Numerical Solution of Fuzzy linear Regression using Fuzzy Neural Network Based on Probability Function

In this work, we consider the development of a fuzzy neural network based on probability function for Estimated output of fuzzy regression models with test real input and fuzzy output. The proposed approach is a fuzzification of the outputs and weights of conventional fuzzy neural network based on probability function. The error of the proposed method is based on total square error is minimized by optimization method in order to be able to obtain the optimal weights of the neural network. The advantage of the proposed approach is its simplicity and computation as well as its performance. To compare the performance of the proposed method with the other traditional methods given in the literature several numerical examples are presented.


Introduction
Regression analysis is one of the most popular methods of estimation.It is applied to evaluate the functional relationship between dependent and independent variables.Fuzzy regression (FR) is a fuzzy type of classical regression analysis in which some elements of the model are represented by fuzzy numbers.After introducing the concept of fuzzy sets by zadeh in 1965 [1,2,3], different researchers developed regression analysis.Fuzzy linear regression (FLR) was first suggested by Tanaka [4], that is the extension of classic regression analysis that has turned into a powerful tool for the discovery of ambiguous relationships [5].Indeed, in fuzzy regression, some of the elements of the regression model have been presented with ambiguous information.Different methods have been presented for solving these type of problems [6][7][8][9][10].These are divided into two approaches: linear programming (LP) based methods and fuzzy least squares (FLS) methods [11].Neural networks (NN) can be valuable when the functional http://www.ispacs.com/journals/jfsva/2017/jfsva-00380/International Scientific Publications and Consulting Services relationship between dependent and independent variables are not known.Probable basis function (PBF) neural network forms one of the essential type of neural networks.In this paper, we proposed simple but powerful method for fuzzy regression analysis using PBF neural network, since the proposed method employ PBF neural network which have higher flexibility and a wider application field than the existing LP based and FLS fuzzy regression methods [12][13][14][15][16].The effectiveness of our method was demonstrated by three examples and a computational experience.Present article consists of following sections: In Section 2, the basic and required concepts have been stated.In Section 3, the proposed model for solving the FLR regression equations is introduced.In Section 4, the analysis of recommended method error has been stated.In Section 5, the method of calculating the weights of PBF neural network has been mentioned.In Section 6, numerical example has been presented.In Section 7, the conclusion has been presented and in Section 8, the references have been stated.

Main section
Consider the general model of FLR as follows: ) is to obtain an optimal model with fuzzy coefficients for describing and analyzing the data and predicting based on it where x ij are real numbers and y i are of Fuzzy type.For estimating regression with above conditions, we define the proposed method as follows:   = .
(3.2) Where   is the proposed solution and Net is the feed forward artificial neural network that consists of two layers.The first layer is the inputs layer and the second layer is the out puts layer with linear transfer function that is introduced in the following form: 3) Where   are artificial neural network weights.o ij are inputs of neural network and correspond x.Now, Let's suppose our proposed method consists of the same density function of the problem, that is, f() = λ − ,  ≤  ≤ .It is clear that the neural network of relation (3.3) has fuzzy value and this means that we have fuzzy weights for real observations, thus equation (3.3) could be written as follows: where the value of   1 is the center and the value of   2 is the fuzzy width of  neural network, so the relation (3.4) could be rewritten as follows: The value of  that is almost the estimated answer   , is close to the value limit of the main answer , to that end, we define the target function of neural network as follows: (3.7)Where in general, for  observations we will have .
Where degree of membership for above relation is introduced with () and is as follows:  .Now, we minimize the total square error given distance  mentioned in previous section by using w  =   . M  =  M(u 0 , … , u n ).Given what was said, for both equation (3.4) we have: The idea of finding the least squares is w i that are obtained by minimizing M i , that is, the total square errors on distance  and this is done using the Matlab soft-ware and fminunc command is based on Quasi-Newton algorithm.

