K=∑ �� ����

When the equations describing the dynamical system are available, one can calculate the entire Lyapunov spectrum. The approach involves numerically solving the system's n equations for n+1 nearby initial conditions. The growth of a corresponding set of vectors is measured, and as the system evolves, the vectors are repeatedly reorthonormalized using the Gram-Schmidt procedure. This guarantees that only one vector has a component in the direction of most rapid expansion, i.e., the vectors maintain a proper phase space orientation. In experimental settings, however, the equations of motion are usually unknown and this approach is not applicable. Furthermore, experimental data often consist of time series from a single observable, and one must employ a technique for attractor reconstruction, e.g., method of delays [60], singular value decomposition.

As suggested above, one cannot calculate the entire Lyapunov spectrum by choosing arbitrary directions for measuring the separation of nearby initial conditions. One must measure the separation along the Lyapunov directions which correspond to the principal axes of the ellipsoid previously considered. These Lyapunov directions are dependent upon the system flow and are defined using the Jacobian matrix, i.e., the tangent map, at each point of interest along the flow [58]. Hence, one must preserve the proper phase space orientation by using a suitable approximation of the tangent map. This requirement, however, becomes unnecessary when calculating only the largest Lyapunov exponent.

A Multi Adaptive Neuro Fuzzy Inference System for

So by calculating its logarithm we have:

Table 1 we can expect system to forecast.

**5. Preparing the input data** 

following form:

elements [15]:

Table 1. Lyapunov exponent for seasons of one year

previous days; 2, 7, and 14 day ago, and 2, 3, 4 day ago.

**6. Adaptive neural- Fuzzy inference system** 

Rule i: If x is Ai and y is Bi then fi = pix+qiy+ri.

O*l,i* is the output of the *i*th node of the layer l.

membership functions;

Short Term Load Forecasting by Using Previous Day Features 345

λ= � � Ln �� ��

There should be at least one Lyapunov exponent bigger than zero to have chaos, the existence of positive value of λ means the chaotic behavior of system. Therefore, in order to

> **Winter Fall Summer Spring**  0.07563 0.05428 0.0444 0.0523

First step in the process of electricity load forecasting is to provide last information of the system load being studied. After preparing the input data matrix, it is turn of classification. The reason of this classification is the existence of completely determined models in different days that were referred to in many references. Among different days of weeks, Saturday to Thursday which are working days in Iran, have the same load model. Fridays have also their own particular model and have a low level of load. Special days have a completely different model, too. So it seems necessary at the first look that each of these classes should be analyzed separately. We consider 2 groups of features that refer to

ANFIS, proposed by Jang [14, 15], is an architecture which functionally integrates the interpretability of a fuzzy inference system with adaptability of a neural network. Loosely speaking ANFIS is a method for tuning an existing rule base of fuzzy system with a learning algorithm based on a collection of training data found in artificial neural network. Due to the less tunable use of parameters of fuzzy system compared with conventional artificial neural network, ANFIS is trained faster and more accurately than the conventional artificial neural network. An ANFIS which corresponds to a Sugeno type fuzzy model of two inputs and single output is shown in Fig. 1. A rule set of first order Sugeno fuzzy system is the

ANFIS structure as shown in Figure 1 is a weightless multi-layer array of five different

Layer 1: Input data are fuzzified and neuron values are represented by parameterized

O*1,i* = µA*i*(x) for *i* = 1, 2, or

Every node *i* in this layer is an adaptive node with a node function

��=����� (4)

(5)

If we assume that there exists an ergodic measure of the system, then the multiplicative ergodic theorem of Oseledec [61] justifies the use of arbitrary phase space directions when calculating the largest Lyapunov exponent with smooth dynamical systems. We can expect that two randomly chosen initial conditions will diverge exponentially at a rate given by the largest Lyapunov exponent [62]. In other words, we can expect that a random vector of initial conditions will converge to the most unstable manifold, since exponential growth in this direction quickly dominates growth (or contraction) along the other Lyapunov directions. Thus, the largest Lyapunov exponent can be defined using the following equation where d(t) is the average divergence at time t and C is a constant that normalizes the initial separation:

$$\mathbf{d}(\mathbf{t}) = \mathbf{C}e^{\lambda \mathbf{1} \mathbf{t}}$$

For experimental applications, a number of researchers have proposed algorithms that estimate the largest Lyapunov exponent [55,59], the positive Lyapunov spectrum, i.e., only positive exponents [59], or the complete Lyapunov spectrum [58]. Each method can be considered as a variation of one of several earlier approaches [59] and as suffering from at least one of the following drawbacks: (1) unreliable for small data sets, (2) computationally intensive, (3) relatively difficult to implement. These drawbacks motivated our search for an improved method of estimating the largest Lyapunov exponent.

