to implement & validate. They employed the statistics like: Cross Correlation (CC), Auto Correlation Function (ACF) and Partial Auto Correlation Function (PACF) for input selection on the basis of standard deviation, variance, mean and slope of input curve etc. The **Figure 1** shows ACF and PACF of hourly Wind Power time series based on these two parameters input

pre-processing means data cleaning data transformation and data reduction input data and converting it into useful information as per dimensions. Data must be classified based on seasonal and weather variable variation. Kalman filter is an appropriate solution to various problems such as: complexity in data, over-fitting and outliers of input data generated during learning process [15, 16]. As Unscented Kalman Filter (UKF) achieves higher efficiency in handling random fluctuations, so it is an economical and adequate choice for non-linear estimation of wind speed [17]. In presented work, in order to investigate the performance of different forecasting models, real wind generation data of Ontario Electricity Market (OEM) from 2011 to 2014 [18] has been considered. For obtaining more accuracy and overtraining avoidance in learning process to achieve greater accuracy, large set of data values have not been considered, as generation of wind power is dependent function on numerous parameters such as: changing season, temperature and weather conditions. As time moves wind capacity (defined as actual energy produced in comparison to energy actually dissipated by turbines under favorable conditions) can fluctuate. The main concern of Wavelet Transform (WT) is to collect the meaningful information with removal of noise & irregularities from the original signal. From the available literature on forecasting and experimental analysis, it has been observed that Daubechies wavelet at different levels performs an appropriate smoothness of the signal with respect to wave-length, which results in an appropri-

7. *A1* Approximate Series 8. *D1, D2, D3, D4, D5, D6* Detailed Series

**S. No. Time lag series No. of time lag** 1. *WP (t-1)* 1 2. *WP (t-1), WP (t-2)* 2 3. *WP (t-1), WP (t-2), WP (t-3)* 3 4. *WP (t-1), WP (t-2), WP (t-3), WP (t-4)* 4 5. *WP (t-1), WP (t-2), WP (t-3), WP (t-4), WP (t-5)* 5 6. *WP (t-1), WP (t-2), WP (t-3), WP (t-4), WP (t-5), WP (t-6)* 6

ate behavior of input data pattern for wind power prediction tool.

frequency signals [25]. The NN models train themselves better with the pre-processed data, as a result of this better prediction performance.

For the past two decades, models based on machine learning have captured

attention & become more sophisticated and reliable contenders in spite of

**3.3 Wind power forecasting tools**

**75**

**Table 2.** *Inputs used.*

*Wind Power Forecasting*

*DOI: http://dx.doi.org/10.5772/intechopen.94550*

The WT implementation is done to decompose wind power series broadly into constitutive series set. This set of constitutive series help in reduction of input data and outperforms original wind series in behavior leading to prediction accuracy improvement. The WT divides wind series signal into two distinguishing signals having low and high frequency, then the decomposed signals are provided to the separate NN model for training. There are four filters (decomposition low pass & high pass filter, reconstruction low & high pass filter) used in Discrete Wavelet Transform (DWT) for scaling the input data pattern into approximate (A) and detailed (D) signals as given in **Table 2** [19–24]. Empirical Model Decomposition (EMD) has also been used to decompose the wind power series into high and low

**Figure 1.** *ACF & PACF for hourly wind power series.*

time lag parameterization of both time series and Artificial Intelligence (AI) take place. The higher is the value of ACF more is correlation between two consecutive series. However, the selection of input variables is one of the most important part of NN based forecasting model on with the accuracy of the model depends and that also determines the input architecture of the model. During the training of NN model, there may be problem of overtraining or over fitting that leads to poor accuracy of model. Therefore, it is necessary to know the relation that exists between present time wind power series along with their past time lag series. The input time lag is given below in **Table 2**. The wind forecast problem aims to find an estimate *WP (t + k)* of the wind vector *WP(t + n)* based on the previous n measurements *WP(t), WP(t-1),*. *.., WP(t-n).*

#### *3.1.3 Hybrid (physical + statistical) models*

It is the combination of NWP and statistical tools for input data selection. In this, on the bases of statistical analysis, the NWP variables are pre-processed to time lag for the prediction of next step.

#### **3.2 Input data collection & pre-processing**

The input data and wind data pattern is accumulated in raw form and does not possesses highly efficient forecasting capability with accurate precision. Raw data is unpredictable, irregular, seasonal and more complex due to changing weather. While prediction computation, over-fitting or over-training of NN is the main issue in time series variation leading to foot fall in accuracy of forecasted values. Data


#### **Table 2.** *Inputs used.*

time lag parameterization of both time series and Artificial Intelligence (AI) take place. The higher is the value of ACF more is correlation between two consecutive series. However, the selection of input variables is one of the most important part of NN based forecasting model on with the accuracy of the model depends and that also determines the input architecture of the model. During the training of NN model, there may be problem of overtraining or over fitting that leads to poor accuracy of model. Therefore, it is necessary to know the relation that exists between present time wind power series along with their past time lag series. The input time lag is given below in **Table 2**. The wind forecast problem aims to find an estimate *WP (t + k)* of the wind vector *WP(t + n)* based on the previous n measurements *WP(t),*

*Forecasting in Mathematics - Recent Advances, New Perspectives and Applications*

It is the combination of NWP and statistical tools for input data selection. In this, on the bases of statistical analysis, the NWP variables are pre-processed to time lag

The input data and wind data pattern is accumulated in raw form and does not possesses highly efficient forecasting capability with accurate precision. Raw data is unpredictable, irregular, seasonal and more complex due to changing weather. While prediction computation, over-fitting or over-training of NN is the main issue in time series variation leading to foot fall in accuracy of forecasted values. Data

*WP(t-1),*. *.., WP(t-n).*

**Figure 1.**

**74**

*3.1.3 Hybrid (physical + statistical) models*

**3.2 Input data collection & pre-processing**

for the prediction of next step.

*ACF & PACF for hourly wind power series.*

pre-processing means data cleaning data transformation and data reduction input data and converting it into useful information as per dimensions. Data must be classified based on seasonal and weather variable variation. Kalman filter is an appropriate solution to various problems such as: complexity in data, over-fitting and outliers of input data generated during learning process [15, 16]. As Unscented Kalman Filter (UKF) achieves higher efficiency in handling random fluctuations, so it is an economical and adequate choice for non-linear estimation of wind speed [17].

In presented work, in order to investigate the performance of different forecasting models, real wind generation data of Ontario Electricity Market (OEM) from 2011 to 2014 [18] has been considered. For obtaining more accuracy and overtraining avoidance in learning process to achieve greater accuracy, large set of data values have not been considered, as generation of wind power is dependent function on numerous parameters such as: changing season, temperature and weather conditions. As time moves wind capacity (defined as actual energy produced in comparison to energy actually dissipated by turbines under favorable conditions) can fluctuate. The main concern of Wavelet Transform (WT) is to collect the meaningful information with removal of noise & irregularities from the original signal. From the available literature on forecasting and experimental analysis, it has been observed that Daubechies wavelet at different levels performs an appropriate smoothness of the signal with respect to wave-length, which results in an appropriate behavior of input data pattern for wind power prediction tool.

The WT implementation is done to decompose wind power series broadly into constitutive series set. This set of constitutive series help in reduction of input data and outperforms original wind series in behavior leading to prediction accuracy improvement. The WT divides wind series signal into two distinguishing signals having low and high frequency, then the decomposed signals are provided to the separate NN model for training. There are four filters (decomposition low pass & high pass filter, reconstruction low & high pass filter) used in Discrete Wavelet Transform (DWT) for scaling the input data pattern into approximate (A) and detailed (D) signals as given in **Table 2** [19–24]. Empirical Model Decomposition (EMD) has also been used to decompose the wind power series into high and low frequency signals [25]. The NN models train themselves better with the pre-processed data, as a result of this better prediction performance.

#### **3.3 Wind power forecasting tools**

For the past two decades, models based on machine learning have captured attention & become more sophisticated and reliable contenders in spite of

traditional statistical models in forecasting. These are non parametric & non-linear models also known as data driven or black box models having usage of historical data patterns to learn the stochastic dependency between past and future. These NN's models always leave behind other traditional statistical models such as: linear regression and Box-Jenkins approaches. The NNs can be successfully used for modeling and forecasting non-linear time series [26].

*3.3.2 Artificial intelligence (AI) models*

*DOI: http://dx.doi.org/10.5772/intechopen.94550*

*Wind Power Forecasting*

Network with Time Delay (LNNTD).

The FFNN architecture, which is also called as Multi Layer Perceptron (MLP), along with back propagation (BP) as the learning algorithm is the most popular choice among researchers. The neural network (NN) and machine learning algorithms structures used by most of the researchers after 2000 in the leading journals are: Feed Forward Neural Network (FFNN), Recurrent Neural Network (RNN), Radial Basis Function Neural Network (RBFNN), Support Vector Machine (SVM), Support Vector Regression (SVR), Adaptive Neuro Fuzzy Inference System (ANFIS), Extreme Learning Machine (ELM), Adaptive Wavelet Neural Network (AWNN), General Regression Neural Network (GRNN), and Linear Neural

In this, wind forecasting has been done by the three different models: (i) Benchmark, (ii) NN and (iii) WT based model. In the first category, only Naïve Predictor has been considered. This is the standard benchmark for wind forecasting applications, in which the previous values of input wind power series have been used for the next lead hour as forecasted values. In the second category, different ANN based models have been taken into consideration with different structure of network and learning algorithms. The NN along with gradient-based optimization techniques is most popular choice among all researchers and associated with the short comings of local minima and sensitivity to initial value persists as a result of poor accuracy. So, as to resolve above said problems, global evolutionary algorithms (EA) such as Genetic Algorithms (GA) [1, 23], Particle Swarm Optimization (PSO) [19, 33, 34] have been utilized. The main advantages of EA lie in its global convergence, inherent parallel search nature, and great robustness. These algorithms gen-

For proper input selection, there is need of complete experimental analysis on the basis of error rate. The input structures of WT based models are different from

combination of Wind Power series and WT based approximated and detailed wind power series. Therefore, the number of input nodes is more as compared to non WT models. The structure of WT based FFNN for wind power prediction has been

**Step 1:** From the raw data of wind power, a time series as input is selected on the

**Step 2:** Supply the created input signal to WT for performing multilevel decom-

**Step 5:** The approximated and detailed wind power series along with six original

**Step 3:** Now extract the multi level approximation A6 and 1, 2, 3,4,5,6 level

**Step 6:** A three layer FFNN, as shown in **Figure 3**, has been selected having thirteen input nodes equal to the number of input variables, twelve hidden neurons with tangential sigmoid transfer function, and one output neuron with pure linear activation function, with each series. The network is trained using Levenberg– Marquardt (LM) training algorithms with architecture [12–11–1]. The momentum constant and learning rate have been kept equal to 0.06 and 0.001, respectively. **Step 7:** For the prediction, one year wind data has been trained and tested for next one month, similar process is continuously repeated up-to next 24 months with one month moving window. The maximum epochs were set equal to 10,000 with

**Step 8:** The output values found by the network has been assessed on the

that of the non WT based models. In the WT based models, the input is the

position on wind power signal by utilizing Daubechies (db10) wavelet.

detailed coefficients D1 to D6 of input wind power series signal.

erate a high quality solution within a short computation time.

shown in **Figure 2** & detailed prediction steps are:

time lags has been used as an input variables.

accuracy criterion with actual wind power data series.

the performance goal of 0.001.

**77**

behalf of ACF.

#### *3.3.1 Statistical models*

The conventional statistical models (persistence, Moving Average & Gray Models) are identical to the direct random time-series model. Based on a number of historical data, pattern identification, parameter estimation, model checking are utilized to make a mathematical model for the prediction problem.

	- a. **Naïve Predictor:** In order to get a significant evaluation of WPF a naïve model should be used. This is one of the old and simple ways to forecast wind power & speed also called persistence model. It is based on the simple assumption that wind power at present time t will be same in a future time (t + x) [27].
	- b. **Simple Moving Average:** The moving average predicts the wind power based on simply the average of past values of wind power. It has also been used as a benchmark for assessing the accuracy criteria of prediction model.
	- c. **Gray Model (1,1) Predictor:** GM (n, m) model is based on the Gray theory as demonstrated by Professor Deng in 1982. GM (n, m) denotes a Gray model where n is the order differential equation and m is the no. of variables. It predicts the future values of time series based on the recent data fluctuations. There are various types of Gray Models as designed by various researchers but because of computational efficiency of GM (1, 1) is generally used.

ii. Linear or Time Series (TS) Models

According to the methods which have been proposed by Jenkins, these models can be further divided as follows: autoregressive model (AR), moving average model (MA), autoregressive moving average model (ARMA), auto regressive integrated moving average model (ARIMA) [28]. Generalized Autoregressive Conditional Heteroskedasticity (GARCH) has been used for interval forecasting to simulate the fluctuating characteristics of the residual series in Mictrogrid China. Fractional-ARIMA method has been proposed to overcome the disadvantage of ARIMA method, which has been characterized by a slow decay in its ACF [29]. The stochastic and seasonality pattern of wind power has been tackled by designing a combined Autoregressive Fractionally Integrated Moving Average (ARFIMA) and GARCH model [30]; whereas, for above said problem ref. [31] demonstrated ARMA with Vector Auto-regression and ref. [32] designed different ARMA models for wind speed and direction tuples prediction (above said problem).

#### *3.3.2 Artificial intelligence (AI) models*

traditional statistical models in forecasting. These are non parametric & non-linear models also known as data driven or black box models having usage of historical data patterns to learn the stochastic dependency between past and future. These NN's models always leave behind other traditional statistical models such as: linear regression and Box-Jenkins approaches. The NNs can be successfully used for

*Forecasting in Mathematics - Recent Advances, New Perspectives and Applications*

The conventional statistical models (persistence, Moving Average & Gray

a. **Naïve Predictor:** In order to get a significant evaluation of WPF a naïve model should be used. This is one of the old and simple ways to forecast wind power & speed also called persistence model. It is based on the simple assumption that wind power at present time t

b. **Simple Moving Average:** The moving average predicts the wind power based on simply the average of past values of wind power. It has also been used as a benchmark for assessing the accuracy criteria

c. **Gray Model (1,1) Predictor:** GM (n, m) model is based on the Gray theory as demonstrated by Professor Deng in 1982. GM (n, m) denotes a Gray model where n is the order differential equation and m is the no. of variables. It predicts the future values of time series based on the recent data fluctuations. There are various types of Gray Models as designed by various researchers but because of computational efficiency of GM (1, 1)

According to the methods which have been proposed by Jenkins, these models can be further divided as follows: autoregressive model (AR), moving average model (MA), autoregressive moving average model (ARMA), auto regressive integrated moving average model (ARIMA) [28]. Generalized Autoregressive Conditional Heteroskedasticity (GARCH) has been used for interval forecasting to simulate the fluctuating characteristics of the residual series in Mictrogrid China. Fractional-ARIMA method has been proposed to overcome the disadvantage of ARIMA method, which has been characterized by a slow decay in its ACF [29]. The stochastic and seasonality pattern of wind power has been tackled by designing a combined Autoregressive Fractionally Integrated Moving Average (ARFIMA) and GARCH model [30]; whereas, for above said problem ref. [31] demonstrated ARMA with Vector Auto-regression and ref. [32] designed different ARMA models for wind speed and direction tuples prediction (above said

Models) are identical to the direct random time-series model. Based on a number of historical data, pattern identification, parameter estimation, model checking are utilized to make a mathematical model for the prediction

will be same in a future time (t + x) [27].

of prediction model.

is generally used.

ii. Linear or Time Series (TS) Models

modeling and forecasting non-linear time series [26].

*3.3.1 Statistical models*

i. Traditional Models

problem.

problem).

**76**

The FFNN architecture, which is also called as Multi Layer Perceptron (MLP), along with back propagation (BP) as the learning algorithm is the most popular choice among researchers. The neural network (NN) and machine learning algorithms structures used by most of the researchers after 2000 in the leading journals are: Feed Forward Neural Network (FFNN), Recurrent Neural Network (RNN), Radial Basis Function Neural Network (RBFNN), Support Vector Machine (SVM), Support Vector Regression (SVR), Adaptive Neuro Fuzzy Inference System (ANFIS), Extreme Learning Machine (ELM), Adaptive Wavelet Neural Network (AWNN), General Regression Neural Network (GRNN), and Linear Neural Network with Time Delay (LNNTD).

In this, wind forecasting has been done by the three different models: (i) Benchmark, (ii) NN and (iii) WT based model. In the first category, only Naïve Predictor has been considered. This is the standard benchmark for wind forecasting applications, in which the previous values of input wind power series have been used for the next lead hour as forecasted values. In the second category, different ANN based models have been taken into consideration with different structure of network and learning algorithms. The NN along with gradient-based optimization techniques is most popular choice among all researchers and associated with the short comings of local minima and sensitivity to initial value persists as a result of poor accuracy. So, as to resolve above said problems, global evolutionary algorithms (EA) such as Genetic Algorithms (GA) [1, 23], Particle Swarm Optimization (PSO) [19, 33, 34] have been utilized. The main advantages of EA lie in its global convergence, inherent parallel search nature, and great robustness. These algorithms generate a high quality solution within a short computation time.

For proper input selection, there is need of complete experimental analysis on the basis of error rate. The input structures of WT based models are different from that of the non WT based models. In the WT based models, the input is the combination of Wind Power series and WT based approximated and detailed wind power series. Therefore, the number of input nodes is more as compared to non WT models. The structure of WT based FFNN for wind power prediction has been shown in **Figure 2** & detailed prediction steps are:

**Step 1:** From the raw data of wind power, a time series as input is selected on the behalf of ACF.

**Step 2:** Supply the created input signal to WT for performing multilevel decomposition on wind power signal by utilizing Daubechies (db10) wavelet.

**Step 3:** Now extract the multi level approximation A6 and 1, 2, 3,4,5,6 level detailed coefficients D1 to D6 of input wind power series signal.

**Step 5:** The approximated and detailed wind power series along with six original time lags has been used as an input variables.

**Step 6:** A three layer FFNN, as shown in **Figure 3**, has been selected having thirteen input nodes equal to the number of input variables, twelve hidden neurons with tangential sigmoid transfer function, and one output neuron with pure linear activation function, with each series. The network is trained using Levenberg– Marquardt (LM) training algorithms with architecture [12–11–1]. The momentum constant and learning rate have been kept equal to 0.06 and 0.001, respectively.

**Step 7:** For the prediction, one year wind data has been trained and tested for next one month, similar process is continuously repeated up-to next 24 months with one month moving window. The maximum epochs were set equal to 10,000 with the performance goal of 0.001.

**Step 8:** The output values found by the network has been assessed on the accuracy criterion with actual wind power data series.

• The ability of better improvement and understanding of forecast. The

sub-spaces & utilize it for improving forecasting model.

important. The various methods for accuracy calculation given below:

Where, *WPt,* is actual observation at time *t*, *Ft,* is forecast for time *t*

1 *n* X*n t*¼0

s

*PE* <sup>¼</sup> ð Þ *WPt* � *Ft Yt*

*MAPE* <sup>¼</sup> <sup>1</sup>

wind power curves with error curve have been shown in **Figure 4**.

*n* X*t n*¼1

The prediction performance of forecasting carried out by the different models used in this research is justified on the basis of forecasting accuracy indices. The methodology described above has been applied to predict the wind power of OEM for two years from November 2012 to October 2014 on MAPE & MAE accuracy criteria. The software used for training and testing of NN is MATLAB version R2011b. The extensive use of WT for data pre-processing makes the results more significant and effective. From the results **Table 3**, it is clear that the results achieved with the help of WT based models have been found to be better up to 40–60% as compare to non WT based models. The 24 hours actual and forecasted

*MAE* <sup>¼</sup> <sup>1</sup> *n* X*n t*¼1

*RMSE* ¼

• **The Mean Absolute Percentage Error (MAPE)**

can be obtained with the comparison of others.

compared with each other.

*DOI: http://dx.doi.org/10.5772/intechopen.94550*

• **The Mean Absolute Error (MAE)**

• **The Root Mean Square Error (RMSE)**

• **Percentage Error (PE)**

**79**

• **The Error**

*Wind Power Forecasting*

evaluation of forecast exposes all those sub-spaces whose forecasting error is more out of model state space. So, a forecaster can take advantage of analyzing

• Justifying the cost associated with resources used in forecasting model. The forecast performance assessment in accuracy terms gives a measure that can be

• The ability of performing model selection so that maximum certainty of results

In most of the forecasting models accuracy is the criterion for selecting a particular method for the forecasting. For a consumer accuracy of forecasting is most

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ð Þ *WPt* � *Ft* <sup>2</sup>

*E* ¼ ð Þ *WPt* � *Ft* (6)

� � <sup>∗</sup> <sup>100</sup> (9)

j j *PE* (10)

*WPt* � *Ft* j j (7)

(8)

directly linked to the utility or forecast user. Then coast and utility are

**Figure 2.** *Hourly curve for load, price & wind power from Ontario electricity market.*

**Figure 3.** *WT based FFNN for wind power forecasting.*

#### **3.4 Evaluation of prediction performance**

The aim of forecast evaluation is to assess, the general quality of a forecast by comparing the forecasted system states to actual observed states. The forecast evaluation provides a forecaster with:


In most of the forecasting models accuracy is the criterion for selecting a particular method for the forecasting. For a consumer accuracy of forecasting is most important. The various methods for accuracy calculation given below:

• **The Error**

$$E = (\mathbf{W}\mathbf{P}\_t - \mathbf{F}\_t) \tag{6}$$

Where, *WPt,* is actual observation at time *t*, *Ft,* is forecast for time *t*

• **The Mean Absolute Error (MAE)**

$$MAE = \frac{1}{n} \sum\_{t=1}^{n} |WP\_t - F\_t| \tag{7}$$

• **The Root Mean Square Error (RMSE)**

$$RMSE = \sqrt{\frac{1}{n} \sum\_{t=0}^{n} \left( WP\_t - F\_t\right)^2} \tag{8}$$

• **Percentage Error (PE)**

$$PE = \left(\frac{(WP\_t - F\_t)}{Y\_t}\right) \* 100\tag{9}$$

• **The Mean Absolute Percentage Error (MAPE)**

$$\text{MAPE} = \frac{1}{n} \sum\_{n=1}^{t} |PE| \tag{10}$$

The prediction performance of forecasting carried out by the different models used in this research is justified on the basis of forecasting accuracy indices. The methodology described above has been applied to predict the wind power of OEM for two years from November 2012 to October 2014 on MAPE & MAE accuracy criteria. The software used for training and testing of NN is MATLAB version R2011b. The extensive use of WT for data pre-processing makes the results more significant and effective. From the results **Table 3**, it is clear that the results achieved with the help of WT based models have been found to be better up to 40–60% as compare to non WT based models. The 24 hours actual and forecasted wind power curves with error curve have been shown in **Figure 4**.

**3.4 Evaluation of prediction performance**

*Hourly curve for load, price & wind power from Ontario electricity market.*

*Forecasting in Mathematics - Recent Advances, New Perspectives and Applications*

evaluation provides a forecaster with:

*WT based FFNN for wind power forecasting.*

**Figure 2.**

**Figure 3.**

**78**

The aim of forecast evaluation is to assess, the general quality of a forecast by comparing the forecasted system states to actual observed states. The forecast


#### **Table 3.**

*Overall prediction comparisons for all models used.*

pre-assumed probabilistic value, the basic aim of interval forecasting is to find out the range of prediction interval in which next hour wind power output lies. This framework has been consequently used for evaluating and analyzing the skill of the models for one lead hour point forecast. Thus, the overall results have been proving the reliability of results and show how the resolution may improve the

The probabilistic forecasting has a wide range of statistical parameters on which the probabilistic outcomes of wind power lies. The prediction intervals (PI) stands for a wide range of possible probabilistic values within which the observed wind power values lies with a certain predefined probability. The basic idea behind the prediction intervals is to estimate the uncertainty associated with observed wind

more enclosed and wider both depending on the value of confidence intervals (CI).

For a given sample size α has been a significant level which has been used to take into account the CI of the certain prediction intervals. The probabilistic stochastic

*<sup>t</sup>* ð Þ *WPi* , *UB<sup>α</sup>*

� �. The prediction intervals range can be much

*<sup>t</sup>* ð Þ *WPi* � � (12)

� � (13)

� � (14)

Confidence Interval CI ð Þ¼ 100 1ð Þ � α % (11)

*=*2 *σ* ffiffiffi *n* p

*=*2 *σ* ffiffiffi *n* p

<sup>2</sup> is the critical value of standard Gaussian distribution,

*i*

*<sup>t</sup>* ð Þ¼ *WPi LB<sup>α</sup>*

In the Eq. (12), the lower bound and upper bound can be expressed as:

*<sup>t</sup>* ð Þ¼ *WPi F WP* ð Þ�*<sup>i</sup> z*<sup>1</sup>�*<sup>σ</sup>*

*<sup>t</sup>* ð Þ¼ *WPi F WP* ð Þþ*<sup>i</sup> z*<sup>1</sup>�*<sup>σ</sup>*

which depends on certain value of CI, *n* is look ahead hour for the prediction sample & *σ* is the standard deviation of predicted values [11, 35–37] which is expressed as:

forecasts skill.

**Figure 5.**

*Wind Power Forecasting*

*DOI: http://dx.doi.org/10.5772/intechopen.94550*

power *WP<sup>t</sup>*

*i*

The CI can be expressed as:

interval (PSI) can be obtained by:

In (13) and (14) *z*<sup>1</sup>�*<sup>σ</sup>*

**81**

� � and forecasted *F WP<sup>t</sup>*

*PI with nominal confidence 95% in 24 hours look ahead.*

*PSI<sup>α</sup>*

*LB<sup>α</sup>*

*UB<sup>α</sup>*

*=*

**Figure 4.** *One day ahead actual & forecasted wind power curve during winter season.*

#### **3.5 Uncertainty of forecasts using probabilistic forecasting**

The uncertainty of forecasts is mainly due to the noise of training data, the misspecification of NN model for regression and input data selection.

**NN Model Uncertainty:** Uncertainty in NN forecasting arises due to misspecification in input parameters and structure of model which occurs due to local minima in the training process, random generation of input weights and so on. In case of global minima, misspecifications lead to non-eligible uncertainties in results related to prediction. The other factor behind model uncertainty is that during training finite samples never guarantee consistent generalization in performance of NN for future days. Basically, in WPF, it has become impossible to gather accurate information for reducing uncertainties while predicting and hence collectively called as model uncertainty. Due to model uncertainty, uncertainty in output should be handled carefully for accurate estimation in NN.

