ساخت فازی برآوردگرهای مربع حداقل در مدل های رگرسیون خطی فازی

A new concept and method of imposing imprecise (fuzzy) input and output data upon the conventional linear regression model is proposed. Under the considerations of fuzzy parameters and fuzzy arithmetic operations (fuzzy addition and multiplication), we propose a fuzzy linear regression model which has the similar form as that of conventional one. We conduct the h-level (conventional) linear regression models of fuzzy linear regression model for the sake of invoking the statistical techniques in (conventional) linear regression analysis for real-valued data. In order to determine the sign (nonnegativity or nonpositivity) of fuzzy parameters, we perform the statistical testing hypotheses and evaluate the confidence intervals. Using the least squares estimators obtained from the h-level linear regression models, we can construct the membership functions of fuzzy least squares estimators via the form of “Resolution Identity” which is well-known in fuzzy sets theory. In order to obtain the membership degree of any given estimate taken from the fuzzy least squares estimator, optimization problems have to be solved. We also provide two computational procedures to deal with those optimization problems

In the real world, the data sometimes cannot be recorded or collected precisely due to human errors, machine errors or some other unexpected situations. For instance, the water level of a river cannot be measured in an exact way because of the fluctuation, and the temperature in a room is also not able to be measured precisely because of similar reasons. With this situation, fuzzy sets theory is naturally an appropriate tool in statistical models when fuzzy data have been observed. The more appropriate way to describe the water level is to say that the water level is around 30 m. The phrase “around 30 m” can be regarded as a fuzzy number View the MathML source30∼. This is the main concern of this paper. Since Zadeh (1965) introduced the concept of fuzzy sets, the applications of considering fuzzy data to the regression models have been proposed in the literature. Tanaka, Uejima, and Asai (1982) initiated this research topic. They also generalized their approaches to more general models in Tanaka and Warada, 1988, Tanaka et al., 1989 and Tanaka and Ishibuchi, 1991. The book on fuzzy regression analysis edited by Kacprzyk and Fedrizzi (1992) gave an insightful survey. Chang and Ayyub (2001) gave the differences between the fuzzy regression and conventional regression analysis and Kim, Moskowitz, and Koksalan (1996) also compared both fuzzy regression and statistical regression conceptually and empirically. In the approach of Tanaka et al. (1982), they considered the L–R fuzzy data and minimized the index of fuzziness of the fuzzy linear regression model. Redden and Woodall (1994) compared various fuzzy regression models and gave the difference between the approaches of fuzzy regression analysis and conventional regression analysis. They also pointed out some weakness of the approaches proposed by Tanaka et al. Chang and Lee (1994) also pointed out another weakness of the approaches proposed by Tanaka et al. Peters (1994) introduced a new fuzzy linear regression models based on Tanaka’s approach by considering the fuzzy linear programming problem. Moskowitz and Kim (1993) proposed a method to assess the H-value in a fuzzy linear regression model proposed by Tanaka et al. Wang and Tsaur (2000) also proposed a new model to improve the predictability of Tanaka’s model. In this paper, we propose a fuzzy linear regression model, and then the h-level linear regression models will be created by taking the h-level set of fuzzy linear regression model. We shall see that the h-level linear regression models are conventional linear regression models. Therefore, the statistical techniques proposed in the conventional linear regression analysis can be invoked to discuss the h-level linear regression models. For the least squares sense, Chang (2001) proposed a method for hybrid fuzzy least squares regression by defining the weighted fuzzy-arithmetic and using the well-accepted least squares fitting criterion. Celminš, 1987 and Celminš, 1991 proposed a methodology for the fitting of differentiable fuzzy model function by minimizing a least squares objective function. Chang and Lee (1996) proposed a fuzzy regression technique based on the least squares approach to estimate the modal value and the spreads of L–R fuzzy number. Jajuga (1986) calculated the linear fuzzy regression coefficients using a generalized version of the least squares method by considering the fuzzy classification of a set of observations and obtaining the homogeneous classes of observations. In this paper, the least squares estimators will be obtained from the h-level linear regression models. Using these least squares estimators, we can construct a fuzzy least squares estimators via the form of “Resolution Identity” which is introduced by Zadeh et al. (1975) and is well-known in fuzzy sets theory. For optimization approach, Sakawa and Yano (1992) introduced three indices for equalities between fuzzy numbers. From these three indices, three types of multiobjective programming problems were formulated. Tanaka and Lee (1998) used the quadratic programming approach to obtain the possibility and necessity regression models simultaneously. The advantage of adopting a quadratic programming approach is to be able to integrate both the property of central tendency in least squares and the possibilistic property in fuzzy regression. In this paper, in order to obtain the membership value (confidence degree) of any given estimate taken from the fuzzy least squares estimator, the optimization problems have to be solved. We also provide two computational procedures to solve those optimization problems. There are also some other interesting articles concerning the fuzzy regression analysis. Näther, 1997, Näther, 2000, Näther and Albrecht, 1990 and Körner et al., 1998 introduced the concept of random fuzzy sets (fuzzy random variables) into the linear regression model, and developed an estimation theory for the parameters. Dunyak and Wunsch (2000) described a method for nonlinear fuzzy regression using a special training technique for fuzzy number neutral networks. Kim and Bishu (1998) used a criterion of minimizing the difference of the membership degrees between the observed and estimated fuzzy numbers. Yager (1982) used a linguistic variable to represent imprecise information for the regression models. Bárdossy (1990) proposed many different measures of fuzziness which must be minimized with respect to some suggested constraints. In Section 2, we give some properties of fuzzy numbers. In Section 3, The techniques for solving fuzzy linear regression problems are proposed. We shall focus on the h-level linear regression models of fuzzy linear regression model, and then apply the conventional linear regression techniques to solve the h-level linear regression models. The membership functions of fuzzy least squares estimators in fuzzy linear regression model will be constructed according to the form of “Resolution Identity” in fuzzy sets theory. In Section 4, we shall develop two computational procedures to obtain the membership degree of any given estimate taken from the fuzzy least squares estimators. We also provide an example to clarify the theoretical results, and show the possible applications in linear regression analysis for imprecise data.

