2. Definition of fuzzy time series

Fuzzy sets, presented by Zadeh [1], have numerous presentations, such as in fuzzy sets, fuzzy decision analysis, and fuzzy time series. The concept is also widely applied in social science article and applications [2–22]. Fuzzy time series is developed rapidly since their introduction by Song and Chissom [18–21]. Current fuzzy time series methods have benefited from both theoretical developments as well as relevant applications in research [2–22], which has led to more diverse uses. This trend indicates that the development of fuzzy time series has markedly improved. The definitions of the fuzzy time series used in this article are described as follows.

Definition 1 [18–21]. A fuzzy number on the real line ℜ is a fuzzy subset of ℜ that is normal and convex.

Definition 2 [18–21]. Let Y tð Þð Þ t ¼ …; 0; 1; 2; … , a subset of ℜ, be the universe of discourse on which the fuzzy sets fi ð Þt ð Þ t ¼ 1; 2; … are defined, and let F tð Þ be the collection of fi ð Þt ð Þ t ¼ 1; 2; … . Then, F tð Þ is called fuzzy time series on Y tð Þð Þ t ¼ …; 0; 1; 2; … .

Definition 3 [18–21]. Let I and J be the index sets for F tð Þ � 1 and F tð Þ, respectively. If for any fj ð Þt ∈ F tð Þ, where j∈ J, there then exists fi ð Þ t � 1 ∈ F tð Þ � 1 , where i ∈ I, such that there exists a fuzzy relation Rijð Þ t; t � 1 and fj ðÞ¼ t fi ð Þ t � 1 ∘ Rijð Þ t; t � 1 , where '∘' is the max–min composition. Then, F tð Þ is said to be caused by only F tð Þ � 1 . Denote this as fi ð Þ! t � 1 fj ð Þt , or equivalently, F tð Þ! � 1 F tð Þ.

Definition 4 [18–21]. If, for any fj ð Þ t � 1 ∈ F tð Þ, where j∈ J, there exists fi ð Þ t � 1 ∈ F tð Þ � 1 , where i∈ I, and a fuzzy relation Rijð Þ t; t � 1 , such that fj ðÞ¼ t fi ð Þ t � 1 ∘ Rijð Þ t; t � 1 . Let R tð Þ¼ ; t � 1 ∪ijRijð Þ t; t � 1 , where ∪ is the union operator. Then, R tð Þ ; t � 1 is called the fuzzy relation between F tð Þ and F tð Þ � 1 . Thus, we define this as the following fuzzy relational equation:

$$F(t) = F(t-1) \bullet R(t, t-1).$$

Definition 5 [18–21]. Suppose that <sup>R</sup>1ð Þ¼ <sup>t</sup>; <sup>t</sup> � <sup>1</sup> <sup>∪</sup>ijR<sup>1</sup> ijð Þ t; t � 1 and <sup>R</sup>2ð Þ¼ <sup>t</sup>; <sup>t</sup> � <sup>1</sup> <sup>∪</sup>ijR<sup>2</sup> ijð Þ t; t � 1 are two fuzzy relations between F tð Þ and F tð Þ � 1 . If, for any fj ð Þt ∈ F tð Þ, where j∈ J, there exists fi ð Þ t � 1 ∈ F tð Þ � 1 , where i ∈ I, and fuzzy relations R<sup>1</sup> ijð Þ <sup>t</sup>; <sup>t</sup> � <sup>1</sup> and <sup>R</sup><sup>2</sup> ijð Þ t; t � 1 such that fi ðÞ¼ t fi ð Þ <sup>t</sup> � <sup>1</sup> <sup>∘</sup> <sup>R</sup><sup>1</sup> ijð Þ t; t � 1 and fi ðÞ¼ t fi ð Þ <sup>t</sup> � <sup>1</sup> <sup>∘</sup> <sup>R</sup><sup>2</sup> ijð Þ t; t � 1 , then define R1ð Þ¼ t; t � 1 R2ð Þ t; t � 1 .

Definition 6 [18–21]. Suppose that F tð Þ is only caused by F tð Þ � 1 , F tð Þ � 2 , ..., or F tð Þ � m ð Þ m . 0 . This relation can be expressed as the following fuzzy relational equation: F tðÞ¼ F tð Þ � 1 ∘ R0ð Þ t; t � m , which is called the first-order model of F tð Þ.

Definition 7 [18–21]. Suppose that F tð Þ is simultaneously caused by F tð Þ � 1 , F tð Þ � 2 , …, and F tð Þ � m ð Þ m . 0 . This relation can be expressed as the following fuzzy relational equation: F tðÞ¼ ð Þ F tð Þ� � 1 F tð Þ� � 2 … � F tð Þ � m ∘ Rað Þ t; t � m , which is called the <sup>m</sup>th-order model of F tð Þ.