**Data Uncertainty:** Not only model uncertainty but also data noise adds to prediction uncertainty. If the data is stochastic in nature, then modeling is deterministically is really difficult. Both model misspecification and data noise are the major sources of uncertainties that affect the forecasting results.

In this, probabilistic forecasting of wind power has been performed in coordination with single step ahead wind power point forecasts. The major emphasis of probabilistic forecasting is to take into account the uncertainty associated with the wind power with probabilistic forecasting attributes such as: sharpness, reliability, resolution and discrimination. It consists of a set of prediction intervals which works in coordination with the best forecasts of single step ahead of wind power for the next coming hour; the interval forecasting has been incorporated. With a

**Figure 5.** *PI with nominal confidence 95% in 24 hours look ahead.*

pre-assumed probabilistic value, the basic aim of interval forecasting is to find out the range of prediction interval in which next hour wind power output lies. This framework has been consequently used for evaluating and analyzing the skill of the models for one lead hour point forecast. Thus, the overall results have been proving the reliability of results and show how the resolution may improve the forecasts skill.

The probabilistic forecasting has a wide range of statistical parameters on which the probabilistic outcomes of wind power lies. The prediction intervals (PI) stands for a wide range of possible probabilistic values within which the observed wind power values lies with a certain predefined probability. The basic idea behind the prediction intervals is to estimate the uncertainty associated with observed wind power *WP<sup>t</sup> i* � � and forecasted *F WP<sup>t</sup> i* � �. The prediction intervals range can be much more enclosed and wider both depending on the value of confidence intervals (CI). The CI can be expressed as:

$$\text{Confidence Interval} \left(\text{CI}\right) = \mathbf{100} (\mathbf{1} - \mathbf{a}) \text{\%} \tag{11}$$

For a given sample size α has been a significant level which has been used to take into account the CI of the certain prediction intervals. The probabilistic stochastic interval (PSI) can be obtained by:

$$PSI\_t^a(WP\_i) = \left[LB\_t^a(WP\_i), UB\_t^a(WP\_i)\right] \tag{12}$$

In the Eq. (12), the lower bound and upper bound can be expressed as:

$$LB\_t^a(\mathsf{WP}\_i) = F(\mathsf{WP}\_i) - z\_{1-\sigma\_2} \left(\frac{\sigma}{\sqrt{n}}\right) \tag{13}$$

$$U\mathcal{B}\_t^a(\mathcal{W}\mathcal{P}\_i) = F(\mathcal{W}\mathcal{P}\_i) + z\_{1-\sigma\_2} \left(\frac{\sigma}{\sqrt{n}}\right) \tag{14}$$

In (13) and (14) *z*<sup>1</sup>�*<sup>σ</sup>=*<sup>2</sup> is the critical value of standard Gaussian distribution, which depends on certain value of CI, *n* is look ahead hour for the prediction sample & *σ* is the standard deviation of predicted values [11, 35–37] which is expressed as:

**3.5 Uncertainty of forecasts using probabilistic forecasting**

*One day ahead actual & forecasted wind power curve during winter season.*

**Table 3.**

**Figure 4.**

**80**

*Overall prediction comparisons for all models used.*

be handled carefully for accurate estimation in NN.

major sources of uncertainties that affect the forecasting results.

misspecification of NN model for regression and input data selection.

The uncertainty of forecasts is mainly due to the noise of training data, the

**Model Naïve FFNN ERNN GANN PSONN GAPSONN GRNN LNNTD WT + FFNN** MAPE 15.016 13.83 13.885 14.015 13.91 13.915 14.48 13.825 5.948 MAE 65.073 58.415 58.145 58.413 58.4675 58.29209 62.285 58.0475 23.225

*Forecasting in Mathematics - Recent Advances, New Perspectives and Applications*

fication in input parameters and structure of model which occurs due to local minima in the training process, random generation of input weights and so on. In case of global minima, misspecifications lead to non-eligible uncertainties in results related to prediction. The other factor behind model uncertainty is that during training finite samples never guarantee consistent generalization in performance of NN for future days. Basically, in WPF, it has become impossible to gather accurate information for reducing uncertainties while predicting and hence collectively called as model uncertainty. Due to model uncertainty, uncertainty in output should

**Data Uncertainty:** Not only model uncertainty but also data noise adds to prediction uncertainty. If the data is stochastic in nature, then modeling is deterministically is really difficult. Both model misspecification and data noise are the

In this, probabilistic forecasting of wind power has been performed in coordination with single step ahead wind power point forecasts. The major emphasis of probabilistic forecasting is to take into account the uncertainty associated with the wind power with probabilistic forecasting attributes such as: sharpness, reliability, resolution and discrimination. It consists of a set of prediction intervals which works in coordination with the best forecasts of single step ahead of wind power for the next coming hour; the interval forecasting has been incorporated. With a

**NN Model Uncertainty:** Uncertainty in NN forecasting arises due to misspeci-

$$
\sigma = \sqrt{\frac{\left(\overline{\mathbf{W}P - \overline{\mathbf{W}P}}\right)^2}{(n-1)}} \tag{15}
$$

**References**

*Wind Power Forecasting*

05.001

2009.

**83**

[1] Pai, P. F. and Hong, W. C.: Forecasting regional electricity load based on recurrent support vector machines with genetic algorithms. International Journal of Electric Power Systems Research. 2005;74:417–425. DOI: 10.1016/j.epsr.2005.01.006

*DOI: http://dx.doi.org/10.5772/intechopen.94550*

forecasting models. Energy Procedia. 2011; 12: 770–778. DOI: https://doi.org/

10.1016/j.egypro.2011.10.103

[9] Togelou, A., Sideratos, G. and Hatziargyriou, N. D.: Wind power forecasting in the absence of historical data. IEEE Transactions on Sustainable Energy. 2012; 3(3): 416–421. DOI: 10.1109/TSTE.2012.2188049

[10] Amjady, N. and Keynia, F.: Dayahead price forecasting of electricity markets by mutual information technique

and cascaded neuro-evolutionary algorithm, IEEE Transactions on Power Systems. 2009; 24(1): 306–318. DOI: 10.1109/TPWRS.2008.2006997

[11] Pinson, P. and Kariniotakis, G.: Conditional prediction intervals of wind power generation. IEEE Transactions on Power Systems. 2010; 25(4): 1845–1856. DOI: 10.1109/TPWRS.2010.2045774

[12] Aggarwal, S. K., Saini, L. M. and Kumar, A.: Day-ahead price forecasting in Ontario electricity market using variablesegmented support vector machine-based model. International Journal of Electric Power Components and Systems. 2009; 37: 495–516. DOI: https://doi.org/10.1080/

[13] Olsson, M., Perninge, M. and Soder, L.: Modeling real-time balancing power demands in wind power systems using stochastic differential equations. International Journal of Electric Power Systems Research. 2010; 80: 966–974. DOI: https://doi.org/10.1016/j.epsr.

[14] Ernst, B., Oakleaf, B., Ahlstrom, M. L., Lange, M., Moehrlen, C., Lange, B., Focken, U. and Rohrig, K.: Predicting the wind. IEEE Power & Energy Magazine. 2007; 78–89. DOI: 10.1109/

[15] Zhao, P., Wang, J. F., Xia, J., Dai, Y., Sheng, Y. and Yue, J.: Performance

15325000802599353

2010.01.004

MPE.2007.906306

[2] Kumar, A., Srivastva, S. C. and Singh, S. N.: Congestion management in

[3] Kumar, A., Srivastva, S. C. and Singh, S. N.: A zonal congestion management approach using real and reactive power rescheduling. IEEE Transactions on Power Systems. 2004;

19(1): 554–562. DOI: 10.1109/

[4] Negnevitsky, M., Mandal, P. Srivastava, A. K.: Machine learning applications for load, price and wind power prediction in power systems. 15th International Conference on Intelligent System Applications to Power Systems.

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[7] Amjady, N. and Hemmati, M. (2006). Energy price forecasting, IEEE Power & Energy Magazine, 2006; 20– 29. DOI: 10.1109/MPAE.2006.1597990

[8] Wang, X., Guo, P. and Huang, X. (2011). A review of wind power

TPWRS.2003.821448

competitive power market: A bibliography survey. International Journal of Electric Power Systems Research. 2005; 76:153–164. DOI: https://doi.org/10.1016/j.epsr.2005.

For the WT based model, the upper bound curve and lower bound curves obtained at 95% of the confidence and the actual measured wind power curve in 24 hours has been shown in **Figure 5**.

#### **4. Conclusions**

The uncertainty, complexity and seasonal aspects associated with the wind contribute high level of uncertainties in wind power generation. Because weather conditions and wind speeds vary very much in different seasons. Therefore, for a perfect efficient forecasting model it is necessary to take care of input variables and their proper selection in time series. Actually, the improper input cause improper training of NN model as a result of that poor accuracy of forecasts. In this chapter, in order to take care of models forecast performance, probabilistic parameters have been taken into consideration.

In order to evaluate the performance on probabilistic forecasting, on the basis of single step reliable Prediction Intervals (PI's) need to be derived. In this, instead of exact values of forecast a range of forecasting interval need to be considered. If the predicted values lie in that range then, the performance of model is good otherwise model is poor one. Furthermore, power system operations require useful efficient forecast values with high level of reference confidence. Therefore, to fulfill the need of power system, more practical data based model should be required with highconfidence-level PI's.

#### **Author details**

Sumit Saroha<sup>1</sup> \*, Sanjeev Kumar Aggarwal<sup>2</sup> and Preeti Rana<sup>3</sup>

1 Assistant Professor, Department of Electrical Engineering, Guru Jambheshwar University of Science and Technology, Hisar, India

2 Associate Professor, Electrical and Instrumentation Engineering Department, Thapar Institute of Engineering and Technology, Patiala, Punjab, India

3 Scholar, MTH 48, Guru Jambheshwar University of Science and Technology, Hisar, India

\*Address all correspondence to: saroha\_sumit0178@yahoo.com

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### **References**

*σ* ¼

24 hours has been shown in **Figure 5**.

been taken into consideration.

confidence-level PI's.

**Author details**

Sumit Saroha<sup>1</sup>

Hisar, India

**82**

**4. Conclusions**

s

*Forecasting in Mathematics - Recent Advances, New Perspectives and Applications*

For the WT based model, the upper bound curve and lower bound curves obtained at 95% of the confidence and the actual measured wind power curve in

The uncertainty, complexity and seasonal aspects associated with the wind contribute high level of uncertainties in wind power generation. Because weather conditions and wind speeds vary very much in different seasons. Therefore, for a perfect efficient forecasting model it is necessary to take care of input variables and their proper selection in time series. Actually, the improper input cause improper training of NN model as a result of that poor accuracy of forecasts. In this chapter, in order to take care of models forecast performance, probabilistic parameters have

In order to evaluate the performance on probabilistic forecasting, on the basis of single step reliable Prediction Intervals (PI's) need to be derived. In this, instead of exact values of forecast a range of forecasting interval need to be considered. If the predicted values lie in that range then, the performance of model is good otherwise model is poor one. Furthermore, power system operations require useful efficient forecast values with high level of reference confidence. Therefore, to fulfill the need of power system, more practical data based model should be required with high-

\*, Sanjeev Kumar Aggarwal<sup>2</sup> and Preeti Rana<sup>3</sup>

University of Science and Technology, Hisar, India

provided the original work is properly cited.

1 Assistant Professor, Department of Electrical Engineering, Guru Jambheshwar

2 Associate Professor, Electrical and Instrumentation Engineering Department,

3 Scholar, MTH 48, Guru Jambheshwar University of Science and Technology,

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

Thapar Institute of Engineering and Technology, Patiala, Punjab, India

\*Address all correspondence to: saroha\_sumit0178@yahoo.com

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *WP* � *WP* � �<sup>2</sup> ð Þ *n* � 1

(15)

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[2] Kumar, A., Srivastva, S. C. and Singh, S. N.: Congestion management in competitive power market: A bibliography survey. International Journal of Electric Power Systems Research. 2005; 76:153–164. DOI: https://doi.org/10.1016/j.epsr.2005. 05.001

[3] Kumar, A., Srivastva, S. C. and Singh, S. N.: A zonal congestion management approach using real and reactive power rescheduling. IEEE Transactions on Power Systems. 2004; 19(1): 554–562. DOI: 10.1109/ TPWRS.2003.821448

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[9] Togelou, A., Sideratos, G. and Hatziargyriou, N. D.: Wind power forecasting in the absence of historical data. IEEE Transactions on Sustainable Energy. 2012; 3(3): 416–421. DOI: 10.1109/TSTE.2012.2188049

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[27] Felice, M. D. and Yao, X.: Shortterm load forecasting with neural network ensembles: A comparative study. IEEE Computational Intelligence Magazine. 2011; 47–56. DOI: 10.1109/

[28] Aggarwal, S. K., Saini, L. M. and Kumar, A.: Short term price forecasting

https://doi.org/10.1016/j.eswa.

using EMD and chaotic theory,

Nonlinear Science Numerical

09.016

*Forecasting in Mathematics - Recent Advances, New Perspectives and Applications*

2013.08.011

06.003

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MCI.2011.941590

[16] Ramirez-Rosado, I. J., Fernandez-Jimenez L. A., Monteiro, C., Sousa, J. and Bessa, R.: Comparison of two new short-term wind-power forecasting systems, International Journal of Renewable Energy, 2009; 34: 1848– 1854. DOI: https://doi.org/10.1016/

[17] Chen, K. and Yu, J.: Short-term wind speed prediction using an unscented kalman filter based statespace support vector regression approach, International Journal of Applied Energy, 2014; 113: 690–705. DOI: https://doi.org/10.1016/j.apenergy.

[18] Ontario electricity market data. Available at: http://www.ieso.ca/ en/power-data/data-directory.

[19] Catalão, J. P. S., Pousinho, H. M. I., Mendes, V. M. F.: Hybrid wavelet-PSO-ANFIS approach for short-term wind power forecasting in Portugal, IEEE Transactions on Sustainable Energy, 2011; 2(1): 50–59. DOI: 10.1109/

[20] Liu, H. and Tian, H. Q., Chen, C. and Li, Y. F.: A hybrid statistical method to predict wind speed and wind power, International Journal of Renewable Energy, 2010; 35(8): 1857–1861. DOI: https://doi.org/10.1016/j.renene.2009.

[21] An, X., Jiang, D., Liu, C. and Zhao, M.: Wind farm power prediction based on wavelet decomposition and chaotic time series, International Journal of Expert Systems with Applications, 2011; 38(9): 11280–11285. DOI: https://doi. org/10.1016/j.eswa.2011.02.176

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j.renene.2008.11.014

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[30] Erdem, E. and Shi, J.: ARMA based approaches for forecasting the tuple of wind speed and direction, International Journal of Applied Energy, 2011; 88(4): 1405–1414. DOI: 10.1016/j. apenergy.2010.10.031

[31] Hill, D. C., McMillan, D., Bell, K. R. W. and Infield, D.: Application of autoregressive models to U.K. wind speed data for power system impact studies, IEEE Transactions on Sustainable Energy, 2012; 3(1):134–141. DOI: 10.1109/TSTE.2011.2163324

[32] Erdem, E. and Shi, J.: ARMA based approaches for forecasting the tuple of wind speed and direction, International Journal of Applied Energy, 2011; 88(4): 1405–1414. DOI:10.1016/j. apenergy.2010.10.031

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[36] Sideratos, G. and Hatziargyriou, N. D.: An advanced statistical method for wind power forecasting, IEEE Transactions on Power Systems, 2007; 22(1): 258–265. DOI: 10.1109/ TPWRS.2006.889078

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**Chapter 5**

**Abstract**

**1. Introduction**

states at every location.

for all *t* and *p θ*1, … *::*, *θT*j*y*1, … *::*, *yt*

**87**

we treat *θ<sup>t</sup>* as a random variable [1, 2].

*Deneshkumar Venugopal,*

*Senthamarai Kannan Kaliyaperumal*

*and Sonai Muthu Niraikulathan*

the steady state probability distribution.

Stock Market Trend Prediction

In Recent years many forecasting methods have been proposed and implemented for the stock market trend prediction. In this Chapter, the trend analyses of the stock market prediction are presented by using Hidden Markov Model with the one day difference in close value for a particular period. The probability values π gives the trend percentage of the stock prices which is calculated for all the observe sequence and hidden sequences. This chapter helps for decision makers to make decisions in case of uncertainty on the basis of the percentage of probability values obtained from

The fundamental idea behind a hidden Markov model is that there is a Markov process we cannot observe that determines the probability distribution for what we do observe. Thus a hidden Markov model is specified by the transition density of the Markov chain and the probability laws that govern what we observe given the state of the Markov chain. Given such a model, we want to estimate any parameters that occur in the model. And also determined the most likely sequence for the hidden process. Finally we may want the probability distribution for the hidden

Let *yt* represents the observed value of the process at location *t* for *t* ¼ 1, *:* … , *T*, *θ<sup>t</sup>* the value of the hidden process at location *t* and let ϕ represents parameters necessary to determine the probability distribution for *yt* given *θ<sup>t</sup>* and *θ<sup>t</sup>* given *θ<sup>t</sup>*�1. In our applications, *yt* will either be an increase or decrease and the hidden process will

<sup>∣</sup> *<sup>θ</sup>t*, <sup>ϕ</sup><sup>Þ</sup>

, *<sup>p</sup> <sup>θ</sup>t*j*y*1, … *:*, *yT*

Using Hidden Markov Model

**Keywords:** stock market, HMM, TPM, EPM and trend prediction

determine the probability distribution of observing different letters.

various distribution in which we are interested are *p* ϕ j*y*1, … *:*, *yT*

Our model is then described by the sets of probability distributions *p yt*

and *p θ<sup>t</sup>* ð Þ j *θ<sup>t</sup>*�1, ϕ . A crucial component of this model is that the *yt* are independent given the set of *θ<sup>t</sup>* and *θ* only depends directly on its neighbors *θ<sup>t</sup>*�<sup>1</sup> and *θ<sup>t</sup>*þ1. The

The measure of best is to find the path that has the maximum probability in the

HMM, given the sequence *X*. Recall that the model gives the joint probabilities

. We will adopt a Bayesian perspective, so that

#### **Chapter 5**

## Stock Market Trend Prediction Using Hidden Markov Model

*Deneshkumar Venugopal, Senthamarai Kannan Kaliyaperumal and Sonai Muthu Niraikulathan*

#### **Abstract**

In Recent years many forecasting methods have been proposed and implemented for the stock market trend prediction. In this Chapter, the trend analyses of the stock market prediction are presented by using Hidden Markov Model with the one day difference in close value for a particular period. The probability values π gives the trend percentage of the stock prices which is calculated for all the observe sequence and hidden sequences. This chapter helps for decision makers to make decisions in case of uncertainty on the basis of the percentage of probability values obtained from the steady state probability distribution.

**Keywords:** stock market, HMM, TPM, EPM and trend prediction

#### **1. Introduction**

The fundamental idea behind a hidden Markov model is that there is a Markov process we cannot observe that determines the probability distribution for what we do observe. Thus a hidden Markov model is specified by the transition density of the Markov chain and the probability laws that govern what we observe given the state of the Markov chain. Given such a model, we want to estimate any parameters that occur in the model. And also determined the most likely sequence for the hidden process. Finally we may want the probability distribution for the hidden states at every location.

Let *yt* represents the observed value of the process at location *t* for *t* ¼ 1, *:* … , *T*, *θ<sup>t</sup>* the value of the hidden process at location *t* and let ϕ represents parameters necessary to determine the probability distribution for *yt* given *θ<sup>t</sup>* and *θ<sup>t</sup>* given *θ<sup>t</sup>*�1. In our applications, *yt* will either be an increase or decrease and the hidden process will determine the probability distribution of observing different letters.

Our model is then described by the sets of probability distributions *p yt* <sup>∣</sup> *<sup>θ</sup>t*, <sup>ϕ</sup><sup>Þ</sup> and *p θ<sup>t</sup>* ð Þ j *θ<sup>t</sup>*�1, ϕ . A crucial component of this model is that the *yt* are independent given the set of *θ<sup>t</sup>* and *θ* only depends directly on its neighbors *θ<sup>t</sup>*�<sup>1</sup> and *θ<sup>t</sup>*þ1. The various distribution in which we are interested are *p* ϕ j*y*1, … *:*, *yT* , *<sup>p</sup> <sup>θ</sup>t*j*y*1, … *:*, *yT* for all *t* and *p θ*1, … *::*, *θT*j*y*1, … *::*, *yt* . We will adopt a Bayesian perspective, so that we treat *θ<sup>t</sup>* as a random variable [1, 2].

The measure of best is to find the path that has the maximum probability in the HMM, given the sequence *X*. Recall that the model gives the joint probabilities

*Pr H*ð Þ , *X* for all sequence, it also gives the posterior probability *Pr H*ð Þ¼ , *X Pr H*ð Þ , *X =Pr X*ð Þ, for every possible state path H through the model, conditioned on the sequence *X* with maximum posterior probability [3, 4]. Given that the denominator *Pr X*ð Þ is constant in the conditional probability formula for a given sequence *X*, maximizing the posterior probability is equivalent to finding the state path H\* that maximizes the joint probability *Pr H*<sup>∗</sup> ð Þ , *<sup>X</sup> :* Nguyen [5] has determined the optimal number of states for the HMM by using the AIC, BIC and HQ information criteria and also discussed the applications of HMM in stock trading. Hassan and Nath [6] have applied HMM to the airlines stock forecast. HMMs have been used for pattern recognition and classification problems and it was suitable for modeling dynamic systems.

**2.1 Evaluation problem**

**2.2 Decoding problem**

**2.3 Learning problem**

**3. Results and discussions**

Interval values: S1 = 9500 to 551. S2 = 550 to 251. S3 = 250 to 249. S4 = 250 to 8500.

below (**Table 2**).

**89**

(**Figure 2** and **Table 3**):

training HMM.

is solved by the Forward Backward Algorithm [7, 8].

*Stock Market Trend Prediction Using Hidden Markov Model*

*DOI: http://dx.doi.org/10.5772/intechopen.93988*

Given the HMM = {A,B,π} and the observation sequence O = o1,o2, … ,oM, the probability that model λ has generated sequence O is calculated. Often this problem

Given the HMM λ = {A,B,π} and the observation sequence O = o1,o2,… ,oM, calculate the most likely sequence of hidden states that produced this observation

Given some training observation sequences O = o1,o2, … ,oM, and general structure of HMM (numbers of hidden and visible states), determine HMM parameters λ = {A,B,π} that best fit training data. The most common solution for this problem is Baum-Welch algorithm [9, 10] which is considered as the traditional method for

In this chapter, the data has been taken from Yahoofinance.com and the NSE daily close value data for a month of January 2020 period is considered for the analysis. Here two observing symbols "I" for Increasing states and the symbols "D" for decreasing states have been used. If the differences of close value greater than 0 its observing that the symbol is "f" and If the differences of close value less than 0 its observing that the symbol is "D". There are six hidden states assumed and are denoted by the symbol S1, S2, S3, S4, S5, S6 are indicates that very low, low, moderate low, moderate high, high and very high respectively. The states are not directly observable. The situations of the stock market are considered hidden. Given a sequence of observation we can find the hidden state sequence that produced those observa-

The various probability values of TPM, EPM and π for difference in one day, two days, three days, four days, five days, six days close value are calculated as given

      

*I D*

*S*10 1 *S*20 1 *S*3 0*:*2849 0*:*7143 *S*4 0*:*5 0*:*5

Probability values of TPM, EPM, and π for difference in one day close value

sequence O. Usually this problem is handled by Viterbi Algorithm [7, 8].

tions. **Table 1** shows the daily close value of the stock market.

*S*1 *S*2 *S*3 *S*4

*S*10 0 1 0 *S*20 0 1 0 *S*3 0*:*071 0*:*071 0*:*4286 0*:*4286 *S*40 0 1 0

#### **2. Hidden Markov model**

Hidden Markov model (HMM) is a stochastic model which is not directly observable, It describes the observable events that are depends on internal factors. The observable events are represented as symbols, where the invisible factor involved in the observation is represented as a state. HMM is a stochastic model where the system is assumed to be a Markov Process with hidden states and it gives better accuracy than the other models. Using the given input values, the parameters of the HMM (λ) denoted by A, B and π are found out. An HMM is defined as λ = (S,O,A,B,π) where S = {s1,s2, … ,sN} is a set of N possible states O = {o1,o2, … ,oM} is a set of M possible observation symbols, A is an N\*N state Transition Probability Matrix (TPM), B is an N\*M observation or Emission Probability Matrix (EPM) and Π is an N dimensional initial state probability distribution vector and A,B and π should satisfy the following conditions (**Figure 1**):

$$\sum\_{j=1}^{N} a\_{\vec{\imath}} = \mathbf{1} \text{ where } \mathbf{1} \le \mathbf{i} \le N;$$

$$\sum\_{j=1}^{M} b\_{\vec{\imath}} = \mathbf{1} \text{ where } \mathbf{1} \le \mathbf{i} \le N;$$

$$\sum\_{i=1}^{N} \pi\_i = \mathbf{1} \text{ where } \pi\_i \ge 0$$

**Figure 1.** *Diagram of HMM.*

#### **2.1 Evaluation problem**

*Pr H*ð Þ , *X* for all sequence, it also gives the posterior probability *Pr H*ð Þ¼ , *X*

*Forecasting in Mathematics - Recent Advances, New Perspectives and Applications*

**2. Hidden Markov model**

**Figure 1.** *Diagram of HMM.*

**88**

should satisfy the following conditions (**Figure 1**):

X *N*

*j*¼1

X *M*

*j*¼1

X *N*

*i*¼1

*Pr H*ð Þ , *X =Pr X*ð Þ, for every possible state path H through the model, conditioned on the sequence *X* with maximum posterior probability [3, 4]. Given that the denominator *Pr X*ð Þ is constant in the conditional probability formula for a given sequence *X*, maximizing the posterior probability is equivalent to finding the state path H\* that maximizes the joint probability *Pr H*<sup>∗</sup> ð Þ , *<sup>X</sup> :* Nguyen [5] has determined the optimal number of states for the HMM by using the AIC, BIC and HQ information criteria and also discussed the applications of HMM in stock trading. Hassan and Nath [6] have applied HMM to the airlines stock forecast. HMMs have been used for pattern recognition and classification problems and it was suitable for modeling dynamic systems.

