Fuzzy linear regression models with least square errors

doi:10.1016/j.amc.2004.05.004

Applied Mathematics and Computation

Volume 163, Issue 2, 15 April 2005, Pages 977-989

https://doi.org/10.1016/j.amc.2004.05.004 Get rights and content

Abstract

To estimate the parameters of fuzzy linear regression models with fuzzy output and crisp inputs, we develop a mathematical programming model in this paper. The method is constructed on the basis of minimizing the square of the total difference between observed and estimated spread values or in other words minimizing the least square errors. The advantage of the proposed approach is its simplicity in programming and computation as well as its performance. To compare the performance of the proposed approach with the other methods, two examples are presented.

Introduction

Regression analysis is of the most popular methods of estimation. It is applied to evaluate the functional relationship between the dependent and independent variables. Fuzzy regression analysis is an extension of the classical regression analysis in which some elements of the model are represented by fuzzy numbers. Fuzzy regression methods have been successfully applied to various problems such as forecasting [1], [3], [7], [18], [19] and engineering [2], [9]. Since in real world, many regression analysis problems have fuzzy elements (output or/and input), a number of researchers focused on different types of fuzzy linear regression models.

Fuzzy regression models can be classified into two classes. The first class includes Tanaka's method and its extension. In this class, the total vagueness of the estimated values for the dependent variable is minimized [11], [15], [16], [17]. The second one adopts fuzzy least square method (FLSM) to minimize the total square of errors in the estimated value [4], [10], [13]. As pointed out by Tsaur and Wang [21], the advantage of Tanaka's model is its simplicity in programming and computation, while that of FLSM is its minimum degree of fuzziness between the observed and estimated values.

The purpose of this paper is to introduce a method to estimate the parameters of a fuzzy linear regression model by applying a mathematical-programming-approach such that both predictability and computability are improved. We assume the dependent variables are crisp while the dependent variable is a symmetric fuzzy number.

The paper is organized as follows. In Section 2, some elementary properties of fuzzy numbers as well as the concept of possibility equality are described. In Section 3, we review fuzzy regression methods in literature and especially Tanaka's model. The proposed method is presented in Section 4. Two numerical examples are illustrated to compare the proposed model with previous ones, in Section 5.

Section snippets

Fuzzy numbers and possibility equality

A fuzzy number $A$ is a convex normalized fuzzy subset of the real line R with an upper semi-continuous membership function of bounded support. A general representation of a fuzzy number, is introduced by Dubois and Prade [5], called the L–R type of fuzzy numbers. In this paper, we only consider symmetric fuzzy numbers.

Definition 2.1

A symmetric fuzzy number $A$ , denoted by $A =(a,α)_{L}$ is defined as $A (t)=L((t−a)/α), α⩾0,$ where a and α are the center and spread of $A$ , respectively. Furthermore, the reference function L(x

Fuzzy linear regression models

The formulation of fuzzy linear regression function has been introduced by Tanaka et al. [15], [16], [17]. There are m independent non-fuzzy variables, X_i,i=1,2,…,m, while the dependent variable is a symmetric fuzzy number.

For ith observation, the vector of independent variable X_i=(X_i0,X_i1,…,X_in)^T results in a fuzzy number $Y_{i} =(y ̄_{i},e_{i})_{L}$ . The objective is to estimate a fuzzy linear regression (FLS) model, expressed as follows: $Y_{i} = A_{0} X_{i0} + A_{1} X_{i1} +…+ A_{n} X_{in} = A X_{i} .$ In model (1), $A =(A_{0}, A_{1},…, A_{n})$ is a vector of

The proposed approach

In this section, we develop a mathematical model to determine the fuzzy parameters $A_{i}$ for i=0,1,…,n, of model (1), where $A_{i}$ , is assumed to be a symmetric fuzzy number as defined by Definition 2.1.

The mathematical model is constructed on the basis of the following concepts:

(i)
The objective function is to minimize the total square of difference between the estimated regression spread and the spread of the given data.
(ii)
The degree of the fitness of the FLR model, as defined below, is greater than or

Numerical examples

Kim and Bishu [8] used the two examples to show that their method has a better performance than Tanaka et al. [15] and Savic and Pedrycz [14] methods. In this section, we use the same two examples to illustrate how the proposed method (denoted by PM) performs. We compare the results of our method in fuzzy regression analysis by Tanaka's method [15] (denoted by TM), Diamond method [4] (denoted by DM), Savic and Pedrycz method [14] (denoted by SP) and Kim and Bishu [8] (denoted by KB). The fuzzy

Conclusion

In this paper, we developed a model such that the predictability of Tanaka's model can be improved and the computation complexity of the FLSM can be decreased. Also, the results from two examples indicate that the proposed method has better performance than the previous studies.

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Fuzzy linear regression models with least square errors

Abstract

Introduction

Section snippets

Fuzzy numbers and possibility equality

Fuzzy linear regression models

The proposed approach

Numerical examples

Conclusion

Fuzzy Sets Syst.

Fuzzy Sets Syst.

Fuzzy Sets Syst.

Inform. Sci.

Infom. Sci.

Fuzzy Sets Syst.

Fuzzy Sets Syst.

Fuzzy Sets Syst.

Fuzzy Sets Syst.

Fuzzy Sets Syst.

Fuzzy Sets Syst.