Definition 8 [3]. F tð Þ is fuzzy time series if F tð Þ is a fuzzy set. The transition is denoted as F tð Þ! � 1 F tð Þ.

Definition 9 [7]. Let d tð Þ be a set of real numbers: d tð Þ⊆ R. We define an exponential function where

Fuzzy Forecast Based on Fuzzy Time Series DOI: http://dx.doi.org/10.5772/intechopen.82843

1. y ¼ exp d tð Þ ⇔ ln y ¼ d tð Þ and

significance level process. A numerical example of STI is shown in Section 4, and

Fuzzy sets, presented by Zadeh [1], have numerous presentations, such as in fuzzy sets, fuzzy decision analysis, and fuzzy time series. The concept is also widely applied in social science article and applications [2–22]. Fuzzy time series is developed rapidly since their introduction by Song and Chissom [18–21]. Current fuzzy time series methods have benefited from both theoretical developments as well as relevant applications in research [2–22], which has led to more diverse uses. This trend indicates that the development of fuzzy time series has markedly improved. The definitions of the fuzzy time series used in this article are described

Definition 1 [18–21]. A fuzzy number on the real line ℜ is a fuzzy subset of ℜ

Definition 2 [18–21]. Let Y tð Þð Þ t ¼ …; 0; 1; 2; … , a subset of ℜ, be the universe of

ð Þt ð Þ t ¼ 1; 2; … . Then, F tð Þ is called fuzzy time series on

Definition 3 [18–21]. Let I and J be the index sets for F tð Þ � 1 and F tð Þ, respec-

ð Þ t � 1 ∘ Rijð Þ t; t � 1 . Let R tð Þ¼ ; t � 1 ∪ijRijð Þ t; t � 1 , where ∪ is the union

ijð Þ t; t � 1 are two fuzzy relations between F tð Þ and F tð Þ � 1 . If, for

ðÞ¼ t fi

ð Þt ∈ F tð Þ, where j∈ J, there then exists fi

Rijð Þ t; t � 1 , where '∘' is the max–min composition. Then, F tð Þ is said to be caused

ð Þ! t � 1 fj

operator. Then, R tð Þ ; t � 1 is called the fuzzy relation between F tð Þ and F tð Þ � 1 .

ijð Þ t; t � 1 such that fi

F tðÞ¼ F tð Þ � 1 ∘ R tð Þ ; t � 1 :

ijð Þ t; t � 1 , then define R1ð Þ¼ t; t � 1 R2ð Þ t; t � 1 . Definition 6 [18–21]. Suppose that F tð Þ is only caused by F tð Þ � 1 , F tð Þ � 2 , ..., or F tð Þ � m ð Þ m . 0 . This relation can be expressed as the following fuzzy relational equation: F tðÞ¼ F tð Þ � 1 ∘ R0ð Þ t; t � m , which is called the first-order model of F tð Þ. Definition 7 [18–21]. Suppose that F tð Þ is simultaneously caused by F tð Þ � 1 , F tð Þ � 2 , …, and F tð Þ � m ð Þ m . 0 . This relation can be expressed as the following fuzzy relational equation: F tðÞ¼ ð Þ F tð Þ� � 1 F tð Þ� � 2 … � F tð Þ � m ∘ Rað Þ t; t � m ,

Definition 8 [3]. F tð Þ is fuzzy time series if F tð Þ is a fuzzy set. The transition is

Definition 9 [7]. Let d tð Þ be a set of real numbers: d tð Þ⊆ R. We define an

ð Þ t � 1 ∈ F tð Þ � 1 , where i∈ I, and a fuzzy relation Rijð Þ t; t � 1 , such that

i ∈ I, such that there exists a fuzzy relation Rijð Þ t; t � 1 and fj

Thus, we define this as the following fuzzy relational equation:

Definition 5 [18–21]. Suppose that <sup>R</sup>1ð Þ¼ <sup>t</sup>; <sup>t</sup> � <sup>1</sup> <sup>∪</sup>ijR<sup>1</sup>

ð Þt ∈ F tð Þ, where j∈ J, there exists fi

ijð Þ <sup>t</sup>; <sup>t</sup> � <sup>1</sup> and <sup>R</sup><sup>2</sup>

which is called the <sup>m</sup>th-order model of F tð Þ.