Hidden Markov model (HMM) is a stochastic model which is not directly observable, It describes the observable events that are depends on internal factors. The observable events are represented as symbols, where the invisible factor involved in the observation is represented as a state. HMM is a stochastic model where the system is assumed to be a Markov Process with hidden states and it gives better accuracy than the other models. Using the given input values, the parameters of the HMM (λ) denoted by A, B and π are found out. An HMM is defined as λ = (S,O,A,B,π) where S = {s1,s2, … ,sN} is a set of N possible states O = {o1,o2, … ,oM} is a set of M possible observation symbols, A is an N\*N state Transition Probability Matrix (TPM), B is an N\*M observation or Emission Probability Matrix (EPM) and Π is an N dimensional initial state probability distribution vector and A,B and π

*aij* ¼ 1 *where* 1≤*i* ≤ *N*;

*bij* ¼ 1 *where* 1≤*i* ≤ *N*;

*π<sup>i</sup>* ¼ 1 *where π<sup>i</sup>* ≥ 0

Given the HMM = {A,B,π} and the observation sequence O = o1,o2, … ,oM, the probability that model λ has generated sequence O is calculated. Often this problem is solved by the Forward Backward Algorithm [7, 8].

#### **2.2 Decoding problem**

Given the HMM λ = {A,B,π} and the observation sequence O = o1,o2,… ,oM, calculate the most likely sequence of hidden states that produced this observation sequence O. Usually this problem is handled by Viterbi Algorithm [7, 8].

#### **2.3 Learning problem**

Given some training observation sequences O = o1,o2, … ,oM, and general structure of HMM (numbers of hidden and visible states), determine HMM parameters λ = {A,B,π} that best fit training data. The most common solution for this problem is Baum-Welch algorithm [9, 10] which is considered as the traditional method for training HMM.

#### **3. Results and discussions**

In this chapter, the data has been taken from Yahoofinance.com and the NSE daily close value data for a month of January 2020 period is considered for the analysis.

Here two observing symbols "I" for Increasing states and the symbols "D" for decreasing states have been used. If the differences of close value greater than 0 its observing that the symbol is "f" and If the differences of close value less than 0 its observing that the symbol is "D". There are six hidden states assumed and are denoted by the symbol S1, S2, S3, S4, S5, S6 are indicates that very low, low, moderate low, moderate high, high and very high respectively. The states are not directly observable.

The situations of the stock market are considered hidden. Given a sequence of observation we can find the hidden state sequence that produced those observations. **Table 1** shows the daily close value of the stock market.

Interval values:

S1 = 9500 to 551.


The various probability values of TPM, EPM and π for difference in one day, two days, three days, four days, five days, six days close value are calculated as given below (**Table 2**).

Probability values of TPM, EPM, and π for difference in one day close value (**Figure 2** and **Table 3**):


#### *Forecasting in Mathematics - Recent Advances, New Perspectives and Applications*


**Table 1.** *Daily close value of NSE.*

Probability values of TPM, EPM, and π for difference in two day close value (**Figure 3** and **Table 4**).


Probability values of TPM, EPM, and π for difference in three day close value (**Figure 4** and **Table 5**):


**S. no**

**91**

1

2

3

 40,676.63 40,869.47

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

**Table 2.** *Daily close value for finding differences*

 *in one day, two day, three days, four days, five days, six days close value.*

 40,723.49

 190.33

 40,913.82

 284.84

 I I

94.51

I

611.15

 D

 1547.85

 I

2634.8

 D

 3456.95

 I

516.64

 I

936.7

I

1086.95

 D

 822.15

 41,198.66

 40,966.86

 188.26 231.8

 D

 420.06

I

150.25

 D

264.8

 D

 1320.48

 I I

498.33

I

381.75

 D

I

269.81

I

415.05

 D

 1055.68

 I

938.73

I

1828.24

 D

 41,155.12

 458.01

 41,613.19

 41,386.4

 41,115.38

 208.43 271.02

226.79

 D I

684.86

 D

 640.63

I

116.95

 D

889.51

 D

 2593.44

 I

44.23

 D

 523.68

 D

479.45

I

482.78

 D

I

1006.46

 D

 1703.93

 I

2423.8

 D

 697.47

I

719.87

 D

99.31

I

 I

3.33

D

 214.69

I

22.4

D

620.56

 D

 1621.48

 I

 41,323.81

 41,528.91

 416.46

205.1

I

211.36

I

192.22

I

642.96

 D

 1000.92

 I

953.6

 D

 I

403.65

I

450.67

 D

 357.96

I

47.32

I

134.22

 D

 41,945.37

 41,932.56

 41,872.73

 41,952.63

 41,759.69

 41,599.72

 41,452.35

 40,817.74

192.84

51.73 634.61

147.37

259.97

92.94

79.9 59.83

12.81

 D

47.02

 D

92.71

 D

 405.28

I

86.9

 D

42.34

I

 D

139.73

I

312.57

 D

 318.38

I

44.56

 D

1108.73

 D

I

172.84

 D

5.81

I

273.82

I

1153.28

 D

 3806.18

 I

 D

167.03

 D

 279.63

 D

112.6

I

599.84

 D

I

879.47

 D

 2652.89

 I

6530.8

 D

 1773.42

 I

3877.91

 D

8138.7

I

 D

487.24

 D

 1173.58

 I

 2104.49

 I

 4260.79

 I

9249.25

 D

*Stock Market Trend Prediction Using Hidden Markov Model*

*DOI: http://dx.doi.org/10.5772/intechopen.93988*

 D

686.84

I

930.91

 D

 2156.3

I

244.57

 D

 1225.39

 I

2882.16

 D I

4988.46

 D

 D

980.82

I

1606.77

 D

 787.98

I

625.95

 D

 41,464.61

 162.03

I

 41,626.64

 **c.v**

 **D in 1 day CV**

 **o.s**

 **D in 2 days CV**

 **o.s**

 **D in3 days CV**

 **o.s**

 **D in 4 day CV**

 **o.s**

 **D in 5 day CV**

 **o.s**

 **D in6 days CV**

 **o.s**


#### **Table 2.** *Daily close value for finding differences in one day, two day, three days, four days, five days, six days close value.*

#### *Stock Market Trend Prediction Using Hidden Markov Model DOI: http://dx.doi.org/10.5772/intechopen.93988*

Probability values of TPM, EPM, and π for difference in two day close value

**S. no Date Close** 01/02/2020 41,626.64 01/03/2020 41,464.61 01/06/2020 40,676.63 01/07/2020 40,869.47 01/08/2020 40,817.74 01/09/2020 41,452.35 01/10/2020 41,599.72 01/13/2020 41,859.69 01/14/2020 41,952.63 01/15/2020 41,872.73 01/16/2020 41,932.56 01/17/2020 41,945.37 01/20/2020 41,528.91 01/21/2020 41,323.81 01/22/2020 41,115.38 01/23/2020 41,386.4 01/24/2020 41,613.19 01/27/2020 41,155.12 01/28/2020 40,966.86 01/29/2020 41,198.66 01/30/2020 40,913.82 01/31/2020 40,723.49

*Forecasting in Mathematics - Recent Advances, New Perspectives and Applications*

Probability values of TPM, EPM, and π for difference in three day close value

*I D*

*S*1 0*:*5 0*:*5 *S*2 0*:*5 0*:*5 *S*3 0*:*4444 0*:*5556 *S*41 0

*I D*

*S*10 1 *S*21 1 *S*3 0*:*6 0*:*4 *S*41 0

*S*1 *S*2 *S*3 *S*4

*S*1 *S*2 *S*3 *S*4

*S*10 0 0 1 *S*20 00*:*75 0*:*25 *S*300*:*6 0*:*2 0*:*2 *S*4 0*:*5 0*:*2 0*:*2 0*:*2

*S*10 0 1 0 *S*20 0 1 0 *S*3 0*:*0111 0 0*:*5555 0*:*3333 *S*400*:*3333 0*:*5 0*:*1667

(**Figure 3** and **Table 4**).

*Daily close value of NSE.*

(**Figure 4** and **Table 5**):

**Table 1.**

**90**

#### *Forecasting in Mathematics - Recent Advances, New Perspectives and Applications*


**Table 3.**

*Transitions with probability values for one day close value.*

**Figure 2.** *Diagram of TPM day 1.*


Probability values of TPM, EPM and π for difference in five days close value

S1 0 0 0 0 0 0 0 1 S2 0 0 0 0 0 0.75 0 0.25 S3 0 0 0.4 0.2 0.2 0 0 0.2 S4 0.5 0 0.2 0 0.2 0 0.2 0

Probability values of TPM, EPM and π for difference in six days close value

**S1 S2 S3 S4 IDI D I D I D**

*I D*

*I D*

*S*10 1 *S*20 1 *S*3 0*:*667 0*:*3333 *S*41 0

*S*10 1 *S*20 1 *S*3 0*:*3333 0*:*6667 *S*41 0

*S*1 *S*2 *S*3 *S*4 *S*1 0*:*1667 0 0*:*1667 0*:*6667 *S*20 0 0 0 *S*3 0 00*:*6667 0*:*3333 *S*4 0*:*7143 0 0 0*:*6667

*Transition table with probability values for difference in three day close value.*

*Stock Market Trend Prediction Using Hidden Markov Model*

*DOI: http://dx.doi.org/10.5772/intechopen.93988*

*S*1 *S*2 *S*3 *S*4

*S*100*:*2 0*:*2 0*:*6 *S*20 0 0 1 *S*3 0*:*6667 0 0*:*3333 0 *S*4 0*:*5 00*:*25 0*:*25

(**Figure 6** and **Table 7**):

(**Figure 7** and **Table 8**):

**93**

**Figure 3.**

**Table 5.**

*Diagram of TPM day 2.*

**Table 4.**

*Transition table with probability values for difference in two day close value.*

Probability values of TPM, EPM and π for difference in four days close value (**Figure 5** and **Table 6**):


*Stock Market Trend Prediction Using Hidden Markov Model DOI: http://dx.doi.org/10.5772/intechopen.93988*

**Figure 3.** *Diagram of TPM day 2.*


#### **Table 5.**

*Transition table with probability values for difference in three day close value.*

Probability values of TPM, EPM and π for difference in five days close value (**Figure 6** and **Table 7**):


Probability values of TPM, EPM and π for difference in six days close value (**Figure 7** and **Table 8**):


Probability values of TPM, EPM and π for difference in four days close value

**S1 S2 S3 S4 ID I D I D I D**

S1 0 0 0 0 0 0 0.5 0.5 S2 0 0 0 0 0.5 0.5 0 0 S3 0 0.111 0 0 0.3333 0.2222 0.1111 0.2222 S4 0 0 0.3333 0 0.5 0 0.1667 0

**S1 S2 S3 S4 IDI D I D I D**

S1 0 0 0 0 1 0 0 0 S2 0 0 0 0 1 0 0 0 S3 0.071 0 0.071 0 0.1429 0.2857 0 0.4286 S4 0 0 0 0.8 0.2 0 0 0

*Forecasting in Mathematics - Recent Advances, New Perspectives and Applications*

*I D S*1 0*:*1429 0*:*9573 *S*2 0*:*5 0*:*5 *S*30 1 *S*41 0

*S*1 *S*2 *S*3 *S*4 *S*1 0*:*3858 0*:*1429 0*:*1429 0*:*4286 *S*2 0*:*500*:*5 0 *S*30 0 1 0 *S*4 0*:*4286 0*:*1429 0 0*:*4286

*Transition table with probability values for difference in two day close value.*

(**Figure 5** and **Table 6**):

**Table 3.**

**Figure 2.**

**Table 4.**

**92**

*Diagram of TPM day 1.*

*Transitions with probability values for one day close value.*


**Table 6.**

*Transition table with probability values for difference in four day close value.*

The various transitions probability values for difference in one day to six days

S1 0 0 0 0.2 0 0.2 0 0.6 S2 0 0 0 0 0 0 0 1 S3 0.3333 0.3333 0 0 0.3333 0 0 0 S4 0.5 0 0 0 0.25 0 0.25 0

**S1 S2 S3 S4 I D ID I D I D**

**S1 S2 S3 S4 I D IDI D I D**

S1 0 0.1667 0 0 0 0.1667 0 0.6667 S2 0 0 0 0 0 0 0 0 S3 0 0 0 0 0 0.6667 0.3333 0 S4 0.7143 0 0 0 0 0 0.2857 0

*Transition table with probability values for difference in five day close value.*

*Stock Market Trend Prediction Using Hidden Markov Model*

*DOI: http://dx.doi.org/10.5772/intechopen.93988*

To generate a random sequence of emission symbols and states are calculated by using the function "Hmmgenerate". The HMM matlab toolbox syntax is: [Sequence, States] = Hmmgenerate(L,TPM,EPM). The length of both sequence and state to be generated is denoted by L [11]. The fitness function used for finding the fitted value

close values are displayed in **Figure 2** to **Figure 7** respectively.

*Transition table with probability values for difference in six day close value.*

Optimum Sequence of States:

**Table 7.**

**Figure 6.**

**Table 8.**

**95**

*Diagram of TPM day 5.*

of sequence of states is defined by

**Figure 5.** *Diagram of TPM day 4.*

*Stock Market Trend Prediction Using Hidden Markov Model DOI: http://dx.doi.org/10.5772/intechopen.93988*


**Table 7.**

**S1 S2 S3 S4 I D I DID I D**

S1 0.1429 0.2429 0 0.1429 0 0.1429 0 0.4286 S2 0.5 0 0 0 0 0.5 0 0 S3 0 0 0 0 0 1 0 0 S4 0.4286 0 0.1429 0 0 0 0.4286 0

*Forecasting in Mathematics - Recent Advances, New Perspectives and Applications*

*Transition table with probability values for difference in four day close value.*

**Table 6.**

**Figure 4.**

*Diagram of TPM day 3.*

**Figure 5.**

**94**

*Diagram of TPM day 4.*

*Transition table with probability values for difference in five day close value.*

**Figure 6.** *Diagram of TPM day 5.*


**Table 8.**

*Transition table with probability values for difference in six day close value.*

The various transitions probability values for difference in one day to six days close values are displayed in **Figure 2** to **Figure 7** respectively.

Optimum Sequence of States:

To generate a random sequence of emission symbols and states are calculated by using the function "Hmmgenerate". The HMM matlab toolbox syntax is: [Sequence, States] = Hmmgenerate(L,TPM,EPM). The length of both sequence and state to be generated is denoted by L [11]. The fitness function used for finding the fitted value of sequence of states is defined by

**4. Conclusion**

*Comparison of six optimum state sequences.*

**Table 9.**

investors.

**Author details**

**97**

and Sonai Muthu Niraikulathan

Abishekapatti, Tamil Nadu, India

provided the original work is properly cited.

Stock prediction is challenging due to its randomness. Hidden Markov Model can be used for stock prediction by finding hidden patterns. Here the Hidden Markov model easily recognized four states of the stock market and also it was used to predict the future values. The highest value in the Optimum State Sequences is the better performance of the particular sequence. Hidden states and sequences have been generated to easily identify the level of the sequence whether the next day value is increasing. And also identified whether the increasing level is moderate high or high or very high and also decreasing level whether moderate low or low or very low. This model will be very much useful for short term as well as long term

**S. no. Comparison of six optimum sequence of states Calculated value Fitness =** <sup>P</sup> **<sup>1</sup>**

 (1,2) + (1,3) + (1,4) 1 1 (2,1) + (2,3) + (2,4) 1.7 0.588 (3,1) + (3,2) + (3,4) 2.425 0.412 (4,1) + (4,2) + (4,3) 3.15 0.32

*Stock Market Trend Prediction Using Hidden Markov Model*

*DOI: http://dx.doi.org/10.5772/intechopen.93988*

*comparision*ð Þ *i***,** *j*

Deneshkumar Venugopal\*, Senthamarai Kannan Kaliyaperumal

\*Address all correspondence to: vdenesh@msuniv.ac.in

Department of Statistics, Manonmaniam Sundaranar University, Tirunelveli,

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

**Figure 7.** *Diagram of TPM day 6.*

$$(\text{Fitness} =) \frac{1}{\sum \text{compare}(i, j)}$$

Using the iterative procedure, for each TPM and EPM framed we get an optimum sequence of states generated.

The length of the sequence taken as L = 4 and the optimum sequence of states obtained from the all six day's differences with TPM and EPM is given in the below and here 'ε' is the start symbol.


Here, the one day difference of TPM and EPM has the shortest path. So the best optimum sequence is found from one day difference in close value. Using the fitness function we compute the fitness value for each of the optimum sequences of states obtained (**Table 9**).

In column four the highest value is the fitness value and the better is the performance of the particular sequence.

*Stock Market Trend Prediction Using Hidden Markov Model DOI: http://dx.doi.org/10.5772/intechopen.93988*


**Table 9.**

<sup>ð</sup>Fitness ¼Þ <sup>1</sup>

*Forecasting in Mathematics - Recent Advances, New Perspectives and Applications*

Using the iterative procedure, for each TPM and EPM framed we get an

The length of the sequence taken as L = 4 and the optimum sequence of states obtained from the all six day's differences with TPM and EPM is given in the below

! D

! I

! I

! D

! I

! D

In column four the highest value is the fitness value and the better is the

S4

S4

s2

S4

S3

S1

Here, the one day difference of TPM and EPM has the shortest path. So the best optimum sequence is found from one day difference in close value. Using the fitness function we compute the fitness value for each of the optimum sequences of states

! I

! I

! D

! I

! D

! I

S3

S4

S3

S3

S2

S3

! D

! D

! D

! D

! D

! D

S4

S3

S1

S4

S4

S4

optimum sequence of states generated.

S4

S1

S4

S1

S3

S4

and here 'ε' is the start symbol.

**Figure 7.**

*Diagram of TPM day 6.*

1. ε ! I

2. ε ! D

3. ε ! I

4. ε ! D

5. ε ! I

6. ε ! I

performance of the particular sequence.

obtained (**Table 9**).

**96**

<sup>P</sup>*compare i*ð Þ , *<sup>j</sup>*

*Comparison of six optimum state sequences.*

#### **4. Conclusion**

Stock prediction is challenging due to its randomness. Hidden Markov Model can be used for stock prediction by finding hidden patterns. Here the Hidden Markov model easily recognized four states of the stock market and also it was used to predict the future values. The highest value in the Optimum State Sequences is the better performance of the particular sequence. Hidden states and sequences have been generated to easily identify the level of the sequence whether the next day value is increasing. And also identified whether the increasing level is moderate high or high or very high and also decreasing level whether moderate low or low or very low. This model will be very much useful for short term as well as long term investors.

#### **Author details**

Deneshkumar Venugopal\*, Senthamarai Kannan Kaliyaperumal and Sonai Muthu Niraikulathan Department of Statistics, Manonmaniam Sundaranar University, Tirunelveli, Abishekapatti, Tamil Nadu, India

\*Address all correspondence to: vdenesh@msuniv.ac.in

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### **References**

[1] Medhi J. Stochastic processes. New Age International; 1994.

[2] Reilly C. Statistics in human genetics and molecular biology. CRC Press; 2009 Jun 19.

[3] Brejová B, Brown DG, Vinaˇr T. ADVANCES IN HIDDEN MARKOV MODELS FOR SEQUENCE. Bioinformatics Algorithms: Techniques and Applications. 2008 Feb 15;3:55.

[4] Gupta A, Dhingra B. Stock market prediction using hidden Markov models. In 2012 Students Conference on Engineering and Systems 2012 Mar 16 (pp. 1-4). IEEE.

[5] Nguyen N. Hidden Markov model for stock trading. International Journal of Financial Studies. 2018 Jun;6(2):36.

[6] Hassan MR, Nath B. Stock market forecasting using hidden Markov model: a new approach. In 5th International Conference on Intelligent Systems Design and Applications (ISDA'05) 2005 Sep 8 (pp. 192-196). IEEE.

[7] Rabiner L. Theory and implementation of hidden Markov models. Fundamentals of speech recognition. 1993.

[8] Rabiner LR. A tutorial on hidden Markov models and selected applications in speech recognition. Proceedings of the IEEE. 1989 Feb;77(2): 257-286.

[9] Lloyd RW. Hidden Markov Models and the Baum-Welch Algorithm. IEEE Information Theory Society Newsletter. 2003 Dec;53(4).

[10] Mandoiu I, Zelikovsky A. Bioinformatics algorithms: techniques and applications. John Wiley & Sons; 2008 Feb 25.

[11] Murphy K. HMM toolbox for Matlab. Internet: http://www.cs.ubc. ca/ murphyk/Software/HMM/hmm. html, [Oct. 29, 2011]. 1998.

**Chapter 6**

*Ismit Mado*

**Abstract**

Indonesia.

**99**

forecasting, time series

**1. Introduction**

Electric Load Forecasting an

Time Series Analysis

Application of Cluster Models

Based on Double Seasonal Pattern

Electricity consumption always changes according to need. This pattern deserves serious attention. Where the electric power generation must be balanced with the demand for electric power on the load side. It is necessary to predict and classify loads to maintain reliable power generation stability. This research proposes a method of forecasting electric loads with double seasonal patterns and classifies electric loads as a cluster group. Double seasonal pattern forecasting fits perfectly with fluctuating loads. Meanwhile, the load cluster pattern is intended to classify seasonal trends in a certain period. The first objective of this research is to propose DSARIMA to predict electric load. Furthermore, the results of the load prediction

are used as electrical load clustering data through a descriptive analytical

**Keywords:** electric loads, DSARIMA model, descriptive analytic, clustering,

Fluctuations in electrical power greatly affect the performance of power generation systems. Changes in electrical power due to variations in demand for electrical power momentarily result in an imbalance of electricity generated by the electric power absorbed. If the power supplied is greater than there will be energy waste. And if the power supplied is smaller then there will be overload which will result in a blackout. This means that the amount of electric power generated must be balanced or not too far from the nominal value of the electrical power requirements at the load center. In fact, the use of electrical energy tends to change at any time.

approach. The best model DSARIMA forecasting is ([1, 2, 5, 6, 7, 11, 16, 18, 35, 46], 1, [1, 3, 13, 21, 27, 46]) (1, 1, 1)<sup>48</sup> (0, 0, 1)336 with a MAPE of 1.56 percent. The cluster pattern consists of four groups with a range of intervals between the minimum and maximum data values divided by the quartile. The presentation of this research data is based on data on the consumption of electricity loads every half hour at the Generating Unit, the National Electricity Company in Gresik City,

#### **Chapter 6**

**References**

Jun 19.

Age International; 1994.

[1] Medhi J. Stochastic processes. New

*Forecasting in Mathematics - Recent Advances, New Perspectives and Applications*

[11] Murphy K. HMM toolbox for Matlab. Internet: http://www.cs.ubc. ca/ murphyk/Software/HMM/hmm. html,

[Oct. 29, 2011]. 1998.

[2] Reilly C. Statistics in human genetics and molecular biology. CRC Press; 2009

[3] Brejová B, Brown DG, Vinaˇr T. ADVANCES IN HIDDEN MARKOV

Bioinformatics Algorithms: Techniques and Applications. 2008 Feb 15;3:55.

[4] Gupta A, Dhingra B. Stock market prediction using hidden Markov models.

[5] Nguyen N. Hidden Markov model for stock trading. International Journal of Financial Studies. 2018 Jun;6(2):36.

[6] Hassan MR, Nath B. Stock market forecasting using hidden Markov model: a new approach. In 5th International Conference on Intelligent Systems Design and Applications (ISDA'05) 2005 Sep 8 (pp. 192-196). IEEE.

[7] Rabiner L. Theory and

recognition. 1993.

257-286.

2003 Dec;53(4).

2008 Feb 25.

**98**

implementation of hidden Markov models. Fundamentals of speech

[8] Rabiner LR. A tutorial on hidden

[9] Lloyd RW. Hidden Markov Models and the Baum-Welch Algorithm. IEEE Information Theory Society Newsletter.

Bioinformatics algorithms: techniques and applications. John Wiley & Sons;

Markov models and selected applications in speech recognition. Proceedings of the IEEE. 1989 Feb;77(2):

[10] Mandoiu I, Zelikovsky A.

In 2012 Students Conference on Engineering and Systems 2012 Mar 16

(pp. 1-4). IEEE.

MODELS FOR SEQUENCE.

## Electric Load Forecasting an Application of Cluster Models Based on Double Seasonal Pattern Time Series Analysis

*Ismit Mado*

### **Abstract**

Electricity consumption always changes according to need. This pattern deserves serious attention. Where the electric power generation must be balanced with the demand for electric power on the load side. It is necessary to predict and classify loads to maintain reliable power generation stability. This research proposes a method of forecasting electric loads with double seasonal patterns and classifies electric loads as a cluster group. Double seasonal pattern forecasting fits perfectly with fluctuating loads. Meanwhile, the load cluster pattern is intended to classify seasonal trends in a certain period. The first objective of this research is to propose DSARIMA to predict electric load. Furthermore, the results of the load prediction are used as electrical load clustering data through a descriptive analytical approach. The best model DSARIMA forecasting is ([1, 2, 5, 6, 7, 11, 16, 18, 35, 46], 1, [1, 3, 13, 21, 27, 46]) (1, 1, 1)<sup>48</sup> (0, 0, 1)336 with a MAPE of 1.56 percent. The cluster pattern consists of four groups with a range of intervals between the minimum and maximum data values divided by the quartile. The presentation of this research data is based on data on the consumption of electricity loads every half hour at the Generating Unit, the National Electricity Company in Gresik City, Indonesia.

**Keywords:** electric loads, DSARIMA model, descriptive analytic, clustering, forecasting, time series

#### **1. Introduction**

Fluctuations in electrical power greatly affect the performance of power generation systems. Changes in electrical power due to variations in demand for electrical power momentarily result in an imbalance of electricity generated by the electric power absorbed. If the power supplied is greater than there will be energy waste. And if the power supplied is smaller then there will be overload which will result in a blackout. This means that the amount of electric power generated must be balanced or not too far from the nominal value of the electrical power requirements at the load center. In fact, the use of electrical energy tends to change at any time.

For this reason, it is necessary to predict the use of electric power that is able to maintain a balance between supply and consumption of electric power in the power generation system. Research of electricity load forecasting is very important in the power plant system operation plan [1]. Load forecasting studies are classified into three categories: long-term, medium-term and short-term predictions. Long-term predictions are needed for planning the peak load capacity and system maintenance schedule [2], medium-term predictions are needed for the planning and operation of the power plant system [3], and short-term predictions are needed to control and schedule the generating system [4]. So that load forecasting studies play a role in ensuring the economic value of financing, system reliability, stability and quality of electricity system services.