A fuzzy linear regression model is proposed in this paper for considering the fuzzy input and output data. In order to apply the conventional techniques in linear regression model. We propose the lower and upper h-level linear regression models. Since those two models are the conventional linear regression models, we can naturally obtain the least squares estimators of the lower and upper h-level linear regression models, respectively, using formula (2). In order to determine the nonnegativity or nonpositivity of fuzzy parameters occurring in the fuzzy linear regression model, the testing hypotheses at the level of significance α and the confidence interval with confidence coefficient 1 − α are invoked. With the help of those, we can determine the nonnegativity or nonpositivity of fuzzy parameters statistically. Using the least squares estimators obtained from the lower and upper h-level linear regression models, the membership functions of the fuzzy least squares estimators can be created via the form of “Resolution Identity”. In order to obtain the membership degree (confidence degree) of any given value r taken from the fuzzy least squares estimators, the optimization problems have to be solved. In this paper, we propose two computational procedures to achieve this purpose. Procedure I needs to solve the constrained optimization problems (MP1-I) and (MP1-II), and Procedure II needs to solve the unconstrained optimization problems with bounded decision variable. From the viewpoint of numerical optimization, the unconstrained optimization problem is easier to be implemented than that of constrained optimization problem. Therefore Procedure II is preferred to be invoked to obtain the membership degree (confidence degree) of any given r taken from the fuzzy least squares estimators. However, we might argue that Procedure I is still worth to be used to obtain the membership degree (although the constrained optimization problems have to be solved), since the objective functions are so simple. Therefore, alternatively, Procedure I is still can be used to obtain the membership degree for some problems when those problems make the form of η(h) so complicated in Eq. (18), since we need to solve the unconstrained optimization problems in η(h) many times in STEP 2 if Procedure II is invoked.