ð Þ <sup>t</sup> � <sup>1</sup> <sup>∘</sup> <sup>R</sup><sup>2</sup>

denoted as F tð Þ! � 1 F tð Þ.

exponential function where

ð Þt ð Þ t ¼ 1; 2; … are defined, and let F tð Þ be the

ð Þ t � 1 ∈ F tð Þ, where j∈ J, there exists

ð Þ t � 1 ∈ F tð Þ � 1 , where

ð Þ t � 1 ∘

ijð Þ t; t � 1 and

ðÞ¼ t fi

ð Þt , or equivalently, F tð Þ! � 1 F tð Þ.

ijð Þ t; t � 1 and

ð Þ t � 1 ∈ F tð Þ � 1 , where i ∈ I, and fuzzy

ð Þ <sup>t</sup> � <sup>1</sup> <sup>∘</sup> <sup>R</sup><sup>1</sup>

concluding remarks are mentioned in conclusion.

Time Series Analysis - Data, Methods, and Applications

2. Definition of fuzzy time series

as follows.

collection of fi

fi

fj ðÞ¼ t fi

any fj

fi ðÞ¼ t fi

26

relations R<sup>1</sup>

Y tð Þð Þ t ¼ …; 0; 1; 2; … .

tively. If for any fj

<sup>R</sup>2ð Þ¼ <sup>t</sup>; <sup>t</sup> � <sup>1</sup> <sup>∪</sup>ijR<sup>2</sup>

that is normal and convex.

discourse on which the fuzzy sets fi

by only F tð Þ � 1 . Denote this as fi

Definition 4 [18–21]. If, for any fj

2. exp lnð Þ¼ d tð Þ d tð Þ, ln exp ð Þ¼ x d tð Þ.

Definition 10 [14]. The universe of discourse U ¼ ½ � DL; DU is defined such that DL <sup>¼</sup> <sup>D</sup>min � stαð Þ <sup>n</sup> <sup>=</sup> ffiffiffi <sup>n</sup> <sup>p</sup> and DU <sup>¼</sup> <sup>D</sup>max <sup>þ</sup> stαð Þ <sup>n</sup> <sup>=</sup> ffiffiffi n p when n ≤ 30 or DL <sup>¼</sup> <sup>D</sup>min � <sup>σ</sup>Zα<sup>=</sup> ffiffiffi <sup>n</sup> <sup>p</sup> and DU <sup>¼</sup> <sup>D</sup>max <sup>þ</sup> <sup>σ</sup>Zα<sup>=</sup> ffiffiffi <sup>n</sup> <sup>p</sup> when <sup>n</sup> . 30, where <sup>t</sup>αð Þ <sup>n</sup> is the 100 1ð Þ � α percentile of the t distribution with n degrees of freedom. z<sup>α</sup> is the 100 1ð Þ � α percentile of the standard normal distribution. Briefly, if Z is an N(0, 1) distribution, then P Zð Þ¼ ≥ z<sup>α</sup> α.

Definition 11 [14]. Assuming that there are m linguistic values under consideration, let Ai be the fuzzy number that represents the i th linguistic value of the linguistic variable, where 1≤ i≤ m. The support of Ai is defined as follows:

$$\begin{cases} D\_L + (i - \mathbf{1}) \frac{D\_U - D\_L}{m}, & D\_L + \frac{i(D\_U - D\_L)}{m}, \mathbf{1} \le i \le m - 1 \\\ D\_L + (i - \mathbf{1}) \frac{D\_U - D\_L}{m}, & D\_L + \frac{i(D\_U - D\_L)}{m}, i = m. \end{cases}$$

Definition 12 [17]. For a test H<sup>0</sup> : nonfuzzy trend against H<sup>1</sup> : fuzzy trend, where the critical region <sup>C</sup><sup>∗</sup> <sup>¼</sup> C C<sup>k</sup> <sup>2</sup> <sup>þ</sup> <sup>C</sup><sup>n</sup>�<sup>k</sup> <sup>2</sup> . <sup>C</sup><sup>λ</sup> <sup>¼</sup> <sup>C</sup><sup>n</sup> <sup>2</sup> � ð Þ <sup>1</sup> � <sup>λ</sup> � � � � , the initial value of the significance level α is 0.2.

Definition 13 [8]. Let d tð Þ be a set of real numbers d tð Þ⊆ R. An upper interval for d tð Þ is a number b such that x≤ b for all x∈ d tð Þ. The set d tð Þ is said to be an interval higher if d tð Þ has an upper interval. A number, max, is the maximum of d tð Þ if max is an upper interval for d tð Þ and max∈ d tð Þ.

Definition 14 [8]. Let d tð Þ<sup>⊆</sup> <sup>R</sup>. The least upper interval of d tð Þ is a number max ! satisfying:

1. max ! is an upper interval for d tð Þ such that <sup>x</sup><sup>≤</sup> max ! for all <sup>x</sup><sup>∈</sup> d tð Þ and

2. max ! is the least upper interval for d tð Þ, that is, <sup>x</sup><sup>≤</sup> <sup>b</sup> for all <sup>x</sup><sup>∈</sup> d tðÞ) max ! ≤ b.