15 minutes or 30 minutes is used. In this study, the load data used is 30-minute

*Electric Load Forecasting an Application of Cluster Models Based on Double Seasonal Pattern…*

The definition of the load factor can be written in the following equation: when you are citing sources, the citations should be set in numbered format. All the references given in the list of references should be cited in the body of the text.

*Load factor* <sup>¼</sup> *average load in a certain period*

The load factor can be known from the load curve. As for the estimation of the magnitude of the burden factor in the future, it can be approached with existing

*Ppeak*

If *T* is in a year, an annual expense factor is obtained. If in 1 month the monthly

Daily load factors vary according to the characteristics of the load area, whether it is a dense residential area, industrial area, trade or a combination of various types

This daily load factor will also affect the weather conditions and certain days

Load curves illustrate the variation of loading on a substation measured by KW or KVA as a function of time. Measurement time intervals are usually determined based on the use of measurement results, for example intervals of 30 minutes,

The load curve shows the demand or load requirements at different time inter-

vals. With the help of this load curve, we can determine the magnitude of the

Peak load or maximum demand is defined as the biggest load of needs that occurs during a certain period. Certain periods can be in the form of daily, monthly or annual periods. Furthermore, the peak load must be interpreted as the average load during a certain interval, where the possibility of such load. For example, the daily load of a distribution transformer where the peak load during an interval of 1 hour, ie between 19:00 (point A) and 20:00 (point B). The average value of the

Keep in mind here that peak needs are not instantaneous needs, but on average

The characteristics of the burden between holidays are different from ordinary

during a certain time interval, usually a certain time interval is 15 minutes,

days so that they have different load variants. Load characteristics can also be

largest load and then the generating capacity can also be determined.

� *T*

*Load factor* <sup>¼</sup> *Paverage*

*peak load in a certain period* (1)

*<sup>T</sup>* (2)

Please set citations in square brackets keeping the below points in mind.

When applied to the power plant, it is formulated into

load factor is obtained, as well as the daily load factor.

interval load data.

**2.2 Daily load**

of customers.

**2.3 Load curve**

**2.4 Peak load**

such as holidays and so on.

60 minutes, 1 day or 1 week.

A - B curve is its peak requirement.

30 minutes or 1 hour.

**101**

statistical data as was done in this study.

*DOI: http://dx.doi.org/10.5772/intechopen.93493*

Fluctuations in electrical power at the load center contain a set of time-based information. The characteristics of the load from the period of use both by household, commercial, industrial and public costs, are needed so that fluctuations can be analyzed. The load characteristics, besides being able to be analyzed also contain a series of load patterns tendencies due to usage. This conduct of using electric loads contains seasonal patterns. Daily use tends to recur on certain days, as well as weekly load patterns. This trend is then analyzed through the load cluster approach to achieve load usage patterns based on seasonal patterns.

The Box-Jenkins time series study approach conducted in this research was able to increase the estimated usage and application of seasonal patterns based on electricity load clusters. The time series prediction model is an accurate choice and continues to grow to this day [5–7]. Researchers have carried out load forecasting study activities with 2.06 percent MAPE [8]. In research, the parameter estimation pattern was developed again with the least squares method which is better. And then the load cluster modeling is developed to classify the trend based on seasonal patterns.

#### **2. Electrical load characteristics**

The main purpose of an electric power distribution system is to distribute electric power from substations or sources to a number of customers or loads. The most important main factor in the distribution system planning is the characteristics of various electrical loads.

The electrical load characteristics are needed so that the system voltage, the thermal effect of loading and the loading pattern can be analyzed properly. The analysis is included in determining the initial projections in the next planning.

The characteristics of the electrical load are very dependent on the type of load it serves. This will be clearly seen from the results of recording the load curve in a time interval. The following are several factors that determine the load characteristics according to the needs of this study [9].

#### **2.1 Load factor**

Load factor is the ratio between average load and peak load measured in a certain period. Average load and peak load can be expressed in KiloWatt (KW), KiloVolt-Ampere (KVA) and so on, but the units of both must be the same. Load factor can be calculated for a certain period usually used in units of daily, monthly or yearly.

The peak load referred to in this study is a momentary peak load or average peak load in a certain interval (maximum demand), generally a maximum demand of

*Electric Load Forecasting an Application of Cluster Models Based on Double Seasonal Pattern… DOI: http://dx.doi.org/10.5772/intechopen.93493*

15 minutes or 30 minutes is used. In this study, the load data used is 30-minute interval load data.

The definition of the load factor can be written in the following equation:

when you are citing sources, the citations should be set in numbered format. All the references given in the list of references should be cited in the body of the text. Please set citations in square brackets keeping the below points in mind.

$$Load\,\,factor = \frac{average\,\,load\,\,in\,\,a\,\, certain\,\,period}{peak\,\,load\,\,in\,\,a\,\, certain\,\,period} \tag{1}$$

The load factor can be known from the load curve. As for the estimation of the magnitude of the burden factor in the future, it can be approached with existing statistical data as was done in this study.

When applied to the power plant, it is formulated into

$$Load\ factor = \frac{P\_{average}}{P\_{peak}} \times \frac{T}{T} \tag{2}$$

If *T* is in a year, an annual expense factor is obtained. If in 1 month the monthly load factor is obtained, as well as the daily load factor.

#### **2.2 Daily load**

For this reason, it is necessary to predict the use of electric power that is able to maintain a balance between supply and consumption of electric power in the power generation system. Research of electricity load forecasting is very important in the power plant system operation plan [1]. Load forecasting studies are classified into three categories: long-term, medium-term and short-term predictions. Long-term predictions are needed for planning the peak load capacity and system maintenance schedule [2], medium-term predictions are needed for the planning and operation of the power plant system [3], and short-term predictions are needed to control and schedule the generating system [4]. So that load forecasting studies play a role in ensuring the economic value of financing, system reliability, stability and quality of

*Forecasting in Mathematics - Recent Advances, New Perspectives and Applications*

Fluctuations in electrical power at the load center contain a set of time-based information. The characteristics of the load from the period of use both by household, commercial, industrial and public costs, are needed so that fluctuations can be analyzed. The load characteristics, besides being able to be analyzed also contain a series of load patterns tendencies due to usage. This conduct of using electric loads contains seasonal patterns. Daily use tends to recur on certain days, as well as weekly load patterns. This trend is then analyzed through the load cluster approach

The Box-Jenkins time series study approach conducted in this research was able to increase the estimated usage and application of seasonal patterns based on electricity load clusters. The time series prediction model is an accurate choice and continues to grow to this day [5–7]. Researchers have carried out load forecasting study activities with 2.06 percent MAPE [8]. In research, the parameter estimation pattern was developed again with the least squares method which is better. And then the load cluster modeling is developed to classify the trend based on seasonal

The main purpose of an electric power distribution system is to distribute electric power from substations or sources to a number of customers or loads. The most important main factor in the distribution system planning is the characteristics of

The electrical load characteristics are needed so that the system voltage, the thermal effect of loading and the loading pattern can be analyzed properly. The analysis is included in determining the initial projections in the next planning.

Load factor is the ratio between average load and peak load measured in a certain period. Average load and peak load can be expressed in KiloWatt (KW), KiloVolt-Ampere (KVA) and so on, but the units of both must be the same. Load factor can be calculated for a certain period usually used in units of daily, monthly

The peak load referred to in this study is a momentary peak load or average peak load in a certain interval (maximum demand), generally a maximum demand of

The characteristics of the electrical load are very dependent on the type of load it serves. This will be clearly seen from the results of recording the load curve in a time interval. The following are several factors that determine the load characteris-

to achieve load usage patterns based on seasonal patterns.

electricity system services.

patterns.

**2. Electrical load characteristics**

tics according to the needs of this study [9].

various electrical loads.

**2.1 Load factor**

or yearly.

**100**

Daily load factors vary according to the characteristics of the load area, whether it is a dense residential area, industrial area, trade or a combination of various types of customers.

This daily load factor will also affect the weather conditions and certain days such as holidays and so on.

#### **2.3 Load curve**

Load curves illustrate the variation of loading on a substation measured by KW or KVA as a function of time. Measurement time intervals are usually determined based on the use of measurement results, for example intervals of 30 minutes, 60 minutes, 1 day or 1 week.

The load curve shows the demand or load requirements at different time intervals. With the help of this load curve, we can determine the magnitude of the largest load and then the generating capacity can also be determined.

#### **2.4 Peak load**

Peak load or maximum demand is defined as the biggest load of needs that occurs during a certain period. Certain periods can be in the form of daily, monthly or annual periods. Furthermore, the peak load must be interpreted as the average load during a certain interval, where the possibility of such load. For example, the daily load of a distribution transformer where the peak load during an interval of 1 hour, ie between 19:00 (point A) and 20:00 (point B). The average value of the A - B curve is its peak requirement.

Keep in mind here that peak needs are not instantaneous needs, but on average during a certain time interval, usually a certain time interval is 15 minutes, 30 minutes or 1 hour.

The characteristics of the burden between holidays are different from ordinary days so that they have different load variants. Load characteristics can also be

distinguished by the factor of loading outside the time of the peak load, or who are at the time of the peak load. So we need load forecasting with the aim of preparing operating generating units. When electricity demand increases, it will be balanced with adequate electricity supply to prevent power outages, otherwise if electricity consumption decreases, electricity supply will be reduced so as not to over supply.

**3.2 ARIMA Box-Jenkins procedure**

*DOI: http://dx.doi.org/10.5772/intechopen.93493*

partial autocorrelation function

iteration, as follows:

v. Forecasting.

system or process being studied.

**3.4 Parameter approximation**

the sum of squared residuals.

Decreasing gradually/bumpy (until lag *q*

is still different from zero)

*PACF and ACF patterns.*

**Table 1.**

**103**

correct the iterative approximation.

There are two basic ways to get this parameter:

**3.3 Identification**

The prediction procedure of ARIMA Box-Jenkins model through five stages of

*Electric Load Forecasting an Application of Cluster Models Based on Double Seasonal Pattern…*

ii. Identification of ARIMA model through autocorrelation function and

Identification requires calculation and general review of the results of the autocorrelation function (ACF) and the parisal autocorrelation function (PACF). The results of these calculations are needed to determine the appropriate ARIMA model, whether ARIMA ð Þ *p*, 0, 0 or AR ð Þ *p* , ARIMA 0, 0, ð Þ*q* or MA ð Þ*q* , ARIMA ð Þ *p*, 0, *q* or ARMA ð Þ *p*, *q* , ARIMA ð Þ *p*, *d*, *q* . Meanwhile, to determine the presence or absence of the *d* model value, it is determined by the data itself. If the data form is stationary, *d* is 0, while the data form is not stationary, the value of *d* is not equal to 0 ð Þ *d*> 0 . Likewise, the dual seasonal ARIMA model also refers to the autocorrelation function (ACF) and partial autocorrelation function (PACF) as well as knowledge of the

Identification can be done after fixed time series data. The application of the model after ACF and PACF data has a tendency according to the reference to **Table 1** and for the seasonal data patterns determined by referring to **Table 2** [11].

a. By trial and error, test several different values and choose one of these values (or a set of values, if more than one parameter is estimated) that minimizes

b. Iterative approach, choosing an initial estimate and then letting the computer

**parameters**

ARIMA ð Þ *p*, *d*, *q*

**ACF patterns PACF patterns ARIMA**

Heading to zero after lag *q* Decreasing gradually/bumpy ARIMA 0, ð Þ *d*, *q* Decreasing gradually/bumpy Heading to zero after lag *q* ARIMA ð Þ *p*, *d*, 0

Decreasing gradually/bumpy (until lag *q*

is still different from zero)

i. Preparation of data, including checking of data stationary

iii. Estimation of ARIMA model parameters: *p*, *d*, and *q*

iv. Determination of ARIMA model equations

#### **3. Electrical load analysis based on time series model**

Box and Jenkins popularized the use of ARIMA models and the Box-Jenkins methodology became highly popular in the 1970s among academics [10]. The ARIMA model is also called the Box-Jenkins time series. A time series is a series of observations taken sequentially based on time [11]. The observation process is carried out at the same interval, for example in hour, daily, weekly, monthly, yearly or other intervals. The purpose of time series analysis is twofold, namely to model the stochastic mechanism found in observations based on time and to predict the value of observations in the future. The value of a variable can be predicted if the nature of the variable is known in the present and in the past.

#### **3.1 ARIMA model classification**

The ARIMA model is divided into several groups, namely: autoregressive (AR), moving average (MA), and ARMA. The ARIMA model is a nonstationary ARMA model that has gone through a differencing process so that it becomes a stationary model. The ARIMA model also contains seasonal patterns. Defined as a pattern that repeats in a fixed time interval. The application of this seasonal pattern has been developed into a double seasonal pattern [12–14]. Double seasonal ARIMA model is written with notation, as follows.

$$\text{ARIMA} \ (p, d, q) (P\_1, D\_1, Q\_1)^{S\_1} (P\_2, D\_2, Q\_2)^{S\_2} \tag{3}$$

This model consists of two components, namely the first level which is usually developed from a linear forecasting model to explain seasonal trends from data or known as potential load. And at the second level developed from the ARIMA model to capture autoregressive patterns from data or called irregular loads. For stationary data, the seasonal factor can be determined by identifying the coefficient of autocorrelation at two or three time intervals that are very different from zero. So that this seasonal pattern can be identified whether it contains a tendency to have a seasonal pattern or multiple seasonal patterns and has the following general form [15]:

$$\Phi\_p(\mathcal{B})\Phi\_{\mathbb{P}\_1}(\mathcal{B}^{\epsilon\_1})\Phi\_{\mathbb{P}\_2}(\mathcal{B}^{\epsilon\_1})(\mathbb{1}-\mathcal{B})^d(\mathbb{1}-\mathcal{B}^{\epsilon\_1})^{D\_1}(\mathbb{1}-\mathcal{B}^{\epsilon\_1})^{D\_2}\mathbb{Z}\_t = \theta\_q(\mathcal{B})\Theta\_{\mathbb{Q}\_1}(\mathcal{B}^{\epsilon\_1})\Theta\_{\mathbb{Q}\_2}(\mathcal{B}^{\epsilon\_2})a\_{\mathbb{P}\_2} \tag{4}$$

With

$$
\begin{split}
\phi\_{p}(B) &= \mathbf{1} - \phi\_{1}B - \phi\_{2}B^{2} - \dots - \phi\_{p}B^{p} \\
\Phi\_{\mathcal{P}\_{1}}(B^{\prime\_{1}}) &= \mathbf{1} - \Phi\_{1\_{1}}B^{\prime\_{1}} - \Phi\_{2\_{1}}B^{2s\_{1}} - \dots - \Phi\_{\mathcal{P}\_{1}}B^{\mathcal{P}\_{1}s\_{1}} \\
\Phi\_{\mathcal{P}\_{2}}(B^{\prime\_{2}}) &= \mathbf{1} - \Pi\_{1\_{2}}B^{\prime\_{2}} - \Pi\_{2\_{2}}B^{2s\_{2}} - \dots - \Pi\_{\mathcal{P}\_{2}}B^{\mathcal{P}\_{2}s\_{2}} \\
\theta\_{q}(B) &= \mathbf{1} - \theta\_{1}B - \theta\_{2}B^{2} - \dots - \theta\_{q}B^{q} \\
\Theta\_{Q\_{1}}(B^{\prime\_{1}}) &= \mathbf{1} - \Theta\_{1\_{1}}B^{\prime\_{1}} - \Theta\_{2\_{1}}B^{2s\_{1}} - \dots - \Theta\_{Q\_{1}}B^{Q\_{1}s\_{1}} \\
\Theta\_{Q\_{2}}(B^{\prime\_{2}}) &= \mathbf{1} - \Psi\_{1\_{2}}B^{\prime\_{2}} - \Psi\_{2\_{2}}B^{2s\_{1}} - \dots - \Psi\_{Q\_{2}}B^{Q\_{2}s\_{2}}.
\end{split}
$$

*Electric Load Forecasting an Application of Cluster Models Based on Double Seasonal Pattern… DOI: http://dx.doi.org/10.5772/intechopen.93493*

#### **3.2 ARIMA Box-Jenkins procedure**

The prediction procedure of ARIMA Box-Jenkins model through five stages of iteration, as follows:


#### **3.3 Identification**

distinguished by the factor of loading outside the time of the peak load, or who are at the time of the peak load. So we need load forecasting with the aim of preparing operating generating units. When electricity demand increases, it will be balanced with adequate electricity supply to prevent power outages, otherwise if electricity consumption decreases, electricity supply will be reduced so as not to over supply.

*Forecasting in Mathematics - Recent Advances, New Perspectives and Applications*

Box and Jenkins popularized the use of ARIMA models and the Box-Jenkins methodology became highly popular in the 1970s among academics [10]. The ARIMA model is also called the Box-Jenkins time series. A time series is a series of observations taken sequentially based on time [11]. The observation process is carried out at the same interval, for example in hour, daily, weekly, monthly, yearly or other intervals. The purpose of time series analysis is twofold, namely to model the stochastic mechanism found in observations based on time and to predict the value of observations in the future. The value of a variable can be predicted if the

The ARIMA model is divided into several groups, namely: autoregressive (AR), moving average (MA), and ARMA. The ARIMA model is a nonstationary ARMA model that has gone through a differencing process so that it becomes a stationary model. The ARIMA model also contains seasonal patterns. Defined as a pattern that repeats in a fixed time interval. The application of this seasonal pattern has been developed into a double seasonal pattern [12–14]. Double seasonal ARIMA model is

This model consists of two components, namely the first level which is usually developed from a linear forecasting model to explain seasonal trends from data or known as potential load. And at the second level developed from the ARIMA model to capture autoregressive patterns from data or called irregular loads. For stationary data, the seasonal factor can be determined by identifying the coefficient of autocorrelation at two or three time intervals that are very different from zero. So that this seasonal pattern can be identified whether it contains a tendency to have a seasonal pattern or multiple seasonal patterns and has the following general form [15]:

*ARIMA p*ð Þ , *<sup>d</sup>*, *<sup>q</sup> <sup>P</sup>*1, *<sup>D</sup>*1, *<sup>Q</sup>*<sup>1</sup> ð Þ*<sup>S</sup>*<sup>1</sup> *<sup>P</sup>*2, *<sup>D</sup>*2, *<sup>Q</sup>*<sup>2</sup> ð Þ*<sup>S</sup>*<sup>2</sup> (3)

*Zt* <sup>¼</sup> *<sup>θ</sup>q*ð Þ *<sup>B</sup>* <sup>Θ</sup>Q1 *Bs*<sup>1</sup> ð ÞΘQ2 *<sup>B</sup><sup>s</sup>*<sup>2</sup> ð Þ*at*

(4)

**3. Electrical load analysis based on time series model**

nature of the variable is known in the present and in the past.

*<sup>ϕ</sup>p*ð Þ *<sup>B</sup>* <sup>Φ</sup>*<sup>P</sup>*<sup>1</sup> *Bs*<sup>1</sup> ð ÞΦ*<sup>P</sup>*<sup>2</sup> *Bs*<sup>2</sup> ð Þð Þ <sup>1</sup> � *<sup>B</sup> <sup>d</sup>* <sup>1</sup> � *Bs*<sup>1</sup> ð Þ*<sup>D</sup>*<sup>1</sup> <sup>1</sup> � *Bs*<sup>2</sup> ð Þ*<sup>D</sup>*<sup>2</sup>

<sup>Φ</sup>*<sup>P</sup>*<sup>1</sup> *Bs*<sup>1</sup> ð Þ¼ <sup>1</sup> � <sup>Φ</sup>11*Bs*<sup>1</sup> � <sup>Φ</sup>21*B*<sup>2</sup>*s*<sup>1</sup> � … � <sup>Φ</sup>*<sup>P</sup>*1*B<sup>P</sup>*1*s*<sup>1</sup> <sup>Φ</sup>*<sup>P</sup>*<sup>2</sup> *Bs*<sup>2</sup> ð Þ¼ <sup>1</sup> � <sup>Π</sup>12*B<sup>s</sup>*<sup>2</sup> � <sup>Π</sup>22*B*<sup>2</sup>*s*<sup>2</sup> � … � <sup>Π</sup>*<sup>P</sup>*2*BP*2*s*<sup>2</sup>

<sup>Θ</sup>*<sup>Q</sup>*<sup>1</sup> *Bs*<sup>1</sup> ð Þ¼ <sup>1</sup> � <sup>Θ</sup>11*Bs*<sup>1</sup> � <sup>Θ</sup>21*B*<sup>2</sup>*s*<sup>1</sup> � … � <sup>Θ</sup>*<sup>Q</sup>*1*BQ*1*s*<sup>1</sup> <sup>Θ</sup>*<sup>Q</sup>*<sup>2</sup> *Bs*<sup>2</sup> ð Þ¼ <sup>1</sup> � <sup>Ψ</sup>12*Bs*<sup>2</sup> � <sup>Ψ</sup>22*B*<sup>2</sup>*s*<sup>2</sup> � … � <sup>Ψ</sup>*<sup>Q</sup>*2*BQ*2*s*<sup>2</sup> .

*<sup>ϕ</sup>p*ð Þ¼ *<sup>B</sup>* <sup>1</sup> � *<sup>ϕ</sup>*1*<sup>B</sup>* � *<sup>ϕ</sup>*2*B*<sup>2</sup> � … � *<sup>ϕ</sup>pBp*

*<sup>θ</sup>q*ð Þ¼ *<sup>B</sup>* <sup>1</sup> � *<sup>θ</sup>*1*<sup>B</sup>* � *<sup>θ</sup>*2*B*<sup>2</sup> � … � *<sup>θ</sup>qB<sup>q</sup>*

**3.1 ARIMA model classification**

written with notation, as follows.

With

**102**

Identification requires calculation and general review of the results of the autocorrelation function (ACF) and the parisal autocorrelation function (PACF). The results of these calculations are needed to determine the appropriate ARIMA model, whether ARIMA ð Þ *p*, 0, 0 or AR ð Þ *p* , ARIMA 0, 0, ð Þ*q* or MA ð Þ*q* , ARIMA ð Þ *p*, 0, *q* or ARMA ð Þ *p*, *q* , ARIMA ð Þ *p*, *d*, *q* . Meanwhile, to determine the presence or absence of the *d* model value, it is determined by the data itself. If the data form is stationary, *d* is 0, while the data form is not stationary, the value of *d* is not equal to 0 ð Þ *d*> 0 . Likewise, the dual seasonal ARIMA model also refers to the autocorrelation function (ACF) and partial autocorrelation function (PACF) as well as knowledge of the system or process being studied.

Identification can be done after fixed time series data. The application of the model after ACF and PACF data has a tendency according to the reference to **Table 1** and for the seasonal data patterns determined by referring to **Table 2** [11].

#### **3.4 Parameter approximation**

There are two basic ways to get this parameter:




#### **Table 2.**

*PACF and ACF seasonal patterns.*

#### **3.5 Parameter testing**

Parameter testing phase is to test whether the selection of parameters p, d, q is true and correct. The model is said to be good if the error value is random, meaning that it no longer has a certain pattern. In other words, the model obtained can capture well the existing data patterns. To see the error value of the test carried out testing the value of the autocorrelation coefficient of the error, using one of the following two statistics:

1.Q Box dan Pierce Test

$$Q = n' \sum\_{k=1}^{m} r\_k^2 \tag{5}$$

(RMSE), Mean Absolute Error (MEA) and Mean Absolute Percentage Error

P*<sup>n</sup> <sup>i</sup>*¼<sup>1</sup> *<sup>Z</sup>*�*Z*^*<sup>i</sup> Zi* � � �

offorecasting result. MAPE is defined as follows [13]

*DOI: http://dx.doi.org/10.5772/intechopen.93493*

predicted values.

variance values.

model.

**105**

**4.1 Parameter identification**

**3.9 Electric load cluster modeling**

as mode, media, and mean or mean count.

**4. DSARIMA-based load forecasting**

MAPE ¼

surement of central tendency, and measurement of variability [17].

distribution classes, distribution in groups, and distribution charts.

(MAPE). In this research. MAPE is used as a standard measurement of the accuracy

*Electric Load Forecasting an Application of Cluster Models Based on Double Seasonal Pattern…*

*n*

Where *Zi* and *Z*^*<sup>i</sup>* is the actual and predicted values, while *n* is the number of

Cluster analysis performed in this study refers to the statistical description of the analysis technique. Descriptive statistics are methods relating to the collection and presentation of a group of data so as to provide useful information [16]. This description analysis includes several things, namely: frequency distribution, mea-

The data that has been obtained from a study which is still in the form of random data that can be made into grouped data is data that has been arranged into certain classes. Lists containing grouped data are called frequency distributions or frequency tables. Frequency distribution is the arrangement of data according to certain interval classes or according to certain categories in a list. Frequency distribution can be presented in groups, distribution based on rank order or ranking of

Measuring central tendency is a statistical analysis that specifically describes a representative score. The central tendency shows the location of the largest part of the value in the distribution including a general description of data frequencies such

While the measurement of variability to describe the degree of dispersion of quantitative data. This measure consists of interquartile range, quartile deviation, mean deviation, standard deviation and coefficient of variation, and variance. Measurement of variability serves to determine the homogeneity or heterogeneity of data. A data may have the same central tendency value but have different

The data used in this study is the consumption of electric power every 30 minutes during January 2, 2009 to November 19, 2011 in the Generating Unit

The data is distributed on: 1. Data for training during January 2, 2009 to November 12, 2011, 2. Data for testing with the assumption of real data compared to

Statistical Analysis System (SAS) is used as a simulation of electricity load forecasting and Minitab programming is used to analyze the electricity load cluster

To identify data, the first step that must be taken is to plot the time series of the data. The time series plot is displayed to see the data patterns and stationarity of the

service, the National Electricity Company in Gresik City, Indonesia.

training data from forecasting results during November 13–19, 2011.