#### **4.2 Calculation of lyapunov exponent for time series**

In order to calculate Lyapunov exponent for those systems which their equation is not determined and their time series is not available, different algorithm is suggested [45-49].

The algorithm proposed by Wolf [48], seeks the time series of close points in the phase space. These points went round the phase space or got divergent rapidly. Close points in the same direction are selected.

The differential coefficient is in the direction of the maximum development and their average logarithm on the route of phase space yields the biggest Lyapunov exponent. Suppose that series of x�, x�, x�,… x� is available and the interval between them is obtained as t�- t�= n� that τ is the interval between two successive measurement. If the system has chaotic behavior, we can explain divergence of the adjacent routes based on the difference range between them, as following.

> . . .

$$d\_0 = |\boldsymbol{\omega}\_f - \boldsymbol{\omega}\_l| \tag{1}$$

$$d\_1 = \left| \boldsymbol{\chi}\_{l+1} - \boldsymbol{\chi}\_{l+1} \right| \tag{2}$$

$$d\_n = \left| \boldsymbol{\chi}\_{l+n} - \boldsymbol{\chi}\_{l+n} \right| \tag{3}$$

It is supposed that d� will increase exponential by n increase:

$$d\_n = d\_0 e^{\lambda n} \tag{4}$$

So by calculating its logarithm we have:

344 Fuzzy Inference System – Theory and Applications

If we assume that there exists an ergodic measure of the system, then the multiplicative ergodic theorem of Oseledec [61] justifies the use of arbitrary phase space directions when calculating the largest Lyapunov exponent with smooth dynamical systems. We can expect that two randomly chosen initial conditions will diverge exponentially at a rate given by the largest Lyapunov exponent [62]. In other words, we can expect that a random vector of initial conditions will converge to the most unstable manifold, since exponential growth in this direction quickly dominates growth (or contraction) along the other Lyapunov directions. Thus, the largest Lyapunov exponent can be defined using the following equation where d(t) is the average divergence at time t and C is a constant that normalizes

d(t) =C���� For experimental applications, a number of researchers have proposed algorithms that estimate the largest Lyapunov exponent [55,59], the positive Lyapunov spectrum, i.e., only positive exponents [59], or the complete Lyapunov spectrum [58]. Each method can be considered as a variation of one of several earlier approaches [59] and as suffering from at least one of the following drawbacks: (1) unreliable for small data sets, (2) computationally intensive, (3) relatively difficult to implement. These drawbacks motivated our search for an

In order to calculate Lyapunov exponent for those systems which their equation is not determined and their time series is not available, different algorithm is suggested [45-49].

The algorithm proposed by Wolf [48], seeks the time series of close points in the phase space. These points went round the phase space or got divergent rapidly. Close points in the

The differential coefficient is in the direction of the maximum development and their average logarithm on the route of phase space yields the biggest Lyapunov exponent. Suppose that series of x�, x�, x�,… x� is available and the interval between them is obtained as t�- t�= n� that τ is the interval between two successive measurement. If the system has chaotic behavior, we can explain divergence of the adjacent routes based on the difference

> . . .

��=��� � ��� (1)

��=����� � ����� (2)

��=����� � ����� (3)

improved method of estimating the largest Lyapunov exponent.

**4.2 Calculation of lyapunov exponent for time series** 

It is supposed that d� will increase exponential by n increase:

the initial separation:

same direction are selected.

range between them, as following.

$$
\lambda = \frac{1}{n} \text{Lm} \frac{d\_n}{d\_0} \tag{5}
$$

There should be at least one Lyapunov exponent bigger than zero to have chaos, the existence of positive value of λ means the chaotic behavior of system. Therefore, in order to Table 1 we can expect system to forecast.


Table 1. Lyapunov exponent for seasons of one year