Definition 15 [8]. Let d tð Þ be a set of real numbers d tð Þ⊆ R. A lower interval for d tð Þ is a number b such that x ≥ b for all x∈ d tð Þ. The set d tð Þ is said to be an interval below if d tð Þ has a lower interval. A number, min, is the minimum of d tð Þ if min is a lower interval for d tð Þ and min∈ d tð Þ.

Definition 16 [8]. Let d tð Þ<sup>⊆</sup> <sup>R</sup>. The least lower interval of d tð Þ is a number min satisfying:

1. min is a lower interval for d tð Þ such that x ≥ min for all x∈ d tð Þ and.

2. min is the least lower interval for d tð Þ, that is, x ≥ b for all x∈ d tðÞ) min ≤ b.

Definition 17 [8]. The long-term predictive value interval (min , max ! ) is called the static long-term predictive value interval.

Definition 18 [2]. Let Ai ¼ αi; βi; γ<sup>i</sup> ð Þ, i ¼ 1, 2, …, n, be n triangular fuzzy numbers. By using the graded mean integration representation (GMIR) method, the GMIR value P Að Þ<sup>i</sup> of Ai is P Að Þ¼<sup>i</sup> α<sup>i</sup> þ 4β<sup>i</sup> þ γ<sup>i</sup> ð Þ=6. P Að Þ<sup>i</sup> and P Aj � � are the GMIR values of the triangular fuzzy numbers Ai and Aj, respectively.

Definition 19 [12]. Set up new triangular fuzzy numbers by S = (min , ^ d tð Þ, max ! ). After GMIR transformation, S becomes a real number ΔS. This is called the long-term significance level with fuzzy time series. The ΔS is a real number satisfying the following:

Step 3. Define Ai by letting its membership function be as follows:

DU � DL

DU � DL

Step 4. Then, F tðÞ¼ Ai if d tð Þ∈ suppð Þ Ai , where suppð Þ� denotes the support. Step 5. Derive the transition rule from period t � 1 to t and denote it as

Step 6. The value of d tð Þ can be predicted using the fuzzy time series F tð Þ as

4. Numerical example of Shipping and Transportation Index in Taiwan

In this study, the Shipping and Transportation Index (STI) in Taiwan is used for a numerical example. The STI reflects the spot rates of the Taiwan Stock Exchange Corporation. The STI data are sourced from the Taiwan Stock Exchange Corporation [23], the historical data for which is defined here as the STI, and monthaveraged data for the period between January, 2015, and June, 2018, was collected. Over these 42 data points, the analysis produces an average of 4.226, with a standard deviation of 0.172, maximum value of 4.571, and minimum value of 4.067. These descriptive statistics show that the STI has largely remained at the 1124.70

<sup>m</sup> ; DL <sup>þ</sup>

<sup>m</sup> ; DL <sup>þ</sup>

� �

� �

i Dð Þ <sup>U</sup> � DL m

i Dð Þ <sup>U</sup> � DL m

be the set of rules fired by d tð Þ,

� �. The

, max ! ).

� � be the median of supp Pj

,^ d tð Þmax ! ) .

1 for x∈ DL þ ð Þ i � 1

1 for x∈ DL þ ð Þ i � 1

F tð Þ! � 1 F tð Þ. Aggregate all transition rules. Let the set of rules be

� � is the support of Pj. Let supp Pj

Step 8. Set up new triangular fuzzy numbers by <sup>Δ</sup>S = (min

level. As shown in Figure 2, its current rate of return is negative.

� �; where rj ∈ R � � � �

suppð Þ Qj j j T tð Þ �<sup>1</sup> . Step 7. The long-term predictive value interval for d tð Þ is given as (min

where 1≤ i ≤ m � 1;

where i ¼ m; 0 otherwise:

uAi ð Þ¼ x

R ¼ ri ri : Pi ! Qi f j g.

where supp Pj

Figure 2.

29

Rate of return of the STI.

8

Fuzzy Forecast Based on Fuzzy Time Series DOI: http://dx.doi.org/10.5772/intechopen.82843

>>>>>>>>>><

>>>>>>>>>>:

follows. Let T tðÞ¼ rj d tð Þ∈ supp Pj

predicted value of d tð Þ is <sup>∑</sup>rj <sup>∈</sup> T tð Þ �<sup>1</sup>

Step 9. Defuzzify S to be ΔS.

1. ΔS is called a long-term significance level up, only if: ΔS > ^ d tð Þ;

2. ΔS is called a long-term significance level down, only if: ΔS < ^ d tð Þ; and

3. ΔS is called a long-term significance level stable, only if: ΔS = ^ d tð Þ.