� � �

� 100% (8)

2.Ljung-Box Test

$$Q = n'(n'+2) \sum\_{k=1}^{m} \frac{r\_k^2}{(n'-k)}\tag{6}$$

Spread by chi squared *<sup>χ</sup>*<sup>2</sup> ð Þ with free degrees ð Þ¼ *db* ð Þ *<sup>m</sup>* � *<sup>p</sup>* � *<sup>q</sup>* � *<sup>P</sup>* � *<sup>Q</sup>* Where

$$n' = n - (d + \text{SD}) \tag{7}$$

#### **3.6 Testing criteria**

If *<sup>Q</sup>* <sup>≤</sup>*χ*<sup>2</sup>ð Þ *<sup>α</sup>*, *db* , meaning: error value is random (model is accepted) If *<sup>Q</sup>* <sup>&</sup>gt;*χ*<sup>2</sup>ð Þ *<sup>α</sup>*, *db* , meaning: error value is not random (model cannot be accepted

#### **3.7 Parameter estimation**

This study uses the least squares method in estimating parameters [15]. The ARIMA model parameters are based on the time series observed with *Z*1, *Z*2, … , *Z*1. The quadratic method assumes that the best curve is the curve that has the least square error of the data set. The parameter values of the ARIMA models *p*, *d*, and *q* are determined through the stationary ACF and PACF chart plots.

#### **3.8 Measuring accuracy level of forecasting result**

Basically, to measure the accuracy of forecasting result can be done by various methods. Some statistical methods such as as Root Mean Square Error *Electric Load Forecasting an Application of Cluster Models Based on Double Seasonal Pattern… DOI: http://dx.doi.org/10.5772/intechopen.93493*

(RMSE), Mean Absolute Error (MEA) and Mean Absolute Percentage Error (MAPE). In this research. MAPE is used as a standard measurement of the accuracy offorecasting result. MAPE is defined as follows [13]

$$\text{MAPE} = \frac{\sum\_{i=1}^{n} \left| \frac{z - \bar{Z}\_i}{Z\_i} \right|}{n} \times 100\text{\%} \tag{8}$$

Where *Zi* and *Z*^*<sup>i</sup>* is the actual and predicted values, while *n* is the number of predicted values.

#### **3.9 Electric load cluster modeling**

**3.5 Parameter testing**

*PACF and ACF seasonal patterns.*

ARMA ð Þ *p*, *q*

**Table 2.**

following two statistics:

2.Ljung-Box Test

Where

**104**

**3.6 Testing criteria**

**3.7 Parameter estimation**

1.Q Box dan Pierce Test

Parameter testing phase is to test whether the selection of parameters p, d, q is true and correct. The model is said to be good if the error value is random, meaning that it no longer has a certain pattern. In other words, the model obtained can capture well the existing data patterns. To see the error value of the test carried out testing the value of the autocorrelation coefficient of the error, using one of the

MA ð Þ*<sup>q</sup>* Cut off after lag *<sup>q</sup><sup>s</sup>* Dies down (decreases exponentially) in

*Forecasting in Mathematics - Recent Advances, New Perspectives and Applications*

**Model ACF PACF**

Dies down (decreases exponentially) in

AR ð Þ *p* Dies down (decreases exponentially) in

seasonal lags

seasonal lags

*Q* ¼ *n*<sup>0</sup>

*<sup>Q</sup>* <sup>¼</sup> *<sup>n</sup>*<sup>0</sup> *<sup>n</sup>*<sup>0</sup> ð Þ <sup>þ</sup> <sup>2</sup> <sup>X</sup>*<sup>m</sup>*

If *<sup>Q</sup>* <sup>≤</sup>*χ*<sup>2</sup>ð Þ *<sup>α</sup>*, *db* , meaning: error value is random (model is accepted)

are determined through the stationary ACF and PACF chart plots.

**3.8 Measuring accuracy level of forecasting result**

If *<sup>Q</sup>* <sup>&</sup>gt;*χ*<sup>2</sup>ð Þ *<sup>α</sup>*, *db* , meaning: error value is not random (model cannot be accepted

This study uses the least squares method in estimating parameters [15]. The ARIMA model parameters are based on the time series observed with *Z*1, *Z*2, … , *Z*1. The quadratic method assumes that the best curve is the curve that has the least square error of the data set. The parameter values of the ARIMA models *p*, *d*, and *q*

Basically, to measure the accuracy of forecasting result can be done by various methods. Some statistical methods such as as Root Mean Square Error

Spread by chi squared *<sup>χ</sup>*<sup>2</sup> ð Þ with free degrees ð Þ¼ *db* ð Þ *<sup>m</sup>* � *<sup>p</sup>* � *<sup>q</sup>* � *<sup>P</sup>* � *<sup>Q</sup>*

X*m k*¼1 *r* 2

*k*¼1

*r*2 *k*

Cut off after lag *ps*

Dies down (decreases exponentially) in

seasonal lags

seasonal lags

*n*<sup>0</sup> ¼ *n* � ð Þ *d* þ *SD* (7)

*<sup>k</sup>* (5)

*<sup>n</sup>*ð Þ <sup>0</sup> � *<sup>k</sup>* (6)

Cluster analysis performed in this study refers to the statistical description of the analysis technique. Descriptive statistics are methods relating to the collection and presentation of a group of data so as to provide useful information [16]. This description analysis includes several things, namely: frequency distribution, measurement of central tendency, and measurement of variability [17].

The data that has been obtained from a study which is still in the form of random data that can be made into grouped data is data that has been arranged into certain classes. Lists containing grouped data are called frequency distributions or frequency tables. Frequency distribution is the arrangement of data according to certain interval classes or according to certain categories in a list. Frequency distribution can be presented in groups, distribution based on rank order or ranking of distribution classes, distribution in groups, and distribution charts.

Measuring central tendency is a statistical analysis that specifically describes a representative score. The central tendency shows the location of the largest part of the value in the distribution including a general description of data frequencies such as mode, media, and mean or mean count.

While the measurement of variability to describe the degree of dispersion of quantitative data. This measure consists of interquartile range, quartile deviation, mean deviation, standard deviation and coefficient of variation, and variance. Measurement of variability serves to determine the homogeneity or heterogeneity of data. A data may have the same central tendency value but have different variance values.

#### **4. DSARIMA-based load forecasting**

The data used in this study is the consumption of electric power every 30 minutes during January 2, 2009 to November 19, 2011 in the Generating Unit service, the National Electricity Company in Gresik City, Indonesia.

The data is distributed on: 1. Data for training during January 2, 2009 to November 12, 2011, 2. Data for testing with the assumption of real data compared to training data from forecasting results during November 13–19, 2011.

Statistical Analysis System (SAS) is used as a simulation of electricity load forecasting and Minitab programming is used to analyze the electricity load cluster model.

#### **4.1 Parameter identification**

To identify data, the first step that must be taken is to plot the time series of the data. The time series plot is displayed to see the data patterns and stationarity of the data which aims to determine the ARIMA model. The pattern of data as shown in **Figure 1** is very volatile. This condition is likely influenced by the integrated power distribution system in the Java-Madura-Bali Indonesia interconnection system.

When referring to **Figure 1(a)**, it can be seen that the data are not stationary in variance or mean. For more details, it will be seen in the autocorrelation function as shown in **Figure 2**. And if it refers to time series patterns there is a tendency for the data to contain seasonal patterns as shown in **Figure 1(b)**.

The data is not stationary in the variance, so it is necessary to transform the data as follows. Testing stationarity in variance if the *p*-value or *λ* ¼ 1. Based on the results of the transformation, the data is not stationary in the variance marked with the value *λ* ¼ �0*:*13 as shown in **Figure 3a**. After going through the process of transformation the data becomes significant with the value *λ* ¼ 1 as shown in **Figure 3b**.

After the data is transformed it will be transformed back to get the active data value, as follows

*Z* ∗

*Zt* <sup>¼</sup> *<sup>Z</sup>* <sup>∗</sup> *t* �<sup>100</sup>

Then

**Figure 3.**

method requires data that is stationary.

*(a) Box-Cox transformation; (b) after transformation.*

*DOI: http://dx.doi.org/10.5772/intechopen.93493*

seasonal lags, ie lags 336, 672, etc.

**Figure 4.**

**107**

*<sup>t</sup>* <sup>¼</sup> *<sup>Z</sup>*�0,13

*Electric Load Forecasting an Application of Cluster Models Based on Double Seasonal Pattern…*

The data is stationary in variance, but the transformation results in **Figure 2b** are not stationary in the mean. Data has not shown a constant value in the middle. The stationarity of the data can also be seen through the plot of the autocorrelation function (ACF). From **Figure 2**, it can be seen that the coefficient of autocorrelation is significantly different from zero and slowly decreases. The pattern shows that the data is not stationary in particular not stationary in the mean, while the ARIMA

The ACF plot also shows that there are strong indications of having a seasonal pattern in both daily and weekly seasonal averages as shown in **Figure 4**, below. In **Figure 4a**, it can be seen that the electricity load data has a seasonal pattern that is the daily seasonal as seen in lags 48, 96, 144, etc. And in **Figure 4b**, the data

Because the data is not stationary in the mean, it is necessary to do differencing

Based on the ACF plot in **Figure 5**, it appears that the nonseasonal data has been stationary. However, seasonal plots are still not stationary with an indication that ACF is still falling slowly in daily seasonal lags, ie lags 48, 96, 144, etc., and weekly

ð Þ *d* ¼ 1 . The ACF plot of differencing data results is shown in **Figure 5** below.

also contains weekly seasonal as seen in lag 336, 672, 1008, 1344, etc.

*ACF plots with seasonal patterns: (a) daily seasonal; (b) weekly seasonal.*

*<sup>t</sup>* (9)

<sup>13</sup> (10)

**Figure 1.**

*(a) Data plot of electricity usage every 30 minutes during January 2, 2009 to November 12, 2011; (b) plot of electrical load data with seasonal patterns (red box).*

**Figure 2.** *ACF plot data.*

*Electric Load Forecasting an Application of Cluster Models Based on Double Seasonal Pattern… DOI: http://dx.doi.org/10.5772/intechopen.93493*

**Figure 3.** *(a) Box-Cox transformation; (b) after transformation.*

$$Z\_t^\* := Z\_t^{-0,13} \tag{9}$$

Then

data which aims to determine the ARIMA model. The pattern of data as shown in **Figure 1** is very volatile. This condition is likely influenced by the integrated power distribution system in the Java-Madura-Bali Indonesia interconnection system.

*Forecasting in Mathematics - Recent Advances, New Perspectives and Applications*

data to contain seasonal patterns as shown in **Figure 1(b)**.

**Figure 3b**.

**Figure 1.**

**Figure 2.** *ACF plot data.*

**106**

*electrical load data with seasonal patterns (red box).*

value, as follows

When referring to **Figure 1(a)**, it can be seen that the data are not stationary in variance or mean. For more details, it will be seen in the autocorrelation function as shown in **Figure 2**. And if it refers to time series patterns there is a tendency for the

The data is not stationary in the variance, so it is necessary to transform the data

After the data is transformed it will be transformed back to get the active data

*(a) Data plot of electricity usage every 30 minutes during January 2, 2009 to November 12, 2011; (b) plot of*

as follows. Testing stationarity in variance if the *p*-value or *λ* ¼ 1. Based on the results of the transformation, the data is not stationary in the variance marked with the value *λ* ¼ �0*:*13 as shown in **Figure 3a**. After going through the process of transformation the data becomes significant with the value *λ* ¼ 1 as shown in

$$Z\_t = \left(Z\_t^\*\right)^{-\frac{100}{13}}\tag{10}$$

The data is stationary in variance, but the transformation results in **Figure 2b** are not stationary in the mean. Data has not shown a constant value in the middle. The stationarity of the data can also be seen through the plot of the autocorrelation function (ACF). From **Figure 2**, it can be seen that the coefficient of autocorrelation is significantly different from zero and slowly decreases. The pattern shows that the data is not stationary in particular not stationary in the mean, while the ARIMA method requires data that is stationary.

The ACF plot also shows that there are strong indications of having a seasonal pattern in both daily and weekly seasonal averages as shown in **Figure 4**, below.

In **Figure 4a**, it can be seen that the electricity load data has a seasonal pattern that is the daily seasonal as seen in lags 48, 96, 144, etc. And in **Figure 4b**, the data also contains weekly seasonal as seen in lag 336, 672, 1008, 1344, etc.

Because the data is not stationary in the mean, it is necessary to do differencing ð Þ *d* ¼ 1 . The ACF plot of differencing data results is shown in **Figure 5** below.

Based on the ACF plot in **Figure 5**, it appears that the nonseasonal data has been stationary. However, seasonal plots are still not stationary with an indication that ACF is still falling slowly in daily seasonal lags, ie lags 48, 96, 144, etc., and weekly seasonal lags, ie lags 336, 672, etc.

**Figure 4.** *ACF plots with seasonal patterns: (a) daily seasonal; (b) weekly seasonal.*

**Figure 5.** *ACF plot after differencing d*ð Þ ¼ 1*:*

It is necessary to do differencing data once more in the seasonal pattern ð Þ *d* ¼ 1, *D* ¼ 1, *s* ¼ 48 . After going through seasonal differencing there are strong indications that the data patterns have been stationary.

Based on the ACF plot for differencing ð Þ *d* ¼ 1, *D* ¼ 1, *s* ¼ 48 it is clear that the data as a whole has been stationary in the mean. The nonseasonal data plot has been stationary in lags 1, 2, 3, … , 40. The data pattern tends to dies down and will be cuts off after lag 7 and lag 8 in **Figure 6a**.

The ACF plot for seasonal patterns *s* ¼ 48 after differencing has also been stationary at lags 48, 96, 144, etc. The data pattern tends to be cuts off after lag 48 in **Figure 6b**. The seasonal pattern *s* ¼ 336 tends to be cuts off after lag 336 in **Figure 6c**.

For PACF plots both seasonal ð Þ *s* ¼ 48 and ð Þ *s* ¼ 336 dies down as shown in **Figure 6d**. Based on the provisions in **Tables 1** and **2**, the parameter identification results can be rewritten in the following **Table 3**.

The ACF and PACF data plots are stationary, the alleged nonseasonal ARIMA models are in accordance with the stationary topology in **Table 1** and the seasonal ARIMA in **Table 2**. The temporary model of ARIMA provisional model is double seasonal based on **Table 3** is DSARIMA 1, 1, 1 ð Þð Þ 0, 1, 1 <sup>48</sup>ð Þ 0, 0, 1 <sup>336</sup> . However, there is a possibility that white noise has not been fulfilled, so it is necessary to add or change the order in accordance with the test.

#### **4.2 Parameter estimation**

AR and MA coefficients in the DSARIMA model are estimated using the least squares method. The initial estimate that has been obtained is used as the initial value of the estimation method iteratively. Obtained initial estimates of AR and MA coefficients from the interim model DSARIMA (1, 1, 1) (0, 1, 1)<sup>48</sup> (0, 0, 1)<sup>336</sup> as shown in **Table 4** in the following.

assumptions which include the white noise assumption and meet the independent

**Models ACF PACF Estimated parameters** Nonseasonal Dies down Dies down ARMA 1, 1 ð Þ Seasonal ð Þ *<sup>s</sup>* <sup>¼</sup> <sup>48</sup> Cuts off Dies down MA 1ð Þ<sup>48</sup> Seasonal ð Þ *<sup>s</sup>* <sup>¼</sup> <sup>336</sup> Dies down Dies down MA 1ð Þ<sup>336</sup>

*Electric Load Forecasting an Application of Cluster Models Based on Double Seasonal Pattern…*

**Parameter Estimate Standard error** *t* **value Approx** Pr **>** j j*t* **Lag** MA 1.1 �0.35184 0.01899 �18.53 <0.0001 1 MA 2.1 0>95734 0.0013007 736.02 <0.0001 48 MA 3.1 �0.04526 0.0045103 �10.03 <0.0001 336 AR 1.1 �0.14,578 0.02006 �7.27 <0.0001 1

Ljung-Box Test is used to check the assumption of independence from residuals

*H*<sup>0</sup> : *ρ*<sup>1</sup> ¼ *ρ*<sup>2</sup> ¼ … ¼ *ρ<sup>K</sup>* ¼ 0

*H*1: there is at least one *ρ<sup>i</sup>* that is not equal to zero for *i* ¼ 1, 2, … , *K*

criteria and are normally distributed 0, *<sup>σ</sup>*<sup>2</sup> ð Þ.

*ACF and PACF plot after differencing d*ð Þ ¼ 1, *D* ¼ 1, *s* ¼ 48

*DOI: http://dx.doi.org/10.5772/intechopen.93493*

with the following hypotheses:

*An output SAS of model with CLS iterative.*

*Identification plots for ACF and PACF.*

**Figure 6.**

**Table 3.**

**Table 4.**

**109**

Based on **Table 4**, AR and MA parameters have met the criteria for white noise with a p-value greater than the error tolerance value α = 5%, with an alpha significance level of less than 0.0001. However, it is necessary to re-test the residual

*Electric Load Forecasting an Application of Cluster Models Based on Double Seasonal Pattern… DOI: http://dx.doi.org/10.5772/intechopen.93493*

#### **Figure 6.**

It is necessary to do differencing data once more in the seasonal pattern ð Þ *d* ¼ 1, *D* ¼ 1, *s* ¼ 48 . After going through seasonal differencing there are strong

*Forecasting in Mathematics - Recent Advances, New Perspectives and Applications*

Based on the ACF plot for differencing ð Þ *d* ¼ 1, *D* ¼ 1, *s* ¼ 48 it is clear that the data as a whole has been stationary in the mean. The nonseasonal data plot has been stationary in lags 1, 2, 3, … , 40. The data pattern tends to dies down and will be cuts

The ACF plot for seasonal patterns *s* ¼ 48 after differencing has also been stationary at lags 48, 96, 144, etc. The data pattern tends to be cuts off after lag 48 in **Figure 6b**. The seasonal pattern *s* ¼ 336 tends to be cuts off after lag 336 in

For PACF plots both seasonal ð Þ *s* ¼ 48 and ð Þ *s* ¼ 336 dies down as shown in **Figure 6d**. Based on the provisions in **Tables 1** and **2**, the parameter identification

The ACF and PACF data plots are stationary, the alleged nonseasonal ARIMA models are in accordance with the stationary topology in **Table 1** and the seasonal ARIMA in **Table 2**. The temporary model of ARIMA provisional model is double seasonal based on **Table 3** is DSARIMA 1, 1, 1 ð Þð Þ 0, 1, 1 <sup>48</sup>ð Þ 0, 0, 1 <sup>336</sup> . However, there is a possibility that white noise has not been fulfilled, so it is necessary to add or

AR and MA coefficients in the DSARIMA model are estimated using the least squares method. The initial estimate that has been obtained is used as the initial value of the estimation method iteratively. Obtained initial estimates of AR and MA coefficients from the interim model DSARIMA (1, 1, 1) (0, 1, 1)<sup>48</sup> (0, 0, 1)<sup>336</sup> as

Based on **Table 4**, AR and MA parameters have met the criteria for white noise with a p-value greater than the error tolerance value α = 5%, with an alpha significance level of less than 0.0001. However, it is necessary to re-test the residual

indications that the data patterns have been stationary.

results can be rewritten in the following **Table 3**.

change the order in accordance with the test.

**4.2 Parameter estimation**

shown in **Table 4** in the following.

off after lag 7 and lag 8 in **Figure 6a**.

*ACF plot after differencing d*ð Þ ¼ 1*:*

**Figure 6c**.

**108**

**Figure 5.**

*ACF and PACF plot after differencing d*ð Þ ¼ 1, *D* ¼ 1, *s* ¼ 48


#### **Table 3.**

*Identification plots for ACF and PACF.*


#### **Table 4.**

*An output SAS of model with CLS iterative.*

assumptions which include the white noise assumption and meet the independent criteria and are normally distributed 0, *<sup>σ</sup>*<sup>2</sup> ð Þ.

Ljung-Box Test is used to check the assumption of independence from residuals with the following hypotheses:

$$H\_0: \rho\_1 = \rho\_2 = \dots = \rho\_K = \mathbf{0}$$

*H*1: there is at least one *ρ<sup>i</sup>* that is not equal to zero for *i* ¼ 1, 2, … , *K*

With an error tolerance of 5%, *H*<sup>0</sup> is rejected if the *ρ*-value <*α*, which means the residual does not meet the assumption of white noise. The initial residual tests are shown in **Table 5** below.

Based on the estimated AR and MA coefficient parameters in **Table 5**, the residual normal probability plot must meet the assumption of white noise with a limit of<sup>&</sup>lt; � <sup>1</sup>*:*96ffiffi *<sup>n</sup>* <sup>p</sup> ≈ � 0*:*009, where n as many as 50,160 training data. Then based on the initial estimation results in **Table 5**, it is necessary to estimate to meet the white noise assumption, namely by including an estimate on the lag 2, 3, 4, 5, 7, 8, 9, 11, 16, 17, 18, 19, 20, 21, 22, 23, 27, 29, 30, 31, 46, 47, and 48. The results of the residual check are shown in **Table 6** below. The estimation results are significant for seasonal lag, which is lag 48.

Based on residual checking, namely by adding and subtracting AR and MA parameters, it can be seen that all lags have met the assumption of white noise with a limit of <sup>&</sup>lt; � 1, 96ffiffi *<sup>n</sup>* <sup>p</sup> ≈ � 0, 009 (see ACF Results). The best iteration results of the AR and MA parameters are shown in **Table 7** below.

Based on **Table 7**, the DSARIMA model is obtained with the coefficients ð Þ ½ � 1, 2, 5, 6, 7, 11, 16, 18, 35, 46 , 1, 1, 3, 13, 21, 27, 46 ½ � ð Þ 1, 1, 1 <sup>48</sup>ð Þ 0, 0, 1 336, which have met the assumption of white noise.


**4.3 Electrical load forecasting results**

*DOI: http://dx.doi.org/10.5772/intechopen.93493*

*An output SAS of model with CLS iterative.*

(2.1) = 0.98, MA (3.1) = � 0.0364.

model equation as follows:

**Table 7.**

**111**

Based on the final results of the estimated parameters in **Table 4** the ARIMA

**Parameter Estimate Standard error** *t* **value Approx** Pr **>** j j*t* **Lag** MA 1.1 0.934 0.01770 52.78 <0.0001 1 MA 1.2 �0.077 0.0072138 �10.64 <0.0001 3 MA 1.3 0.008 0.0038171 2.18 0.0293 13 MA 1.4 0.00685 0.0031724 2.16 0.0309 21 MA 1.5 0.017 0.0027856 5.92 <0.0001 27 MA 1.6 0.059 0.0067600 8.67 <0.0001 46 MA 2.1 0.98 0.0009744 1003.38 <0.0001 48 MA 3.1 �0.0364 0.0045572 �7.98 <0.0001 336 AR 1.1 1.1464 0.01855 61.81 <0.0001 1 AR 1.2 �0.295 0.0087427 �33.79 <0.0001 2 AR 1.3 �0.0104 0.0052195 �2.00 0.0454 5 AR 1.4 0.0189 0.0067496 2.80 0.0051 6 AR 1.5 �0.0234 0.0047509 �4.93 <0.0001 7 AR 1.6 �0.004 0.030582 �1.29 0.1958 11 AR 1.7 �0.0083 0.0033299 �2.49 0.0126 16 AR 1.8 �0.0125 0.0033252 �3.77 0.0002 18 AR 1.9 �0.007 0.0022520 �3.26 0.0011 35 AR 1.10 0.07 0.0067089 10.62 <0.0001 46 AR 2.1 0.03 0.0050410 5.86 <0.0001 48

*Electric Load Forecasting an Application of Cluster Models Based on Double Seasonal Pattern…*

(1.2) = � 0.295, AR (1.3) = � 0.0104, AR (1, 4) = 0.0189, AR (1.5) = � 0.0234, AR (1.6) = � 0.004, AR (1.7) = � 0.0083, AR (1.8) = � 0.0125, AR (1.9) = � 0.0074, AR (1.10) = 0.07, AR (2.1) = 0.03, MA (1.1) = 0.934, MA (1.2) = � 0.077, MA (1.3) = 0.008, MA (1.4) = 0.00685, MA (1.5) = 0.017, MA (1.6) = 0.059, MA

<sup>1</sup> � <sup>1</sup>*:*1464*<sup>B</sup>* <sup>þ</sup> <sup>0</sup>*:*295*B*<sup>2</sup> <sup>þ</sup> <sup>0</sup>*:*0104*B*<sup>5</sup> � <sup>0</sup>*:*0189*B*<sup>6</sup> <sup>þ</sup> <sup>0</sup>*:*0234*B*<sup>7</sup> <sup>þ</sup> <sup>0</sup>*:*004*B*<sup>11</sup>

<sup>1</sup> � <sup>0</sup>*:*934*<sup>B</sup>* <sup>þ</sup> <sup>0</sup>*:*077*B*<sup>3</sup> � <sup>0</sup>*:*008*B*<sup>13</sup> � <sup>0</sup>*:*00685*B*<sup>21</sup> � <sup>0</sup>*:*017*B*<sup>27</sup> �0*:*059*B*<sup>46</sup> <sup>1</sup> � <sup>0</sup>*:*98*B*<sup>48</sup> <sup>1</sup> <sup>þ</sup> <sup>0</sup>*:*0364*B*<sup>336</sup> *at*

After going through a reverse transformation *Zt* electrical load for the compari-

son of predicted results with actual data (testing) in **Figure 7** below.

*<sup>t</sup>* ¼

<sup>þ</sup>0*:*0083*B*<sup>16</sup> <sup>þ</sup> <sup>0</sup>*:*0125*B*<sup>18</sup> <sup>þ</sup> <sup>0</sup>*:*0074*B*<sup>35</sup> � <sup>0</sup>*:*07*B*<sup>46</sup> <sup>1</sup> � <sup>0</sup>*:*03*B*<sup>48</sup> *<sup>Z</sup>* <sup>∗</sup>

Based on the prediction model parameters obtained DSARIMA models ð Þ ½ � 1, 2, 5, 6, 7, 11, 16, 18, 35, 46 , 1, 1, 3, 13, 21, 27, 46 ½ � ð Þ 1, 1, 1 <sup>48</sup> ð Þ 0, 0, 1 <sup>336</sup> with the

coefficient parameters are obtained as follows: AR (1.1) = 1.1464, AR

**Table 5.** *An output SAS of model with ACF check of residuals.*


#### **Table 6.**

*An output SAS of model with ACF check of residuals.*


*Electric Load Forecasting an Application of Cluster Models Based on Double Seasonal Pattern… DOI: http://dx.doi.org/10.5772/intechopen.93493*

**Table 7.** *An output SAS of model with CLS iterative.*

#### **4.3 Electrical load forecasting results**

Based on the final results of the estimated parameters in **Table 4** the ARIMA coefficient parameters are obtained as follows: AR (1.1) = 1.1464, AR (1.2) = � 0.295, AR (1.3) = � 0.0104, AR (1, 4) = 0.0189, AR (1.5) = � 0.0234, AR (1.6) = � 0.004, AR (1.7) = � 0.0083, AR (1.8) = � 0.0125, AR (1.9) = � 0.0074, AR (1.10) = 0.07, AR (2.1) = 0.03, MA (1.1) = 0.934, MA (1.2) = � 0.077, MA (1.3) = 0.008, MA (1.4) = 0.00685, MA (1.5) = 0.017, MA (1.6) = 0.059, MA (2.1) = 0.98, MA (3.1) = � 0.0364.

