Fuzzy regression methods - a comparative assessment

...In order to integrate both random and fuzzy types of uncertainty into one regression model, a fuzzy least-squares regression was introduced by Chang (Chang 2001, Chang and Ayyub 2001) to model manufacturing processes which can address both the fuzziness and the randomness of a system in the development of a fuzzy least-squares regression model...

...The deviation between the observed data and the estimated data encountered in classical regression is because of the measurement error or random variations of parameters [7]–[9]...
...Such random variations can be represented as a normal distribution of some variance and zero mean, which makes statistical techniques effective in determining the functional relationship for such types of data [7]...
...The FR model is referred to as a fuzzy or possibilistic model of classical regression, while classical regression is based on the principles of statistics [7]...
...In the former approach, the aim is to minimize the overall fuzziness by minimizing the total spread of the fuzzy coefficients, while the estimated outputs and the observed ones are within a certain h-level of confidence [7], [20]...
...The term h expresses the fitness between the estimated fuzzy outputs and the observed ones [7]...
...On the other hand, in the LS-based model, a similarity measure between observed and estimated outputs is used as the measurement for the fitness of the model [7], [8]...

...In order to integrate both random and fuzzy types of uncertainty into one regression model, a fuzzy least-squares regression was introduced by Chang (Chang 2001, Chang and Ayyub 2001) to model manufacturing processes which can address both the fuzziness and the randomness of a system in the development of a fuzzy least-squares regression model...

...As pointed out by Wang and Tsaur [37, 38], the advantage of this approach is its simplicity in programming and computation, but it has been criticized to provide too wide ranges in estimation which could not give much help in application [38] and not to utilize the concept of leastsquares [15]...
...Different aspects of this approach were investigated by Celmins [12, 13], Diamond [20], Savic and Pedrycz [33], and Chang and Ayyub [15] Celmins [12, 13] defines a compatibility measure between fuzzy data and a model, and uses this measure as a model-fitting criterion...
...Chang and Ayyub [15] discussed reliability issues of FLSRA such as standard error and correlation coefficient...
...8i ¼ 1; �� ;M; aj 2 R;cj � 0; j ¼ 1;2; �� ;N;xi0 ¼ 1; 0 � l � 1: Chang et al. [15] introduced a FLR model...
...Chang and Ayyub [15] (Eq. 10) The input and output data are crisp but coefficients are triangle fuzzy numbers...

...Ayyub [11] in current literature, where the coefficients of input variables are assumed to be fuzzy numbers...

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