Weight Calculation Algorithm for the third state
In this section, the algorithm for the calculation of neural network weights for third state is studied.The first and second states resemble the third state, too.To that end, we first explain the method of calculating u i .Suppose Y  and N i be as Y  = (y  , s  ) T and N i = (u  , σ  ) T .Where y i and N i are the centers, s  and σ  the widths of symmetric triangular fuzzy numbers y  and N  , respectively.Given the equation (3.6) we have: (5.12)Where N One = u 0 + u 1  1 + ⋯ + u n o n , ( (e [Net  ] − e S i ) 2 ).m i=1 (5.15) Now, by minimizing equation (5.15) is with initial weights u i = 0, weights in the direction that the target function (the performance function) decreases, that is, contrary to its slope, they are updated, the algorithm of the neural network for the purpose of calculating the weights in this article is a Quasi-Newton algorithm or BFGs [19].The basic step in Quasi-Newton methods is calculated based on Newton formula.
(5.16)In a way that  1 () is respectively the matrices of the second derivatives from performance functions F ̂ 1(k) for present values of the weights.Also (5.17In above equations  is the number of repetitions.The disadvantage of Newton's method is that it is too complex and computationally too expensive and as a result they are not appropriate for neural networks, of course, there is a type of algorithm based on Newton's methods that since it does not require calculating the second derivative, its computational cost is lower.These methods are called Qauss-Newton's methods.They update the algorithm of approximate Hessian matrix each repetition.Updating is conducted through a function from the slope.The Quasi-Newton method that has considerably been successful, consists of BFGs method.This algorithm is usually converged faster and in lower number of repetitions.In the end, after obtaining weights, that is, u i using the mentioned algorithm (that is by order of fminunc in subject), we substitute them in equation (5.12) and achieve relation (3.6).For the first and second states the same process is applied.

Numerical examples
Example 6.1.For fuzzy variables, consider dependent variable  and independent real variable   , the values given in Table 1 (information in Table 1 has been adopted from reference [20]).

Conclusion
In this article, we introduced a Mathematical model from regression with fuzzy coefficients and generalized neural system based on the given number.Then, we calculated the FLR regression coefficients using artificial neural network, the optimization technique and the least square error method based on the distance between two fuzzy numbers.The error of the method was studied and it was shown that the method error is of fuzzy type.It was proved that the neural network weight section and regression coefficient section (FLR) are convergent.

2 )
−(  1  1 ) −  −   1 ) −(  2  2 ) +  −   2 ) That is, with the assumption that Y = (, ) where  be the center, b the fuzzy widths, Y.So we have( ≤  ≤ ) = ∫ f()   = ∫ λ −    = − − |   =  − −  − (3.8)Where for  = ( 1 ,  2 ) = (, ), in (3.8) will be as follows: 1,  ≤  − −  − ≤  0, ℎ.And so (  ) = ∫    ()  (), Where   is the membership function of fuzzy set  and u is a part of y.   () is the probability density function of y and P(y = u) is the probability function of y.Where as we do not know the basic probability distribution.It is clear from this information that probability distribution is itself a fuzzy number.By minimizing equation (3.7) four weights of the neural network, namely,  0 1 ,  1 1 ,  0 2 and  1 2 are obtained.By substituting the weights obtained in equation (3.5), the values of   1 and   2 are obtained and eventually the value of  in equation (3.4) will be obtained.

Figure 1 :
Figure 1: Fuzzy Neural Network Based on Probability Function.

Figure 4 :
Figure 4: The convergence of neural network weights Example 6.3.
. The amount (  ,   ) measures the distance between  −  of fuzzy numbers from  and .() could be interpreted as weight of  2 (  ,   ).Therefore, to calculate sensitivities for the different layers of neurons in the MLP network the Derivative of conversion neurons functions is required.So functions used that have derivative.One of these functions is Linear function which The characteristics of this function was explained in the previous section.The Error function is described in the following sections.To minimize this unconstrained optimization problem, minimization techniques such as the steepest descent method and the conjugate gradient or Quasi-Newton methods can be employed.The Newton method is one of the important algorithms in nonlinear optimization.The main disadvantage of the Newton method is that it is necessary to evaluate the second derivative matrix (Hessian matrix).Quasi-Newton methods were originally proposed by Davidon in 1959 and were later developed by Fletcher and Powell 1 ,  1 2 )o i1 + ⋯ (3.5)Where   1 and   2 are in the following form: {  1 =  0 1 +  1 1 o i1 + ⋯   2 = | 0 2 | + | 1 2 |o i1 + ⋯ (3.6)Now, we should find the four weights of relation (3.5) that is met following conditions: http://www.ispacs.com/journals/jfsva/2017/jfsva-00380/International Scientific Publications and Consulting Services

Table 2 :
Crisp input-fuzzy output data set from

Table 3 :
Fuzzy estimates (  ,   ) and SSE values for different methods.

Table 5 :
Crisp input-Fuzzy output data set from

Table 6 :
Fuzzy estimates (  ,   ) and SSE values for different methods.
, ,  )  The convergence of neural network weights Example 6.2.For fuzzy variables, consider dependent variable  and independent real variable   , the values given in Table Example 6.3.