Based on the prediction model parameters obtained DSARIMA models ð Þ ½ � 1, 2, 5, 6, 7, 11, 16, 18, 35, 46 , 1, 1, 3, 13, 21, 27, 46 ½ � ð Þ 1, 1, 1 <sup>48</sup> ð Þ 0, 0, 1 <sup>336</sup> with the model equation as follows:

$$(1 - 1.1464B + 0.295B^2 + 0.0104B^5 - 0.0189B^6 + 0.0234B^7 + 0.004B^{11})$$

$$+ 0.0083B^{16} + 0.0125B^{18} + 0.0074B^{35} - 0.07B^{46}) \\ \left(1 - 0.03B^{48}\right) \\ Z\_t^\* = 0$$

$$\left(1 - 0.934B + 0.077B^3 - 0.008B^{13} - 0.00685B^{21} - 0.017B^{27}\right)$$

$$- 0.059B^{46} \\ \left(1 - 0.98B^{48}\right) \\ \left(1 + 0.0364B^{336}\right)a\_t$$

After going through a reverse transformation *Zt* electrical load for the comparison of predicted results with actual data (testing) in **Figure 7** below.

With an error tolerance of 5%, *H*<sup>0</sup> is rejected if the *ρ*-value <*α*, which means the residual does not meet the assumption of white noise. The initial residual tests are

*<sup>n</sup>* <sup>p</sup> ≈ � 0*:*009, where n as many as 50,160 training data. Then based on

*<sup>n</sup>* <sup>p</sup> ≈ � 0, 009 (see ACF Results). The best iteration results of the AR

Based on the estimated AR and MA coefficient parameters in **Table 5**, the residual normal probability plot must meet the assumption of white noise with a

*Forecasting in Mathematics - Recent Advances, New Perspectives and Applications*

the initial estimation results in **Table 5**, it is necessary to estimate to meet the white noise assumption, namely by including an estimate on the lag 2, 3, 4, 5, 7, 8, 9, 11, 16, 17, 18, 19, 20, 21, 22, 23, 27, 29, 30, 31, 46, 47, and 48. The results of the residual check are shown in **Table 6** below. The estimation results are significant for sea-

Based on residual checking, namely by adding and subtracting AR and MA parameters, it can be seen that all lags have met the assumption of white noise with

Based on **Table 7**, the DSARIMA model is obtained with the coefficients ð Þ ½ � 1, 2, 5, 6, 7, 11, 16, 18, 35, 46 , 1, 1, 3, 13, 21, 27, 46 ½ � ð Þ 1, 1, 1 <sup>48</sup>ð Þ 0, 0, 1 336, which have

6 153.39 2 <0.0001 �0.002 �0.019 �0.041 �0.017 �0.028 �0.008 12 274.15 8 <0.0001 �0.033 �0.027 �0.0114 �0.009 �0.014 �0.007 18 342.13 14 <0.0001 �0.009 �0.009 �0.008 �0.017 �0.016 �0.023 24 422>74 20 <0.0001 �0.023 �0.018 �0.020 �0.011 �0.013 �0.003 30 43.05 26 <0.0001 �0.009 �0.008 �0.017 �0.008 �0.017 �0.014 36 489.03 32 <0.0001 �0.011 �0.009 �0.002 0.000 �0.010 0.000 42 497.60 38 <0.0001 �0.007 �0.008 0.002 �0.005 �0.004 0.003 48 804.03 44 <0.0001 0.001 0.002 0.006 0.018 0.044 0.060

6 — 0 — 0.000 0.000 �0.002 0.004 0.001 0.002 12 — 0 — �0.005 �0.002 0.008 0.000 �0.007 �0.004 18 — 0 — 0.005 0.001 �0.008 0.001 �0.003 �0.003 24 18>10 5 0.0028 �0.007 �0.000 0.001 0.003 �0.001 0.002 30 24.78 11 0.0098 �0.005 �0.005 �0.002 0.009 0.0011 �0.001 36 31.03 17 0.0198 �0.005 �0.004 �0.000 �0.001 �0.004 0.008 42 33.77 23 0.0686 �0.000 �0.005 0.004 0.000 0.000 0.004 48 37.61 29 0.1314 0.005 0.006 �0.002 �0.002 0.003 �0.000

**To Lag ChiSq DF Pr > ChiSq ACF results**

**To Lag ChiSq DF Pr > ChiSq ACF results**

shown in **Table 5** below.

sonal lag, which is lag 48.

and MA parameters are shown in **Table 7** below.

met the assumption of white noise.

*An output SAS of model with ACF check of residuals.*

*An output SAS of model with ACF check of residuals.*

a limit of <sup>&</sup>lt; � 1, 96ffiffi

**Table 5.**

**Table 6.**

**110**

limit of<sup>&</sup>lt; � <sup>1</sup>*:*96ffiffi

**Figure 7.** *Comparison of actual power with forecast power.*

#### **4.4 Model testing and measuring forecasting accuracy**

Accuracy testing between actual power data and prediction results. Test using the MAPE procedure and obtained at 1.56 percent.

#### **5. Electric load modeling**

The application of descriptive analytic methods in this book is presented to obtain significant information in managing optimal electrical energy as the author did [18]. Through frequency distribution, data can be arranged based on certain criteria. Data categories are presented based on rank orders that contain ranking data from the top or highest load to the lowest data value.

#### **5.1 Data distribution forecasting results**

This electricity load forecasting data is a usage data for a week at intervals every half hour measurement at the power generation. This electricity load forecasting data sample is 336 (N = 336) with mean of 370.56 MWh, meaning that the value is centered at 370.526 MWh. Standard deviation of 36.2582 or the value of this deviation is not too large, this shows the diversity of data is not too large, which means the data is homogeneous.

Visualizations in other forms can be displayed in the form of boxplot graphics. **Figure 9** shows of range (in a box) every hour of measurement and the average

*Electric Load Forecasting an Application of Cluster Models Based on Double Seasonal Pattern…*

*DOI: http://dx.doi.org/10.5772/intechopen.93493*

**Figure 9** shows that data tend to be at the minimum level, first quartile and the median value. Electricity load increases at third quartile intervals and the maximum

Each measurement of electric power absorption at the load center has a peak load. Based on the measurement data, it can be seen that the peak power load absorption occurs at 19:00 and generally the peak load tendency occurs at

value line of every half hour of measurement.

that hour.

**113**

**Figure 9.**

*Graph forecast boxplot.*

**Figure 8.**

*Plot data forecasting.*

load. This condition occurs between 18:30 until 21:30 at night.

Furthermore, forecasting the data shown in the time measurements every halfhour of electric power consumption in the load center in **Figure 8** below.

*Electric Load Forecasting an Application of Cluster Models Based on Double Seasonal Pattern… DOI: http://dx.doi.org/10.5772/intechopen.93493*

**Figure 8.** *Plot data forecasting.*

**4.4 Model testing and measuring forecasting accuracy**

*Forecasting in Mathematics - Recent Advances, New Perspectives and Applications*

data from the top or highest load to the lowest data value.

**5.1 Data distribution forecasting results**

the MAPE procedure and obtained at 1.56 percent.

**5. Electric load modeling**

*Comparison of actual power with forecast power.*

**Figure 7.**

the data is homogeneous.

**112**

Accuracy testing between actual power data and prediction results. Test using

The application of descriptive analytic methods in this book is presented to obtain significant information in managing optimal electrical energy as the author did [18]. Through frequency distribution, data can be arranged based on certain criteria. Data categories are presented based on rank orders that contain ranking

This electricity load forecasting data is a usage data for a week at intervals every half hour measurement at the power generation. This electricity load forecasting data sample is 336 (N = 336) with mean of 370.56 MWh, meaning that the value is centered at 370.526 MWh. Standard deviation of 36.2582 or the value of this deviation is not too large, this shows the diversity of data is not too large, which means

Furthermore, forecasting the data shown in the time measurements every half-

hour of electric power consumption in the load center in **Figure 8** below.

**Figure 9.** *Graph forecast boxplot.*

Visualizations in other forms can be displayed in the form of boxplot graphics. **Figure 9** shows of range (in a box) every hour of measurement and the average value line of every half hour of measurement.

**Figure 9** shows that data tend to be at the minimum level, first quartile and the median value. Electricity load increases at third quartile intervals and the maximum load. This condition occurs between 18:30 until 21:30 at night.

Each measurement of electric power absorption at the load center has a peak load. Based on the measurement data, it can be seen that the peak power load absorption occurs at 19:00 and generally the peak load tendency occurs at that hour.


**Table 8.**

*Daily data samples.*

Henceforth processing this distribution data through seasonal data that can be presented in the form of daily data, as follows.

Furthermore, seasonal electricity load data on a daily scale can be restated in the

**N = 336**

**Clusters Interval range Frequency** Min–Q1 97 Q1–Median 83 Median–Q3 86 Q3–Max 70

*Electric Load Forecasting an Application of Cluster Models Based on Double Seasonal Pattern…*

In descriptive analysis, frequency distribution, measurement of central tendencies and measurement of variability can be presented in the frequency distribution graph. The purpose of the presentation and information provided in addition to being able to describe the tendency of the data to form certain patterns, this analysis can also be used as a reference for changes in electric power in the power generation

It can be seen that the data sample with N = 336 has an average of 370.53 MWh which means that the centralized data distribution is rated median. Standard deviation of 36.26 or the value of this deviation is not too large, this shows the diversity

Quartile intervals that divide data over median values form a cluster pattern,

An important aspect of this data sample analysis is the presentation of data with seasonal variants. Data development by taking into account the seasonal variants of the hours and daily helped to optimize the management and operational decisions

One of the research trends in electrical engineering is time series analysis. This research includes forecasting studies and modeling of electrical load clusters. The time series analysis method is very suitable with the characteristics of the electrical load that is always fluctuating. This method is also able to produce different data or

of data is not too large, which means the data is homogeneous.

with the distribution of data presented in **Table 9** below

of the generating system both in scheduling and controlling.

The degree of data dispersion can be determined based on the range of interquartile intervals that indicate the homogeneity of the data. In this study, the electrical load cluster is defined as the range of quartile intervals to median value or

form of **Table 8** below.

system.

**Table 9.**

**6. Conclusion**

**115**

**5.2 Predicted cluster data**

*Range of clusters in the data variant.*

*DOI: http://dx.doi.org/10.5772/intechopen.93493*

is shown in the electrical load data below.

not included in the training data process.

The sample data used is Friday data and then the data will be presented in **Table 8** below.

**Friday's electricity load data**—samples of electric load data are 48 (N = 48) with mean of 375.143 MWh, meaning that the value is centered at 375.143 MWh. Standard deviation of 35.4253 or the value of this deviation is not too large, this shows the diversity of data is not too large, which means the data is homogeneous.

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On Friday shown in **Figure 10**, the peak load occurred at 19:00 amounting to 444.234 MWh with a minimum electric absorption range of 327.509 MWh. On Friday, the data has mean of 375.143 MWh.

**Figure 10.** *Data plot on Friday.*

*Electric Load Forecasting an Application of Cluster Models Based on Double Seasonal Pattern… DOI: http://dx.doi.org/10.5772/intechopen.93493*


**Table 9.**

Henceforth processing this distribution data through seasonal data that can be

**No Days Mean StDev Median Minimum Peak Load Time of peak load** 1 Friday 375.143 35.4253 375.832 327.509 444.234 19:00 2 Saturday 373.635 36.2699 375.208 325.378 445.746 19:00 3 Sunday 361.193 36.8101 357.005 312.912 438.985 19:00 4 Monday 368.672 36.5413 368.417 320.639 440.478 19:00 5 Tuesday 370.616 36.2821 371.793 321.685 439.731 19:30 6 Wednesday 371.619 36.3137 372.718 322.735 441.976 19:00 7 Thursday 372.806 36.6616 374.899 323.262 442.727 19:30

*Forecasting in Mathematics - Recent Advances, New Perspectives and Applications*

The sample data used is Friday data and then the data will be presented in

**Friday's electricity load data**—samples of electric load data are 48 (N = 48) with mean of 375.143 MWh, meaning that the value is centered at 375.143 MWh. Standard deviation of 35.4253 or the value of this deviation is not too large, this shows the diversity of data is not too large, which means the data is homogeneous.

On Friday shown in **Figure 10**, the peak load occurred at 19:00 amounting to 444.234 MWh with a minimum electric absorption range of 327.509 MWh. On

presented in the form of daily data, as follows.

Friday, the data has mean of 375.143 MWh.

**Table 8** below.

**Figure 10.** *Data plot on Friday.*

**114**

*Daily data samples.*

**Table 8.**

*Range of clusters in the data variant.*

Furthermore, seasonal electricity load data on a daily scale can be restated in the form of **Table 8** below.

#### **5.2 Predicted cluster data**

In descriptive analysis, frequency distribution, measurement of central tendencies and measurement of variability can be presented in the frequency distribution graph. The purpose of the presentation and information provided in addition to being able to describe the tendency of the data to form certain patterns, this analysis can also be used as a reference for changes in electric power in the power generation system.

The degree of data dispersion can be determined based on the range of interquartile intervals that indicate the homogeneity of the data. In this study, the electrical load cluster is defined as the range of quartile intervals to median value or is shown in the electrical load data below.


It can be seen that the data sample with N = 336 has an average of 370.53 MWh which means that the centralized data distribution is rated median. Standard deviation of 36.26 or the value of this deviation is not too large, this shows the diversity of data is not too large, which means the data is homogeneous.

Quartile intervals that divide data over median values form a cluster pattern, with the distribution of data presented in **Table 9** below

An important aspect of this data sample analysis is the presentation of data with seasonal variants. Data development by taking into account the seasonal variants of the hours and daily helped to optimize the management and operational decisions of the generating system both in scheduling and controlling.

#### **6. Conclusion**

One of the research trends in electrical engineering is time series analysis. This research includes forecasting studies and modeling of electrical load clusters. The time series analysis method is very suitable with the characteristics of the electrical load that is always fluctuating. This method is also able to produce different data or not included in the training data process.

For the purposes of this electrical load research, forecasting study using the DSARIMA method is an appropriate choice. This method accurately considers the seasonal parameters of the electricity load with MAPE of 1.56 percent when compared with the actual data.

Whereas the modeling of electrical load clusters based on descriptive analytic methods, obtained knowledge of the dynamics of electrical loads. The electrical load pattern has seasonal characteristics at daily and weekly intervals. This pattern forms a unique load characteristic at all times.

So, forecasting studies and modeling of electricity load clusters are able to answer the challenges of electricity energy utilization policies and the operation of generating systems that are able to maintain the balance of supply and demand.

#### **Nomenclature**


#### **Greek symbols**


**Author details**

University Borneo Tarakan, Tarakan City, Indonesia

provided the original work is properly cited.

\*Address all correspondence to: ismitmado@borneo.ac.id

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

*Electric Load Forecasting an Application of Cluster Models Based on Double Seasonal Pattern…*

*DOI: http://dx.doi.org/10.5772/intechopen.93493*

Ismit Mado

**117**

#### **Abbreviations**


*Electric Load Forecasting an Application of Cluster Models Based on Double Seasonal Pattern… DOI: http://dx.doi.org/10.5772/intechopen.93493*

### **Author details**

For the purposes of this electrical load research, forecasting study using the DSARIMA method is an appropriate choice. This method accurately considers the seasonal parameters of the electricity load with MAPE of 1.56 percent when com-

*Forecasting in Mathematics - Recent Advances, New Perspectives and Applications*

Whereas the modeling of electrical load clusters based on descriptive analytic methods, obtained knowledge of the dynamics of electrical loads. The electrical load pattern has seasonal characteristics at daily and weekly intervals. This pattern forms

So, forecasting studies and modeling of electricity load clusters are able to answer the challenges of electricity energy utilization policies and the operation of generating systems that are able to maintain the balance of supply and demand.

pared with the actual data.

**Nomenclature**

*Z* ∗

a unique load characteristic at all times.

*T* period of time (hours)

*D*1, *D*2, *d* order of differences

*m* maximum lag time

*α<sup>t</sup>* white noise

MAPE mean absolute percentage error

Q1 quartile 1 Q3 quartile 3

**Greek symbols**

, <sup>Θ</sup>*<sup>Q</sup>*<sup>2</sup> *BS*<sup>2</sup>

, Φ*<sup>P</sup>*<sup>2</sup> *BS*<sup>2</sup>

**Abbreviations**

MWh mega watt hours

Θ*<sup>Q</sup>*<sup>1</sup> *BS*<sup>1</sup>

Φ*<sup>P</sup>*<sup>1</sup> *BS*<sup>1</sup>

**116**

*S* number of period per season

*rk* autocorrelation or time-lag , 2, 3, … , *k Zt* time series process in period *T*

*<sup>t</sup>* forecasting process in transformation in period *T*

*λ* Box-Cox transformation number

*θq*ð Þ *B* regular MA polynomials of order *q*

 MA polynomials of orders *φp*ð Þ *B* regular AR polynomials of orders *p*

AR polynomials of orders

*Paverage* average load in period *T* (watts) *Ppeak* peak load in the *T* (watts) *p*, *d*, *q* nonseasonal parts of the model *P*, *D*, *Q* seasonal parts of the model *S*1, *S*<sup>2</sup> 1st and 2nd period seasonal

> Ismit Mado University Borneo Tarakan, Tarakan City, Indonesia

\*Address all correspondence to: ismitmado@borneo.ac.id

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### **References**

[1] Tsekouras GJ, Dialynas EN, Hatziargyriou ND, Kavatza S. A onlinier multivariable regression model for midterm energy forecasting of power systems. Electric Power Systems Research. 2007;**77**(12):1560-1568

[2] McSharry PE, Bouwman S, Bloemhof G. Probabilistic forecasts of the magnitude and timing of peak electricity demand. IEEE Transactions on Power Systems. 2005;**20**(2):1166- 1172

[3] Gonzalez-Romera E, Jaramillo-Moran MA, Carmona-Fernandez D. Monthly electric energy demand forecasting based on trend extraction. IEEE Transactions on Power Systems. 2006;**21**(4):1946-1953

[4] Taylor JW, Mc Sharry PE. Shortterm load forecasting methods: An evaluation based on european data. IEEE Transactions on Power Systems. 2007;**22**(4):2213-2219

[5] Hahn H, Meyer-Nieberg S, Pickl S. Electric load forecasting methods: Tool for decision making. European Journal of Operational Research. 2009;**199**(3): 902-907

[6] Nia MM, Din J, Lam HY, Panagopoulos AD. Stochastic approach to a rain attenuation time series synthesizer for heavy rain regions. International Journal of Electrical and Computer Engineering. 2016;**6**(5):2379

[7] Kamley S, Jaloree S, Thakur RS. Performance forecasting of share market using machine learning techniques: A review. International Journal of Electrical and Computer Engineering. 2016;**6**(6):2088-8708

[8] Mado I, Soeprijanto A, Suhartono S. Applying of double seasonal ARIMA model for electrical power demand forecasting at PT. PLN Gresik Indonesia. International Journal of Electrical and Computer Engineering. 2018;**8**(6):4892- 4901. ISSN: 2088-8708. DOI: 10.11591/ ijece.v8i6

[17] Wiyono BB. Statistik Pendidikan, Buku Ajar Mata Kuliah Statistik. 2001

*DOI: http://dx.doi.org/10.5772/intechopen.93493*

*Electric Load Forecasting an Application of Cluster Models Based on Double Seasonal Pattern…*

[18] Mado I, Soeprijanto A, Suhartono S. Electrical load adat clustering in PJB UP Gresik based on time series analysis approach. In: Proceedings of the International Conference on Vocational Education and Electrical Engineering. Surabaya, Indonesia; 2015. pp. 261-266. Available from: http://digilib.unimed.ac.

id/23841/1/Fulltext.pdf

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[9] Suswanto D. Karakteristik beban tenaga listrik. In: Suswanto D, editor. Sistem Distribusi Tenaga Listrik. Indonesia: Departement of Electrical Engineering, Universitas Negeri Padang; 2009

[10] Makridakis S, Hibon M. ARMA models and the Box-Jenkins methodology. Journal of Forecasting. 1997;**16**(3):147-163

[11] Wei WW. Time Series Analysis: Univariante and Multivariante Methods. Vol. 2. Boston, MA: Pearson Addison Wesley; 2006;1:1:108-109

[12] Soares LJ, Medeiros MC. Modeling and forecasting short-term electricity load: A comparison of methods with an application to Brazilian data. International Journal of Forecasting. 2008;**24**(4):630-644

[13] Mohamed N, Ahmad MH, Ismail Z, Suhartono S. Short term load forecasting using double seasonal ARIMA model. In: Proceedings of the Regional Conference on Statistical Sciences. Vol. 10. 2010. pp. 57-73

[14] Kim SY, Jung HW, Park JD, Baek SM, Kim WS, Chon KH, et al. Weekly maximum electric load forecasting for 104 weeks by seasonal ARIMA model. Journal of the Korean Institute of Illuminating and Electrical Installation Engineers. 2014;**28**(1):50-56

[15] Cryer JD, Chan KS. Time Series Analysis:With Application. R. Springer Science & Business Media; 2008;**7**:154-158

[16] Walpole RE. Pengantar Statistika. Jakarta: PT Gramedia Pustaka Utama; 1993. ISBN 979-403-313-8

*Electric Load Forecasting an Application of Cluster Models Based on Double Seasonal Pattern… DOI: http://dx.doi.org/10.5772/intechopen.93493*

[17] Wiyono BB. Statistik Pendidikan, Buku Ajar Mata Kuliah Statistik. 2001

**References**

1172

[1] Tsekouras GJ, Dialynas EN, Hatziargyriou ND, Kavatza S. A onlinier multivariable regression model for midterm energy forecasting of power systems. Electric Power Systems Research. 2007;**77**(12):1560-1568

G. Probabilistic forecasts of the magnitude and timing of peak

[2] McSharry PE, Bouwman S, Bloemhof

International Journal of Electrical and Computer Engineering. 2018;**8**(6):4892- 4901. ISSN: 2088-8708. DOI: 10.11591/

[9] Suswanto D. Karakteristik beban tenaga listrik. In: Suswanto D, editor. Sistem Distribusi Tenaga Listrik. Indonesia: Departement of Electrical Engineering, Universitas Negeri

[10] Makridakis S, Hibon M. ARMA

methodology. Journal of Forecasting.

[11] Wei WW. Time Series Analysis: Univariante and Multivariante Methods. Vol. 2. Boston, MA: Pearson Addison

[12] Soares LJ, Medeiros MC. Modeling and forecasting short-term electricity load: A comparison of methods with an

International Journal of Forecasting.

[13] Mohamed N, Ahmad MH, Ismail Z, Suhartono S. Short term load forecasting using double seasonal ARIMA model. In: Proceedings of the Regional Conference on Statistical Sciences. Vol. 10. 2010. pp.

[14] Kim SY, Jung HW, Park JD, Baek SM, Kim WS, Chon KH, et al. Weekly maximum electric load forecasting for 104 weeks by seasonal ARIMA model. Journal of the Korean Institute of Illuminating and Electrical Installation

Engineers. 2014;**28**(1):50-56

1993. ISBN 979-403-313-8

[15] Cryer JD, Chan KS. Time Series Analysis:With Application. R. Springer Science & Business Media; 2008;**7**:154-158

[16] Walpole RE. Pengantar Statistika. Jakarta: PT Gramedia Pustaka Utama;

models and the Box-Jenkins

Wesley; 2006;1:1:108-109

application to Brazilian data.

2008;**24**(4):630-644

57-73

ijece.v8i6

*Forecasting in Mathematics - Recent Advances, New Perspectives and Applications*

Padang; 2009

1997;**16**(3):147-163

electricity demand. IEEE Transactions on Power Systems. 2005;**20**(2):1166-

[3] Gonzalez-Romera E, Jaramillo-Moran MA, Carmona-Fernandez D. Monthly electric energy demand forecasting based on trend extraction. IEEE Transactions on Power Systems.

[4] Taylor JW, Mc Sharry PE. Shortterm load forecasting methods: An evaluation based on european data. IEEE Transactions on Power Systems.

[5] Hahn H, Meyer-Nieberg S, Pickl S. Electric load forecasting methods: Tool for decision making. European Journal of Operational Research. 2009;**199**(3):

Panagopoulos AD. Stochastic approach to a rain attenuation time series synthesizer for heavy rain regions. International Journal of Electrical and Computer Engineering. 2016;**6**(5):2379

[7] Kamley S, Jaloree S, Thakur RS. Performance forecasting of share market using machine learning techniques: A review. International Journal of Electrical and Computer Engineering. 2016;**6**(6):2088-8708

[8] Mado I, Soeprijanto A, Suhartono S. Applying of double seasonal ARIMA model for electrical power demand forecasting at PT. PLN Gresik Indonesia.

2006;**21**(4):1946-1953

2007;**22**(4):2213-2219

[6] Nia MM, Din J, Lam HY,

902-907

**118**

[18] Mado I, Soeprijanto A, Suhartono S. Electrical load adat clustering in PJB UP Gresik based on time series analysis approach. In: Proceedings of the International Conference on Vocational Education and Electrical Engineering. Surabaya, Indonesia; 2015. pp. 261-266. Available from: http://digilib.unimed.ac. id/23841/1/Fulltext.pdf

**121**

**Chapter 7**

**Abstract**

forecasting

models.

at all levels.

flows in order to satisfy the end consumer.

build the forecasts are therefore multiple.

