Abstract
The aim of this study was to determine the best distance measure for estimating the fuzzy linear regression model parameters with Monte Carlo (MC) methods. It is pointed out that only one distance measure is used for fuzzy linear regression with MC methods within the literature. Therefore, three different definitions of distance measure between two fuzzy numbers are introduced. Estimation accuracies of existing and proposed distance measures are explored with the simulation study. Distance measures are compared to each other in terms of estimation accuracy; hence this study demonstrates that the best distance measures to estimate fuzzy linear regression model parameters with MC methods are the distance measures defined by Kaufmann and Gupta (Introduction to fuzzy arithmetic theory and applications. Van Nostrand Reinhold, New York, 1991), Heilpern-2 (Fuzzy Sets Syst 91(2):259---268, 1997) and Chen and Hsieh (Aust J Intell Inf Process Syst 6(4):217---229, 2000). One the other hand, the worst distance measure is the distance measure used by Abdalla and Buckley (Soft Comput 11:991---996, 2007; Soft Comput 12:463---468, 2008). These results would be useful to enrich the studies that have already focused on fuzzy linear regression models.
Highlights
In many cases in real life, most of the data are approximately known
Fuzzy set theory introduced by Zadeh [28] has found important application areas in different field of science as well as in regression analysis, because fuzzy set theory helps to define the vague relationship between variables or the observations that are reported as imprecise quantities for regression analysis
The best distance measure and the one that should not be used for the estimation of fuzzy linear regression parameters with Monte Carlo (MC) methods are identified without using any mathematical programming or heavy fuzzy arithmetic operations
Summary
In many cases in real life, most of the data are approximately known. Fuzzy set theory introduced by Zadeh [28] has found important application areas in different field of science as well as in regression analysis, because fuzzy set theory helps to define the vague relationship between variables or the observations that are reported as imprecise quantities for regression analysis. Distance measure between two fuzzy numbers plays a key role in estimating fuzzy linear regression model parameters with MC methods. Current studies about MC methods in fuzzy linear regression within the literature do not account for different definitions of distance measure between fuzzy numbers. The main contribution of this study to literature is to figure out the appropriate distance measure between two fuzzy numbers for the estimation of fuzzy linear regression model parameters with MC methods. The best distance measure and the one that should not be used for the estimation of fuzzy linear regression parameters with MC methods are identified without using any mathematical programming or heavy fuzzy arithmetic operations. 5. After the decision of the best and the worst distance measures in MC methods for fuzzy linear regression models, these distance measures are used for the real data sets in Sect.
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