Some Approaches for Estimating the Parameters of Fuzzy/Intuitionistic Fuzzy Regression Models

Abstract

In the last few years, several approaches have been proposed in the literature to estimate the parameters (regression coefficients) of fuzzy/ intuitionistic fuzzy regression models (regression models in which some or all the variables and parameters are represented as fuzzy/ intuitionistic fuzzy numbers). In this thesis, limitations and flaws of some existing approaches have been pointed out. Also, to resolve the flaws/ to overcome the limitations of the existing approaches, new approaches have been proposed. The chapter-wise summary of the thesis is as follows: Chapter 1 Introduction In this chapter, the need of interval/ fuzzy set/ intuitionistic fuzzy set in regression analysis is discussed. Also, a brief review of some existing approaches for estimating the parameters of interval/ fuzzy / intuitionistic fuzzy regression models is discussed. Chapter 2 Mehar approach for estimating the parameters of fully interval linear regression models. Souza et al. [158] proposed an approach for estimating the parameters of fully interval linear regression models (linear regression models in which the values of all the input variables, output variable and the coefficients are represented as intervals). In this chapter, limitations of Souza et al. approach [158] are pointed put. Also, it is pointed out that Souza et al. [158] have used some mathematical incorrect assumptions in their proposed approach. Therefore, it is mathematically incorrect to use Souza et al. approach [158]. Furthermore, a new approach (named as Mehar approach) is proposed for estimating the parameters of fully Interval linear regression models. Finally, the proposed Mehar approach has been illustrated with the help of numerical examples. Chapter 3 Modified mathematical programming method for estimating the parameters of FLR model of type-I Chen and Hsueh [41] pointed out the flaws of some existing methods for estimating the parameters of fuzzy linear regression (FLR) models of type-I (linear regression models in which the values of the input variables are real numbers, whereas, the values of the output variables are unrestricted triangular fuzzy numbers) and proposed a mathematical programming method, based on distance criteria, to estimate the parameters. After a deep study, it is observed that Chen and Hsueh [41] have considered some mathematical incorrect assumptions in their proposed method. Hence, it is mathematically incorrect to use Chen and Hsueh’s method [41]. In this chapter the mathematical incorrect assumptions, considered by Chen and Hsueh [41], are pointed out. Also, Chen and Hsueh’s method [41] has been modified. Chapter 4 Modified fuzzy least absolute linear regression method for estimating the parameters of FLR models of type-I Zeng et al. [193] proposed an expression to evaluate the absolute distance between two triangular fuzzy numbers. Also, using the proposed expression, Zeng et al. [193] proposed a method to estimate the parameters of FLR models of type-I. It is pertinent to mention that as Zeng et al.’s method [193], in its present form, cannot be used to estimate the parameters of such a FLR model of type-I in which some or all values of the input variables are negative real numbers. So, one may use the multiplication of a real number with a triangular fuzzy number, proposed by Zeng et al. [193], to generalize Zeng et al.’s method [193]. However, in this chapter, it is shown that the multiplication of a real number with a triangular fuzzy number, proposed by Zeng et al. [193], is not valid. Therefore, it is mathematically incorrect to use it for generalizing Zeng et al.’s method [193]. Also, using the valid multiplication of a real number with an unrestricted triangular fuzzy number, Zeng et al.’s method [193] is generalized to estimate the parameters of FLR models of type-I. Chapter 5 Modified fuzzy least absolute linear regression method for estimating the parameters of FLR models of type-II and fully FLR models Li et al. [116] proposed an expression to evaluate the absolute distance between two trapezoidal fuzzy numbers. Also, using the proposed expression, Li et al. [116] proposed (i) A method to estimate the parameters of FLR of type-II (linear regression models in which the values of the independent variables are unrestricted real numbers, whereas, the values of the dependent variables are unrestricted trapezoidal fuzzy numbers) (ii) A method to estimate the parameters of FLR models of type-II (linear regression models in which the values of the independent and dependent variables, are unrestricted trapezoidal fuzzy numbers, whereas, the regression coefficients are unrestricted real numbers) (iii) A method to estimate the parameters of FLR models (FLR models in which the values of the independent variables, dependent variables and the regression coefficients are unrestricted trapezoidal fuzzy numbers). In this chapter, it is shown that (i) Li et al.’s method [116] to estimate the parameters of FLR models of type-II is not valid (ii) Li et al.’s method [116] to estimate the parameters of fully FLR models is not valid. Also, valid methods are proposed to estimate the parameters of FLR models of type-II and fully FLR models. Chapter 6 Modified approach for estimating the parameters of fully interval-valued FLR models Rabiei et al. [143] proposed an approach for estimating the parameters of fully interval-valued FLR models (linear regression models in which the observation of the response, independent variables as well as the parameters are triangular interval valued fuzzy numbers. However, after a deep study, it is observed that a mathematical incorrect assumption has been considered in this approach. Furthermore, it is observed that to resolve this mathematical incorrect assumption, there is need to propose the multiplication of an unrestricted triangular interval valued fuzzy number with a restricted triangular interval valued fuzzy number (observed values of independent variable). Keeping the same in mind, in this chapter, the same type of multiplication is proposed. Also, with the help of the proposed multiplication, a modified approach is proposed to estimate the parameters of fully interval-valued FLR models. Chapter 7 Mehar approach for estimating the parameters of fully intuitionistic FLR models. Arefi and Taheri [7] proposed an approach for estimating the parameters of fully intuitionistic FLR models (linear regression models in which all the variables and parameters are considered as intuitionistic fuzzy numbers). In future, other researchers may use this approach in real life problems. However, after a deep study, it is observed that some mathematical incorrect assumptions have been considered in this approach. Therefore, it is scientifically incorrect to use this approach. Keeping the same in mind, in this paper, a new approach (named as Mehar approach) is proposed estimate the parameters of fully intuitionistic FLR models. Furthermore, the proposed Mehar approach has been illustrated with the help of a numerical example. Chapter 7 Future Scope In this chapter, some open research problems are discussed.