**1. Introduction**

Seeking Accuracy in Forecasting

Comparison of Various Methods

The need for a good forecast estimate is imperative for managing flows in a supply

**Keywords:** forecasts, accuracy, quality of forecasts, demand forecasting, selling price

For most companies, forecasting is a prerequisite for effective supply chain management. As explained by Lai et al. [1], forecasting is the basis of all production management systems. The entire supply chain is based on the data from forecast

In Ref. [2], the authors show the usefulness of forecasting and planning as a decision-making tool for organizing the supply chain across all horizons of time and

In the academic field, forecasting occupies an important place. Given the primordial role of forecasting, we understand why many models have been developed since the beginning of the twentieth century. Research mainly developed from the 1950s onward with the use of mathematical models. A review of the literature was carried out by Stadtler [3]. We find there the interest of forecasting for the global supply chain in order to integrate the different organizations and coordinate their

The various sources for making these forecasts are located throughout the supply chain, including the commercial part of the business. It is the analysis of this source that will help build the basis for future forecasting. In the end, the sources used to

chain. For this, it is necessary to make forecasts and integrate them into the flow control models, in particular in contexts where demand is very variable. However, forecasts are never reliable, hence the need to give a measure of the quality of these forecasts, by giving a measure of the forecast uncertainty linked to the estimate made. Different forecasting models have been developed in the past, particularly in the statistical area. Before going to our application on real industrial cases which highlights a prospective study of demand forecasting and a comparative study of sales price forecasts, we begin, in the first section of this chapter, by presenting the

Demand and Selling Prices:

*Zineb Aman, Latifa Ezzine, Yassine Erraoui,* 

forecasting models, as well as their validation and monitoring.

*Younes Fakhradine El Bahi and Haj El Moussami*

#### **Chapter 7**

## Seeking Accuracy in Forecasting Demand and Selling Prices: Comparison of Various Methods

*Zineb Aman, Latifa Ezzine, Yassine Erraoui, Younes Fakhradine El Bahi and Haj El Moussami*

#### **Abstract**

The need for a good forecast estimate is imperative for managing flows in a supply chain. For this, it is necessary to make forecasts and integrate them into the flow control models, in particular in contexts where demand is very variable. However, forecasts are never reliable, hence the need to give a measure of the quality of these forecasts, by giving a measure of the forecast uncertainty linked to the estimate made. Different forecasting models have been developed in the past, particularly in the statistical area. Before going to our application on real industrial cases which highlights a prospective study of demand forecasting and a comparative study of sales price forecasts, we begin, in the first section of this chapter, by presenting the forecasting models, as well as their validation and monitoring.

**Keywords:** forecasts, accuracy, quality of forecasts, demand forecasting, selling price forecasting

#### **1. Introduction**

For most companies, forecasting is a prerequisite for effective supply chain management. As explained by Lai et al. [1], forecasting is the basis of all production management systems. The entire supply chain is based on the data from forecast models.

In Ref. [2], the authors show the usefulness of forecasting and planning as a decision-making tool for organizing the supply chain across all horizons of time and at all levels.

In the academic field, forecasting occupies an important place. Given the primordial role of forecasting, we understand why many models have been developed since the beginning of the twentieth century. Research mainly developed from the 1950s onward with the use of mathematical models. A review of the literature was carried out by Stadtler [3]. We find there the interest of forecasting for the global supply chain in order to integrate the different organizations and coordinate their flows in order to satisfy the end consumer.

The various sources for making these forecasts are located throughout the supply chain, including the commercial part of the business. It is the analysis of this source that will help build the basis for future forecasting. In the end, the sources used to build the forecasts are therefore multiple.

#### **2. Application to prospective approach: modeling and forecasting demand using the ARIMA models**

In the manufacturing sector, forecasting demand is one of the most crucial problems in inventory management [4]; it can be used in various operational planning activities during the production process: capacity planning and management of used product acquisitions [5].

For both types of push/pull supply chain processes, demand forecasting forms the basis of all CS planning. The "pull" processes in the SC are performed in response to the client's request, while all the "push" processes are performed in anticipation of the client's request [6]. A business needs to know many factors related to forecasting demand. Some of these factors are listed below:


Businesses need to understand these factors before they can choose an appropriate forecasting method as it can be difficult to decide which method is the most suitable for forecasting. Forecasting methods are classified into the following types: time series, causal, qualitative, and simulation [6].

A time series is considered to be a set of observations cited in chronological order [7]. To forecast demand, time series forecasting models are based on historical data. These mathematical models used are based on the assumption that the future is an expansion of the past [8].

Numerous studies on demand forecasting by time series analysis have been carried out in several fields. They include demand forecasts for food sales [9], tourism [10], spare parts [4, 11], electricity [12, 13], automobiles [14], and some other goods and services [15–17].

In this section, we forecast the demand for a product in a food manufacturing operation based on real data, as well as the precision and characteristics of these forecasts.

Our study will be carried out according to the three stages of the Box-Jenkins approach: identification, estimation, and verification. We present the model relating to product demand from January 2010 to December 2015 as shown in **Figure 1**.

#### **2.1 Identification of model**

This refers to the initial preprocessing of the data to make it stationary and to the choice of *p* and *q* values that can be adjusted during model fitting.

We present the ACF and PACF diagrams of the series in **Figures 2** and **3**, respectively. We find that this series oscillates, respectively, around an average value, and its autocorrelation function decreases to zero point rapidly, which proves the stationarity of the time series studied.

**123**

**Figure 2.**

**Figure 1.**

*Evolution of the final product's sales.*

*Seeking Accuracy in Forecasting Demand and Selling Prices: Comparison of Various Methods*

Moreover, to assess whether the data come from a stationary process, we can perform the unit root test: Dickey-Fuller test for stationarity. After carrying out the

The null hypothesis H0 cannot be rejected since the calculated *p* value is greater than the significance level *α* set at 0.05. We calculated the risk of rejecting the null

test on the Xlstat software, the results are grouped in **Table 1**.

hypothesis H0, while it is true. The risk is 84.38%.

H1: The series does not have a unit root. The series is stationary.

H0: The series has a unit root.

*ACF correlograms of the demand series.*

*DOI: http://dx.doi.org/10.5772/intechopen.93171*

*Seeking Accuracy in Forecasting Demand and Selling Prices: Comparison of Various Methods DOI: http://dx.doi.org/10.5772/intechopen.93171*

**Figure 1.** *Evolution of the final product's sales.*

*Forecasting in Mathematics - Recent Advances, New Perspectives and Applications*

**demand using the ARIMA models**

of used product acquisitions [5].

• past requests;

• product delivery time;

• state of the economy;

is an expansion of the past [8].

and services [15–17].

**2.1 Identification of model**

stationarity of the time series studied.

forecasts.

**Figure 1**.

• price reduction planned; and

• actions undertaken by competitors.

series, causal, qualitative, and simulation [6].

• planned advertising or marketing efforts;

**2. Application to prospective approach: modeling and forecasting** 

In the manufacturing sector, forecasting demand is one of the most crucial problems in inventory management [4]; it can be used in various operational planning activities during the production process: capacity planning and management

For both types of push/pull supply chain processes, demand forecasting forms the basis of all CS planning. The "pull" processes in the SC are performed in response to the client's request, while all the "push" processes are performed in anticipation of the client's request [6]. A business needs to know many factors

Businesses need to understand these factors before they can choose an appropriate forecasting method as it can be difficult to decide which method is the most suitable for forecasting. Forecasting methods are classified into the following types: time

A time series is considered to be a set of observations cited in chronological order [7]. To forecast demand, time series forecasting models are based on historical data. These mathematical models used are based on the assumption that the future

Numerous studies on demand forecasting by time series analysis have been carried out in several fields. They include demand forecasts for food sales [9], tourism [10], spare parts [4, 11], electricity [12, 13], automobiles [14], and some other goods

In this section, we forecast the demand for a product in a food manufacturing operation based on real data, as well as the precision and characteristics of these

Our study will be carried out according to the three stages of the Box-Jenkins approach: identification, estimation, and verification. We present the model relating to product demand from January 2010 to December 2015 as shown in

This refers to the initial preprocessing of the data to make it stationary and to the

We present the ACF and PACF diagrams of the series in **Figures 2** and **3**, respec-

tively. We find that this series oscillates, respectively, around an average value, and its autocorrelation function decreases to zero point rapidly, which proves the

choice of *p* and *q* values that can be adjusted during model fitting.

related to forecasting demand. Some of these factors are listed below:

**122**

#### **Figure 2.**

*ACF correlograms of the demand series.*

Moreover, to assess whether the data come from a stationary process, we can perform the unit root test: Dickey-Fuller test for stationarity. After carrying out the test on the Xlstat software, the results are grouped in **Table 1**.

H0: The series has a unit root.

H1: The series does not have a unit root. The series is stationary.

The null hypothesis H0 cannot be rejected since the calculated *p* value is greater than the significance level *α* set at 0.05. We calculated the risk of rejecting the null hypothesis H0, while it is true. The risk is 84.38%.

**Figure 3.** *PACF correlograms of the demand series.*


#### **Table 1.** *Test results.*

In our study, we checked the stationarity of the series, and we noted from the ACF and PACF correlograms that our model cannot be pure RA or pure MA. Therefore, we tested several models to identify the most suitable for our series.

#### **2.2 Estimation of model coefficients**

Using the ARIMA procedure of the SPSS time series module [18], we can estimate the coefficients of our model by providing the parameters *p*, *q*, and *d* [19–22].

The best model is as simple as possible and minimizes certain criteria, namely AIC criteria (Akaike criterion), SBC (Bayesian criterion of Schwarz), variance, and maximum likelihood [23–25]. The chosen model is that of ARIMA (0, 1, 1). For other models, either the Student "T-RATIO" test values are found in the range of ±1.96, or one of the values of the minimization criteria is higher than that found for the ARIMA model (1, 0, 1) with the constant value.

**Table 2** presents the values of the different models. From this table, we choose the appropriate model on which we will base ourselves to make our forecasts.

It is clear from **Table 2** that the ARIMA model (1,0,1) is selected because all the coefficients are significantly different from 0 according to the Student test (|T-RATIO|) ≥ 1.96) with an acceptable level of adjustment.

The model residue is stationary and follows a white noise process in the range of ±40. The residue histogram shows whether the distribution of residues approximates a normal distribution. In our case, we have residues that distribute relatively normal around zero and with a relatively low dispersion at a 5% risk.

**125**

**Characteristics**

AR (1)

*α*1 SEB *T* value *p* value

> MA (1)

*θ*<sup>1</sup> SEB *T* value *p* value

> AR (2)

*α*2 SEB *T* value *p* value

> MA (2)

*θ*<sup>2</sup> SEB *T* value *p* value

> Constant

Cte SEB *T* value *p* value

> AIC

0.00000000

688.86593

9.8689581

12.608189

124.42969

0.23455708

1.1993429

0.142258

0.17062

0.00258711

3.1284995

0.167073

0.52269

0.00000000

8.8813204

0.104616

0.92913

**ARIMA (1,0,2)**

**ARIMA (2,0,2)**

0.71371 0.761758 0.9369292 0.35216008

0.31779 0.741595 0.4285186 0.66964815

0.19759

0.659442

0.2996279

0.76538859

0.30202

0.409353

0.7377864

0.46322050

124.52640 12.601296 9.8820312 0.00000000

690.82312

125.53260 11.785537 10.651411 0.00000000

688.77347

128.53887 6.5715235 19.559981 0.00000000

689.37103

129.22650

4.8971088

26.388326

0.00000000

693.59055

692.04831

**ARIMA (1,0,1)**

0.90792 0.094852

9.571955 0.000000

0.63880 0.161531 3,954655 0.00018319

**ARIMA (1,0,0)**

0.49434 0.1074471 4.600820 0.00001823

**ARIMA (0,0,1)**

−0.41704 0.1119989 −3.723567 0.00039384

**ARIMA (1,0,1) without constant**

0.99755623 0.00444769

224.28617

0.0000000

0.71392452

0.08579173

8.32160

0.0000000

**Models**

*Seeking Accuracy in Forecasting Demand and Selling Prices: Comparison of Various Methods*

*DOI: http://dx.doi.org/10.5772/intechopen.93171*


*Seeking Accuracy in Forecasting Demand and Selling Prices: Comparison of Various Methods DOI: http://dx.doi.org/10.5772/intechopen.93171*

*Forecasting in Mathematics - Recent Advances, New Perspectives and Applications*

In our study, we checked the stationarity of the series, and we noted from the ACF and PACF correlograms that our model cannot be pure RA or pure MA. Therefore, we tested several models to identify the most suitable for our series.

Tau (observed value) −1.350 Tau (critical value) −0.717 *p* (unilateral) 0.844 *α* 0.05

Using the ARIMA procedure of the SPSS time series module [18], we can estimate the coefficients of our model by providing the parameters *p*, *q*, and *d* [19–22]. The best model is as simple as possible and minimizes certain criteria, namely AIC criteria (Akaike criterion), SBC (Bayesian criterion of Schwarz), variance, and maximum likelihood [23–25]. The chosen model is that of ARIMA (0, 1, 1). For other models, either the Student "T-RATIO" test values are found in the range of ±1.96, or one of the values of the minimization criteria is higher than that found for

**Table 2** presents the values of the different models. From this table, we choose

The model residue is stationary and follows a white noise process in the range of ±40. The residue histogram shows whether the distribution of residues approximates a normal distribution. In our case, we have residues that distribute relatively

the appropriate model on which we will base ourselves to make our forecasts. It is clear from **Table 2** that the ARIMA model (1,0,1) is selected because all the coefficients are significantly different from 0 according to the Student test

**2.2 Estimation of model coefficients**

*PACF correlograms of the demand series.*

the ARIMA model (1, 0, 1) with the constant value.

(|T-RATIO|) ≥ 1.96) with an acceptable level of adjustment.

normal around zero and with a relatively low dispersion at a 5% risk.

**124**

**Figure 3.**

**Table 1.** *Test results.*


**Table 2.** *Coefficients of different models.*

**127**

**Table 3.**

*ARIMA model parameters.*

*Seeking Accuracy in Forecasting Demand and Selling Prices: Comparison of Various Methods*

*t t tt* =δ+ − + α

*t t* : residuals of periods *t* and *t*–1, and constitute a white nose.

• *α*1, *θ*1: coefficients of autoregressive and moving average processes,

We can easily extract from **Table 3** the coefficients of the autoregressive processes and moving averages and inject them into Eq. (1), which becomes:

=+ − 125,524 0,90792 0.6388 − − 1 1

In order to assess the accuracy of the developed model, we compare the experimental and simulated sales during the same period. This comparison is drawn up in **Table 4** and reveals that the model selected has great precision and an ability to simulate dynamic sales behavior. Therefore, this model can be used to analyze and

**Figure 4** shows that the model is validated since the predicted demand fluctuates around the adjustment and the forecast demand, which remained between the

The error varies, but it is within the tolerance range. In order to minimize this

Once the appropriate model is defined and validated, we must do the forecasting,

SEB 0.094852 *T* value 9.571955 *p* value 0.00000000

SEB 0.161531 *T* value 3.954655 *p* value 0.00018319

using the IBM SPSS forecasting. **Table 4** and **Figure 5** present the results of the

AR (1) α<sup>1</sup> 0.90792

MA (1) θ<sup>1</sup> 0.63880

Constant Δ 125.53260

*<sup>t</sup> t t y y* (2)

 θε

 ε1 1 11 − − *y y* (1)

ε

The chosen model parameters are presented in **Table 3**.

• *<sup>t</sup> y* , *t*−<sup>1</sup> *y* : sales of periods *t* and *t*–1, respectively.

*DOI: http://dx.doi.org/10.5772/intechopen.93171*

**2.3 Accuracy of ARIMA (1, 0, 1) model**

model the demand in this food manufacturing.

error, we are opting for other approaches in our future work.

upper limit and the lower limit.

**2.4 Forecast**

With:

• <sup>1</sup> , <sup>−</sup> ε ε

respectively.

The developed model is given by Eq. 1.

The chosen model parameters are presented in **Table 3**. The developed model is given by Eq. 1.

$$\mathcal{Y}\_t = \delta + \alpha\_1 \mathcal{Y}\_{t-1} - \theta\_1 \varepsilon\_{t-1} + \varepsilon\_t \tag{1}$$

With:

*Forecasting in Mathematics - Recent Advances, New Perspectives and Applications*

**126**

**Characteristics**

SBC Log likelihood

Error

**Table 2.**

*Coefficients of different models.*

**ARIMA (1,0,2)**

697.9726 −340.43297

28.034898

**ARIMA (2,0,2)**

702.20645 −340.41156

28.221721

**ARIMA (1,0,1)**

695.60347 −341.38674

28.214812

**ARIMA (1,0,0)**

693.92437 −342.68552

28.576249

**ARIMA (0,0,1)**

698.14388 −344.79527

29.443759

**ARIMA (1,0,1) without constant**

696.60164

−344.02415

28.461048

**Models**


We can easily extract from **Table 3** the coefficients of the autoregressive processes and moving averages and inject them into Eq. (1), which becomes:

$$y\_t = \mathbf{1}25, \mathbf{524} + \mathbf{0}, \mathbf{90792} \ y\_{t-1} - \mathbf{0}. \mathbf{6388} \ \varepsilon\_{t-1} \tag{2}$$

#### **2.3 Accuracy of ARIMA (1, 0, 1) model**

In order to assess the accuracy of the developed model, we compare the experimental and simulated sales during the same period. This comparison is drawn up in **Table 4** and reveals that the model selected has great precision and an ability to simulate dynamic sales behavior. Therefore, this model can be used to analyze and model the demand in this food manufacturing.

**Figure 4** shows that the model is validated since the predicted demand fluctuates around the adjustment and the forecast demand, which remained between the upper limit and the lower limit.

The error varies, but it is within the tolerance range. In order to minimize this error, we are opting for other approaches in our future work.

#### **2.4 Forecast**

Once the appropriate model is defined and validated, we must do the forecasting, using the IBM SPSS forecasting. **Table 4** and **Figure 5** present the results of the


**Table 3.** *ARIMA model parameters.*


#### **Table 4.**

*Forecast sales from January 2016 to October 2016.*

**Figure 4.** *Sales, fit, LCL, and UCL.*

**Figure 5.** *Sales, fit, LCL, UCL, and forecasting.*

sales forecasts that we obtained by applying our ARIMA model (1, 0, 1) for the next 10 months from January 2016 to October 2016.

The chosen model can therefore be used to model and forecast future demand in this food manufacturing. However, each time we have to feed historical data with new data to enrich it and thus improve the new model and forecasts.

The accrue forecasts presented facilitated the production decision in this business. Indeed, the model allowed us to forecast demand and make precise forecasts. Once we have a forecast of demand, it will be much easier to clearly plan the production and thus eliminate the heavy cost losses.

**129**

**Figure 6.**

*Selling price of "SSP."*

*Seeking Accuracy in Forecasting Demand and Selling Prices: Comparison of Various Methods*

**3. Application to comparative approach: comparison of the quality of forecasts obtained in the context of forecasting selling prices**

Our second industrial application is devoted to a modeling study and comparative forecast of sales prices using ARIMA models, artificial neural networks, and

In this section, we will model the actual fuel price data named "SSP" in order to make important predictions to determine future selling prices. The model shown in **Figure 6** is based on the price of "SSP" fuel in a petroleum production from January

Under SPSS, we have drawn the autocorrelation function (ACF) and the partial autocorrelation function (PACF), the results found are presented in **Figures 7** and **8**. The series has a large number of positive shifts for the autocorrelation function,

The next step is to differentiate the series. You have to differentiate it enough to make it immobile but not drag with an excessive differentiation, which will cause a loss of information and therefore unstable models. In our case, we just had to take

Besides, to decide if the data come from a stationary process or not, we can carry out the unit root test: Dickey-Fuller test for stationarity. After performing the test

The null hypothesis H0 must be rejected, and the alternative hypothesis H1 must be accepted since the calculated *p* value is less than the significance level *α* set at 0.05. We calculated the risk of rejecting the null hypothesis H0, while it is true. The

We conclude that our model will have an order of differentiation *d* = 1. We also note that the T-RATIO for the constant of model *μ* is less than 2 in absolute value. We must therefore deduct it from the model before determining the parameters *p* and *q*.

*DOI: http://dx.doi.org/10.5772/intechopen.93171*

**3.1 Forecasting using ARIMA models**

*d* = 1 because of the linearity of the trend.

H0: The series has a unit root.

risk is less than 0.92%.

on the Xlstat software, we grouped the results in **Table 5**.

H1: The series does not have a unit root. The series is stationary.

*3.1.1 Determination of the differentiation parameter*

support vector machines.

2012 to December 2016.

so it must be differentiated.

*Seeking Accuracy in Forecasting Demand and Selling Prices: Comparison of Various Methods DOI: http://dx.doi.org/10.5772/intechopen.93171*

#### **3. Application to comparative approach: comparison of the quality of forecasts obtained in the context of forecasting selling prices**

Our second industrial application is devoted to a modeling study and comparative forecast of sales prices using ARIMA models, artificial neural networks, and support vector machines.

In this section, we will model the actual fuel price data named "SSP" in order to make important predictions to determine future selling prices. The model shown in **Figure 6** is based on the price of "SSP" fuel in a petroleum production from January 2012 to December 2016.

#### **3.1 Forecasting using ARIMA models**

*Forecasting in Mathematics - Recent Advances, New Perspectives and Applications*

**Model 73 74 75 76 77 78 79 80 81 82**

Prévision 95.12 97.92 100.46 102.77 104.86 106.77 108.49 110.06 111.49 112.78 UCL 151.41 156.21 160.35 163.95 167.08 169.83 172.25 174.38 176.26 177.93 LCL 38.83 39.63 40.57 41.59 42.64 43.70 44.74 45.75 46.71 47.63

sales forecasts that we obtained by applying our ARIMA model (1, 0, 1) for the next

The chosen model can therefore be used to model and forecast future demand in this food manufacturing. However, each time we have to feed historical data with

The accrue forecasts presented facilitated the production decision in this business. Indeed, the model allowed us to forecast demand and make precise forecasts. Once we have a forecast of demand, it will be much easier to clearly plan the

new data to enrich it and thus improve the new model and forecasts.

10 months from January 2016 to October 2016.

*Sales, fit, LCL, UCL, and forecasting.*

production and thus eliminate the heavy cost losses.

**128**

**Figure 5.**

**Figure 4.**

Sales-Model\_1

**Table 4.**

*Forecast sales from January 2016 to October 2016.*

*Sales, fit, LCL, and UCL.*

#### *3.1.1 Determination of the differentiation parameter*

Under SPSS, we have drawn the autocorrelation function (ACF) and the partial autocorrelation function (PACF), the results found are presented in **Figures 7** and **8**.

The series has a large number of positive shifts for the autocorrelation function, so it must be differentiated.

The next step is to differentiate the series. You have to differentiate it enough to make it immobile but not drag with an excessive differentiation, which will cause a loss of information and therefore unstable models. In our case, we just had to take *d* = 1 because of the linearity of the trend.

Besides, to decide if the data come from a stationary process or not, we can carry out the unit root test: Dickey-Fuller test for stationarity. After performing the test on the Xlstat software, we grouped the results in **Table 5**.

H0: The series has a unit root.

H1: The series does not have a unit root. The series is stationary.

The null hypothesis H0 must be rejected, and the alternative hypothesis H1 must be accepted since the calculated *p* value is less than the significance level *α* set at 0.05. We calculated the risk of rejecting the null hypothesis H0, while it is true. The risk is less than 0.92%.

We conclude that our model will have an order of differentiation *d* = 1. We also note that the T-RATIO for the constant of model *μ* is less than 2 in absolute value. We must therefore deduct it from the model before determining the parameters *p* and *q*.

**Figure 6.** *Selling price of "SSP."*

**Figure 7.** *ACF correlogram for the sales price series.*

**Figure 8.** *PACF correlogram for the sales price series.*

#### *3.1.2 Determination of the autoregressive parameter*

**Figures 9** and **10** show the residue curve and the ACF and PACF diagrams of the residues of the ARIMA model (0, 1, 0), respectively.

We can clearly see from **Figures 9** and **10** that the partial autocorrelation has a significant peak at offset 2, and we can then deduce that the differentiated series comprises an autoregressive signature. The parameter *p* is therefore equal to 1.

However, the T-RATIO for the autoregressive parameter *φ*1 is lower in absolute value than 2. So, we cannot retain this model. Similarly, the ARIMA model (2, 1, 0) presents the autoregressive parameters whose T-RATIO is less than 2 in absolute value.

**131**

value.

**Figure 9.** *Residue curve.*

**Table 5.** *Test results.*

*3.1.4 Mixed ARIMA model*

use to make forecasts.

retained as follows:

*Seeking Accuracy in Forecasting Demand and Selling Prices: Comparison of Various Methods*

**Tau (observed value)** −4.0325 **Tau (critical value)** −0.7648 *p* **(unilateral)** 0.0092 *α* 0.05

*DOI: http://dx.doi.org/10.5772/intechopen.93171*

*3.1.3 Determination of the moving average parameter*

Now, the T-RATIO for the moving average parameter *θ*1 is lower in absolute value than 2. So we cannot retain this model. Similarly, the ARIMA model (0,1,2) presents moving average parameters whose T-RATIO is less than 2 in absolute

After several iterations and tests, we concluded that only the ARIMA model (1,1,1) had higher T-RATIOS in absolute value than 2. This is the model we should

With the coefficients obtained now, we can write the equation of the model

=− − + + <sup>−</sup>1 12 1 0.928( −− − ) 0.873

The graph in **Figure 11** proves the adequacy of the ARIMA model (1,1,1) devel-

**Table 2** allows us to admit that the chosen model can be used to model and

**Table 6** lists the forecasts obtained for the first quarter of 2017.

oped, which is very close to the real model.

forecast future sales in this petroleum production.

ε

*t t t t t t yy y y* (3)

 ε *Seeking Accuracy in Forecasting Demand and Selling Prices: Comparison of Various Methods DOI: http://dx.doi.org/10.5772/intechopen.93171*


**Table 5.** *Test results.*

*Forecasting in Mathematics - Recent Advances, New Perspectives and Applications*

**130**

**Figure 8.**

**Figure 7.**

*ACF correlogram for the sales price series.*

*3.1.2 Determination of the autoregressive parameter*

*PACF correlogram for the sales price series.*

residues of the ARIMA model (0, 1, 0), respectively.

**Figures 9** and **10** show the residue curve and the ACF and PACF diagrams of the

We can clearly see from **Figures 9** and **10** that the partial autocorrelation has a significant peak at offset 2, and we can then deduce that the differentiated series comprises an autoregressive signature. The parameter *p* is therefore equal to 1. However, the T-RATIO for the autoregressive parameter *φ*1 is lower in absolute value than 2. So, we cannot retain this model. Similarly, the ARIMA model (2, 1, 0) presents the autoregressive parameters whose T-RATIO is less than 2 in absolute value.

**Figure 9.** *Residue curve.*

#### *3.1.3 Determination of the moving average parameter*

Now, the T-RATIO for the moving average parameter *θ*1 is lower in absolute value than 2. So we cannot retain this model. Similarly, the ARIMA model (0,1,2) presents moving average parameters whose T-RATIO is less than 2 in absolute value.

#### *3.1.4 Mixed ARIMA model*

After several iterations and tests, we concluded that only the ARIMA model (1,1,1) had higher T-RATIOS in absolute value than 2. This is the model we should use to make forecasts.

With the coefficients obtained now, we can write the equation of the model retained as follows:

$$y\_t = y\_{t-1} - 0.928(y\_{t-1} - y\_{t-2}) + 0.873x\_{t-1} + \varepsilon\_t \tag{3}$$

**Table 6** lists the forecasts obtained for the first quarter of 2017.

The graph in **Figure 11** proves the adequacy of the ARIMA model (1,1,1) developed, which is very close to the real model.

**Table 2** allows us to admit that the chosen model can be used to model and forecast future sales in this petroleum production.

**Figure 10.** *ACF and PACF diagrams of the residues of the ARIMA model (0,1,0).*


#### **Table 6.**

*Forecast results for the ARIMA model (1,1,1) [26].*

#### *3.1.5 Forecasting using artificial neural networks*

The goal here is to develop a relationship between experimental data collected from authentic sources to estimate the selling prices of fuel. We are trying to apply RBF radial-based neural networks, which are based on machine learning approaches

**133**

**Figure 12.**

*Visualization of the RBF network.*

*Seeking Accuracy in Forecasting Demand and Selling Prices: Comparison of Various Methods*

due to the complex relationships between the input parameter and the output parameter. In this section, we present the modeling approach using this technique to

ing the back propagation algorithm to minimize the mean squared error with one parameter (time) as the input and the desired output (fuel selling price). As presented on the visualization of the network shown in **Figure 12**, the first layer has radial basis transfer functions with the maximum number of 80 neurons, and the second layer has a linear transfer function, in order to build a consistent model for

The radial basis ANN model (comprising two layers) is trained for implement-

Feature selection is one of the core concepts in machine learning, which hugely impacts the performance of our model. Irrelevant or partially relevant features can negatively impact model performance. Feature selection and data cleaning should be the first and most important step of our model designing. However, in our case, this step may be omitted as long as our point cloud is significant. Subsequently, the dataset was randomly divided into two disjoint subsets of training set (60% of total dataset), which help us train our dataset to find the adequate model and testing set (40% of total dataset) to validate the model found. The training set is applied in order to develop the network. After the training phase, the reliability and accuracy

precisely compare it with the ARIMA model used in the previous section.

*DOI: http://dx.doi.org/10.5772/intechopen.93171*

*3.1.6 Model development*

*Results of the ARIMA model (1,1,1) [26].*

**Figure 11.**

providing accurate forecasts [27].

*Seeking Accuracy in Forecasting Demand and Selling Prices: Comparison of Various Methods DOI: http://dx.doi.org/10.5772/intechopen.93171*

#### **Figure 11.** *Results of the ARIMA model (1,1,1) [26].*

*Forecasting in Mathematics - Recent Advances, New Perspectives and Applications*

**132**

**Table 6.**

**Figure 10.**

*3.1.5 Forecasting using artificial neural networks*

*Forecast results for the ARIMA model (1,1,1) [26].*

*ACF and PACF diagrams of the residues of the ARIMA model (0,1,0).*

The goal here is to develop a relationship between experimental data collected from authentic sources to estimate the selling prices of fuel. We are trying to apply RBF radial-based neural networks, which are based on machine learning approaches

**Fortnight Real price Model % error** 1Q January 1072 1042.49 −2.752798507 2Q January 1074 1043.05 −2.881750466 1Q February 1072 1043.59 −2.650186567 2Q February 1082 1044.21 −3.492606285 1Q March 1084 1044.81 −3.615313653 2Q March 1064 1045.48 −1.740601504 due to the complex relationships between the input parameter and the output parameter. In this section, we present the modeling approach using this technique to precisely compare it with the ARIMA model used in the previous section.

#### *3.1.6 Model development*

The radial basis ANN model (comprising two layers) is trained for implementing the back propagation algorithm to minimize the mean squared error with one parameter (time) as the input and the desired output (fuel selling price). As presented on the visualization of the network shown in **Figure 12**, the first layer has radial basis transfer functions with the maximum number of 80 neurons, and the second layer has a linear transfer function, in order to build a consistent model for providing accurate forecasts [27].

Feature selection is one of the core concepts in machine learning, which hugely impacts the performance of our model. Irrelevant or partially relevant features can negatively impact model performance. Feature selection and data cleaning should be the first and most important step of our model designing. However, in our case, this step may be omitted as long as our point cloud is significant. Subsequently, the dataset was randomly divided into two disjoint subsets of training set (60% of total dataset), which help us train our dataset to find the adequate model and testing set (40% of total dataset) to validate the model found. The training set is applied in order to develop the network. After the training phase, the reliability and accuracy

**Figure 12.** *Visualization of the RBF network.*

of the network were perused with the test data. Besides, in our study, we implemented radial basis network of the MATLAB toolbox (i.e., "nwrb"). Furthermore, the Gaussian function is the main kernel function implemented here with the width parameter of 1 [27].

After executing the learning phase, we obtain **Figures 13** and **14** that represent the learning of our database. **Figure 15** represents the error in the training phase. During the test phase, we gave values to the input variable to visualize the results of the output and thus simulate our model.

#### *3.1.7 Error optimization*

Optimizing the error consists of a compromise to be made between the various parameters of the network, namely the speed, the objective, the number of neurons, and the number of neurons to be added to the hidden layer. This compromise is made on the basis of several tests of the different combinations carried out. Some of these combinations are presented in **Table 7**.

After making different combinations, we find that the error is considerable for all the compromises. Consequently, no model can adapt to the time series, especially in the long term. The reason behind this result is not only the large fluctuations in

**135**

**Table 7.**

*Part of different combinations made.*

**Figure 14.**

**Figure 15.** *The graph of error.*

**Parameters**

*Training graph of the RBF network.*

*Seeking Accuracy in Forecasting Demand and Selling Prices: Comparison of Various Methods*

the selling price of the fuel but also the percentage of the total dataset used in the training stage (60%). In fact, this percentage will not allow us to predict 40% of the total dataset. We will have to increase the percentage of training. In the next step,

**Goal Spread MN DF** 0 .01 1.5 25 25 0.01 1 25 30 0.01 2 30 30 0.01 0.8 12 30 0.01 1.57 10 30

*DOI: http://dx.doi.org/10.5772/intechopen.93171*

**Figure 13.** *Training of the RBF network.*

*Seeking Accuracy in Forecasting Demand and Selling Prices: Comparison of Various Methods DOI: http://dx.doi.org/10.5772/intechopen.93171*

**Figure 14.** *Training graph of the RBF network.*

*Forecasting in Mathematics - Recent Advances, New Perspectives and Applications*

parameter of 1 [27].

*3.1.7 Error optimization*

the output and thus simulate our model.

of these combinations are presented in **Table 7**.

of the network were perused with the test data. Besides, in our study, we implemented radial basis network of the MATLAB toolbox (i.e., "nwrb"). Furthermore, the Gaussian function is the main kernel function implemented here with the width

After executing the learning phase, we obtain **Figures 13** and **14** that represent the learning of our database. **Figure 15** represents the error in the training phase. During the test phase, we gave values to the input variable to visualize the results of

Optimizing the error consists of a compromise to be made between the various parameters of the network, namely the speed, the objective, the number of neurons, and the number of neurons to be added to the hidden layer. This compromise is made on the basis of several tests of the different combinations carried out. Some

After making different combinations, we find that the error is considerable for all the compromises. Consequently, no model can adapt to the time series, especially in the long term. The reason behind this result is not only the large fluctuations in

**134**

**Figure 13.**

*Training of the RBF network.*


#### **Table 7.**

*Part of different combinations made.*

the selling price of the fuel but also the percentage of the total dataset used in the training stage (60%). In fact, this percentage will not allow us to predict 40% of the total dataset. We will have to increase the percentage of training. In the next step,

we will consider 80% of the total dataset for the training phase and 20% for testing the model. **Table 8** summarizes the different combinations [27].

The combination that minimizes the error is therefore:


We can conclude that learning with 80% of the database gives increased results in comparison with the other case (learning with 60%) since the error is minimized. The output is calculated and presented in **Table 9**.

From **Table 9**, we can clearly see that the selected model can be used to model and forecast future sales in this petroleum manufacturing. As a last part, we will use the methodology of support vector machines to see that this is going to give a result.

#### *3.1.8 Forecasting using support vector machines (SVMs)*

The aim of our current work is to develop a relationship between experimental data collected from authentic sources to estimate the selling price of fuel. We are trying to apply support vector machines based on machine learning approaches because of the complex relationships between the input parameter and the output.

We prepared our database and then developed the program in Python language, which will be compiled on Spyder software.

We imported our dataset, which is the actual price of our fuel studied, created, and indexed the location of values from the database. Then, we standardized the data so that it corresponds to the learning process that will be carried out using the SVR function. In fact, we have divided our database into a learning part and another for the test. We tried two main distributions: (1) 60% of our database used in the learning phase and 40% used in the testing phase and (2) 80% of our database used in the learning phase and 20% used in the testing phase. We have kept the second distributions based on the results obtained after compiling the program. After that, we learned "Train X" and "Train Y" and executed the test to finally calculate the average of the errors and obtained the values predicted in the test phase, which are grouped in **Figure 16**.

The average error is equal to 26.882361, which represents 2.53%. The error graph is shown in **Figure 17**.


**137**

**Figure 16.**

*Results of the SVR function.*

**Table 9.**

*Predicted value of output after using the RBF model.*

*Seeking Accuracy in Forecasting Demand and Selling Prices: Comparison of Various Methods*

It is clear that the model chosen can be used to model and forecast future sales for this petroleum industry since the error observed (2.53%) respects the allowable margin of error set by the company at 3%. In addition, the SVR function is a useful tool, which guarantees good precision and minimizes the error compared to the ARIMA model.

**Input (time) Real value of output Predicted value of output % error** 1038 1027.2 1.04046243 1043 1044.8 −0.1725791 1035 1033.4 0.15458937 1040 1034.3 0.54807692 1016 1034.7 −1.84055118 1015 1034.8 −1.95073892 1010 1034.9 −2.46534653 1001 1034.9 −3.38661339 1031 1034.9 −0.37827352 1033 1034.9 −0.1839303 1036 1034.9 0.10617761 1030 1034.9 −0.47572816 1000 1034.9 −3.49 1042 1034.9 0.68138196 1072 1034.9 3.4608209 1074 1034.9 3.6405959 1072 1034.9 3.4608209 1082 1034.9 4.35304991 1084 1034.9 4.5295203 1064 1034.9 2.73496241

*DOI: http://dx.doi.org/10.5772/intechopen.93171*

#### **Table 8.**

*Error comparison for several combinations of parameters.*

It is clear that the model chosen can be used to model and forecast future sales for this petroleum industry since the error observed (2.53%) respects the allowable margin of error set by the company at 3%. In addition, the SVR function is a useful tool, which guarantees good precision and minimizes the error compared to the ARIMA model.


#### **Table 9.**

*Forecasting in Mathematics - Recent Advances, New Perspectives and Applications*

the model. **Table 8** summarizes the different combinations [27]. The combination that minimizes the error is therefore:

mized. The output is calculated and presented in **Table 9**.

*3.1.8 Forecasting using support vector machines (SVMs)*

which will be compiled on Spyder software.

which are grouped in **Figure 16**.

**Goal Spread MN DF**

*Error comparison for several combinations of parameters.*

is shown in **Figure 17**.

• goal = 0.01;

• spread = 1;

• DF = 30.

• MN = 20; and

we will consider 80% of the total dataset for the training phase and 20% for testing

We can conclude that learning with 80% of the database gives increased results

From **Table 9**, we can clearly see that the selected model can be used to model and forecast future sales in this petroleum manufacturing. As a last part, we will use the methodology of support vector machines to see that this is going to give a result.

The aim of our current work is to develop a relationship between experimental data collected from authentic sources to estimate the selling price of fuel. We are trying to apply support vector machines based on machine learning approaches because of the complex relationships between the input parameter and the output. We prepared our database and then developed the program in Python language,

We imported our dataset, which is the actual price of our fuel studied, created, and indexed the location of values from the database. Then, we standardized the data so that it corresponds to the learning process that will be carried out using the SVR function. In fact, we have divided our database into a learning part and another for the test. We tried two main distributions: (1) 60% of our database used in the learning phase and 40% used in the testing phase and (2) 80% of our database used in the learning phase and 20% used in the testing phase. We have kept the second distributions based on the results obtained after compiling the program. After that, we learned "Train X" and "Train Y" and executed the test to finally calculate the average of the errors and obtained the values predicted in the test phase,

The average error is equal to 26.882361, which represents 2.53%. The error graph

**Parameters Relative error (%)**

0.01 1 10 30 9.37 0.01 0.25 25 25 7.61 0.01 0.5 30 20 5.29 0.01 0.8 30 20 3.21 0.01 1 20 30 **1.95**

in comparison with the other case (learning with 60%) since the error is mini-

**136**

**Table 8.**

*Predicted value of output after using the RBF model.*

**Figure 16.** *Results of the SVR function.*

#### **3.2 Synthesis**

In the first industrial application of this chapter, we modeled demand using ARIMA models. The model we have obtained will allow the company to forecast demand and make precise forecasts.

In the second application, we studied the selling prices of the SSP via three methodologies: ARIMA, RBF, and SVMs.

First, we developed an ARIMA model based on historical data. This study allowed us to determine the ARIMA model (1,1,1), which gives gasoline price forecasts close to the margin to reach for the first quarter of the current year with an average margin of error 2.855%. Second, we used the RBF technique to improve the modeling and forecasting of the selling price of fuel. It was found that this technique has proven its strength manifested in the error, which has been further minimized: 1.95% instead of 2.85% for the ARIMA model. Finally, we used the SVM function. The forecasts made are quite satisfactory because they respect the margin tolerated by the company. The error of the SVM function is around 2.53%.

As a summary, the SVM function has proven its strength manifesting itself in the error, which has been further minimized: 2.53% instead of 2.885% for the ARIMA model, but which remains higher than the error obtained using the RBF technique.

#### **4. Conclusion**

For most companies, forecasting is a prerequisite for effective supply chain management. Forecasting is the basis of all production management systems. The entire supply chain is based on data from forecast models.

In this chapter, we have presented the study of forecasting demand and selling prices in industrial companies. We also carried out a comparative study aimed at minimizing the error to guarantee increased forecasts.

In the first part, we modeled the future demand for a food company using ARIMA models based on the Box-Jenkins methodology. The model we have obtained will allow the company to forecast demand and make precise forecasts.

**139**

**Author details**

and Haj El Moussami1

for the ARIMA model.

University, Meknes, Morocco

Meknes, Morocco

Settat, Morocco

\*, Latifa Ezzine2

provided the original work is properly cited.

, Yassine Erraoui1

comparing it with the error obtained if we had opted for neural networks.

2 Modeling, Control Systems and Telecommunications, EST, Moulay Ismail

3 Industrial Management and Innovation, Faculty of Science and Technology,

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

\*Address all correspondence to: zineb.aman@gmail.com

1 Mechanics and Integrated Engineering, ENSAM School, Moulay Ismail University,

, Younes Fakhradine El Bahi3

Zineb Aman1

*Seeking Accuracy in Forecasting Demand and Selling Prices: Comparison of Various Methods*

We can clearly see that the chosen model can be used to model and forecast future demand for this agribusiness, but each time we need to populate the historical data

Second, we carried out a study, which consists in comparing the quality of the forecasts obtained in the context of forecasting selling prices. We presented the application of three different methodologies allowing us to make sales forecasts in a

We have developed an ARIMA model based on historical data. This study allowed

Finally, we studied the SSP selling prices via the SVM function. We prepared our database and then developed the program in Python language, which will be compiled on Spyder software. The forecasts made are quite satisfactory with regard to the constraint imposed by the company (plus or minus 3% margin of error). The error of the SVM function is around 2.53%. Consequently, the SVM function has proven its strength manifesting itself in the error, which has been further minimized: 2.53% instead of 2.855% for the ARIMA model, but which remains higher by

us to determine the optimal autoregressive, moving average, and differentiation parameters in order to make predictions. We found that the ARIMA model (1,1,1) gives gasoline price forecasts close to the margin to reach for the first quarter of the current year with an average margin of error of 2.855% included within the margin of error tolerated by the company (plus or minus 3% as margin of error). In addition, the hypothesis that the residues are white Gaussian noise has always been verified. Then, we tried forecasting selling prices via the RBF technique in order to improve the modeling and forecasting done before. To do this, we have developed an RBF network based on historical data to come up with conclusions in terms of superiority of forecast performance. Consequently, the use of this technique has proven itself and has allowed us to minimize the error, which is 1.95% versus 2.85%

*DOI: http://dx.doi.org/10.5772/intechopen.93171*

company operating in the petroleum sector.

with the new data.

#### *Seeking Accuracy in Forecasting Demand and Selling Prices: Comparison of Various Methods DOI: http://dx.doi.org/10.5772/intechopen.93171*

We can clearly see that the chosen model can be used to model and forecast future demand for this agribusiness, but each time we need to populate the historical data with the new data.

Second, we carried out a study, which consists in comparing the quality of the forecasts obtained in the context of forecasting selling prices. We presented the application of three different methodologies allowing us to make sales forecasts in a company operating in the petroleum sector.

We have developed an ARIMA model based on historical data. This study allowed us to determine the optimal autoregressive, moving average, and differentiation parameters in order to make predictions. We found that the ARIMA model (1,1,1) gives gasoline price forecasts close to the margin to reach for the first quarter of the current year with an average margin of error of 2.855% included within the margin of error tolerated by the company (plus or minus 3% as margin of error). In addition, the hypothesis that the residues are white Gaussian noise has always been verified.

Then, we tried forecasting selling prices via the RBF technique in order to improve the modeling and forecasting done before. To do this, we have developed an RBF network based on historical data to come up with conclusions in terms of superiority of forecast performance. Consequently, the use of this technique has proven itself and has allowed us to minimize the error, which is 1.95% versus 2.85% for the ARIMA model.

Finally, we studied the SSP selling prices via the SVM function. We prepared our database and then developed the program in Python language, which will be compiled on Spyder software. The forecasts made are quite satisfactory with regard to the constraint imposed by the company (plus or minus 3% margin of error). The error of the SVM function is around 2.53%. Consequently, the SVM function has proven its strength manifesting itself in the error, which has been further minimized: 2.53% instead of 2.855% for the ARIMA model, but which remains higher by comparing it with the error obtained if we had opted for neural networks.

#### **Author details**

*Forecasting in Mathematics - Recent Advances, New Perspectives and Applications*

In the first industrial application of this chapter, we modeled demand using ARIMA models. The model we have obtained will allow the company to forecast

In the second application, we studied the selling prices of the SSP via three

First, we developed an ARIMA model based on historical data. This study allowed us to determine the ARIMA model (1,1,1), which gives gasoline price forecasts close to the margin to reach for the first quarter of the current year with an average margin of error 2.855%. Second, we used the RBF technique to improve the modeling and forecasting of the selling price of fuel. It was found that this technique has proven its strength manifested in the error, which has been further minimized: 1.95% instead of 2.85% for the ARIMA model. Finally, we used the SVM function. The forecasts made are quite satisfactory because they respect the margin

tolerated by the company. The error of the SVM function is around 2.53%.

As a summary, the SVM function has proven its strength manifesting itself in the error, which has been further minimized: 2.53% instead of 2.885% for the ARIMA model, but which remains higher than the error obtained using the RBF

For most companies, forecasting is a prerequisite for effective supply chain management. Forecasting is the basis of all production management systems. The

In the first part, we modeled the future demand for a food company using ARIMA models based on the Box-Jenkins methodology. The model we have obtained will allow the company to forecast demand and make precise forecasts.

In this chapter, we have presented the study of forecasting demand and selling prices in industrial companies. We also carried out a comparative study aimed at

entire supply chain is based on data from forecast models.

minimizing the error to guarantee increased forecasts.

**138**

**3.2 Synthesis**

**Figure 17.** *Error graph.*

technique.

**4. Conclusion**

demand and make precise forecasts.

methodologies: ARIMA, RBF, and SVMs.

Zineb Aman1 \*, Latifa Ezzine2 , Yassine Erraoui1 , Younes Fakhradine El Bahi3 and Haj El Moussami1

1 Mechanics and Integrated Engineering, ENSAM School, Moulay Ismail University, Meknes, Morocco

2 Modeling, Control Systems and Telecommunications, EST, Moulay Ismail University, Meknes, Morocco

3 Industrial Management and Innovation, Faculty of Science and Technology, Settat, Morocco

\*Address all correspondence to: zineb.aman@gmail.com

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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[4] Ghobbar AA, Friend CH. Evaluation of forecasting methods for intermittent parts demand in the field of aviation: A predictive model. Computers and Operations Research. 2003;**30**:2097-2114

[5] Toktay B. Forecasting product returns. In: Guide VDR Jr, van Wassenhove LN, editors. Business Aspects of Closed-Loop Supply Chains. Pittsburgh: Carnegie Mellon University Press; 2003

[6] Chopra S, Meindl P. Supply chain management. Strategy, planning & operation. In: Das summa summarum des management. Gabler; 2007. pp. 265-275

[7] Bozarth CB, Handfield RB. Introduction to Pperations and Supply Chain Management. 4th ed. Raleigh: North Carolina State University; 2016

[8] Wisner JD, Tan KC, Leong GK. Principles of Supply Chain Management: A Balanced Approach. Thomson South-Western: Mason; 2011

[9] Miller JJ, McCahon CS, Miller JL. Foodservice forecasting with simple time series models. Journal of Hospitality and Tourism Research. 1991;**14**:9-21

[10] Song H, Li G. Tourism demand modeling and forecasting—A review of recent research. Tourism Management. 2008;**29**:203-220

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[15] Gardner ES Jr. Exponential smoothing: the state of the art—Part II. International Journal of Forecasting. 2006;**22**:637-666

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[27] Aman Z, Ezzine L, El Bahi YF, El Moussami H. Improving the modeling and forecasting of fuel selling price using the radial basis function technique: A case study. Journal of Algorithms & Computational

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Statistique deDonnées avec SPSS 6. Slatkine: Genève; 1996

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[21] Mélard G, Roy R. Modèles de séries chronologiques avec seuils. Revue de StatistiqueAppliquée. 1988;**4**(36):5-23

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[24] Brockwell PJ, Davis RA. Time Series: Theory and Method. Springer-Verlag; 1987

[25] Hamilton JD. Time Series Analysis. Princeton: Princeton University Press; 1994

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[10] Song H, Li G. Tourism demand modeling and forecasting—A review of recent research. Tourism Management.

[11] Willemain TR, XSmart CN, Schwarz HF. A new approach to forecasting intermittent demand for service parts inventories. International Journal of Forecasting. 2004;**20**:375-387

[12] Taylor JW, de Menezes LM, McSharry PE. A comparison of univariate methods for forecasting electricity demand up to a day ahead. International Journal of Forecasting.

[13] Weron R. Electricity price forecasting: A review of the state-ofthe-art with a look into the future. International Journal of Forecasting.

[15] Gardner ES Jr. Exponential

[14] Mitsutaka M, Akira I. Examination of demand forecasting by time series analysis for auto parts remanufacturing. Journal of Remanufacturing. 2015;**5**:1. DOI: 10.1186/s13243-015-0010-y

smoothing: the state of the art—Part II. International Journal of Forecasting.

[16] Liu Q, Wang H. Research on the forecast and development of China's public fiscal revenue based on ARIMA model. Theoretical Economics Letters.

[17] Shen S, Shen Y. ARIMA model in the application of Shanghai and Shenzhen stock index. Applications of

[18] Stafford J, Bodson P. L'Analyse multivariée avec SPSS. Presses de l'Universitéde Québec; 2006

[19] Bavaud F, Capel R, Crettaz de Roten F, Müller JP. Guide de l'nalyse

Mathematics. 2016;**7**:171-176

2008;**29**:203-220

2006;**1**:1-16

2014;**30**:1030-1081

2006;**22**:637-666

2015;**5**:482-493

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2006;**4**(4):331-354

[3] Stadtler H. Supply chain

[4] Ghobbar AA, Friend CH. Evaluation of forecasting methods for intermittent parts demand in the field of aviation: A predictive model. Computers and Operations Research.

[5] Toktay B. Forecasting product returns. In: Guide VDR Jr, van Wassenhove LN, editors. Business Aspects of Closed-Loop Supply Chains. Pittsburgh: Carnegie Mellon University

[6] Chopra S, Meindl P. Supply chain management. Strategy, planning & operation. In: Das summa summarum des management. Gabler; 2007.

[7] Bozarth CB, Handfield RB.

Introduction to Pperations and Supply Chain Management. 4th ed. Raleigh: North Carolina State University; 2016

[8] Wisner JD, Tan KC, Leong GK. Principles of Supply Chain

Management: A Balanced Approach. Thomson South-Western: Mason; 2011

[9] Miller JJ, McCahon CS, Miller JL. Foodservice forecasting with simple time series models. Journal of Hospitality and Tourism Research.

2003;**30**:2097-2114

Press; 2003

pp. 265-275

[1] Lai F, Zhao X, Lee T. Selecting forecasting model parameters in material requirement planning systems. International Journal of Internet and Enterprise Management.

[2] Fleischmann B, Meyr H, Wagner M. Supply Chain Management & Advanced Planning. Dans Berlin: Springler; 2002

management and advanced planning— Basics, overview and challenges. European Journal of Operational Research. 2005;**163**(3):575-588

## *Edited by Abdo Abou Jaoude*

Mathematical probability and statistics are an attractive, thriving, and respectable part of mathematics. Some mathematicians and philosophers of science say they are the gateway to mathematics' deepest mysteries. Moreover, mathematical statistics denotes an accumulation of mathematical discussions connected with efforts to most efficiently collect and use numerical data subject to random or deterministic variations. Currently, the concept of probability and mathematical statistics has become one of the fundamental notions of modern science and the philosophy of nature. This book is an illustration of the use of mathematics to solve specific problems in engineering, statistics, and science in general.

Published in London, UK © 2021 IntechOpen © ddukang / iStock

Forecasting in Mathematics - Recent Advances, New Perspectives and Applications

Forecasting in Mathematics

Recent Advances, New Perspectives

and Applications

*Edited by Abdo Abou Jaoude*