A fuzzy nonlinear univariate regression model with exact predictors and fuzzy responses
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METHODOLOGIES AND APPLICATION
A fuzzy nonlinear univariate regression model with exact predictors and fuzzy responses G. Hesamian1 · M. G. Akbari2
© Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract In this paper, a fuzzy nonlinear univariate regression model with nonfuzzy predictors and fuzzy responses is proposed. For this purpose, both nonlinear parametric and nonparametric methods were utilized. The left and right spreads of unknown fuzzy smooth function were estimated via a popular kernel-based curve-fitting method, while the center was estimated using both parametric and nonparametric curve-fitting methods. In fact, two techniques were suggested and compared in terms of estimating the center of fuzzy smooth function: (1) nonparametric method (similar to the left and right spreads) and (2) parametric method adopted with a common nonlinear regression model called truncated spline regression. Each stage was separately estimated the unknown components were addressed via the conventional statistical regression methods. The proposed method managed to provide a simple and fast estimation/prediction approach for the fuzzy univariate regression analysis for any types of L R-fuzzy numbers. Some common goodness-of-fit criteria were also employed to evaluate the performance of the proposed method. The effectiveness of the developed method was further illustrated through three numerical examples including a simulation study based on a common kernel. The proposed method was also compared with several common fuzzy linear/nonlinear regression models. The numerical evaluations indicated that the proposed parametric method for centers exhibited more accurate results as compared with the nonparametric method. Keywords Fuzzy data · Nonlinear · Goodness-of-fit measure · Spline · Kernel fitting
1 Introduction Regression analysis is a powerful tool for the prediction of the unknown values of the response variables from the known predictors. In real-life problems, such a relationship is usually unknown but can be estimated from a series of observations. Regression-based methods can be classified into linear (including parametric) (Ratkowsky 1990; Ritz and Streibig 2008; Huet et al. 2006) and nonlinear (including nonparametric) approaches (Garson 2012; Györfi et al. 2002; Wasserman 2007). The popular nonlinear regression methCommunicated by V. Loia.
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M. G. Akbari [email protected] G. Hesamian [email protected]
1
Department of Statistics, Payame Noor University, Tehran 19395-3697, Iran
2
Department of Statistics, University of Birjand, Birjand 615-97175, Iran
ods rely on polynomials, spline, or basis functions, while the common nonparametric regression techniques can be classified into the kernel estimators, local polynomial estimators, spline regression, and wavelet thresholding. It should be noted that the fuzzy regression analysis extends the classical regression analysis in cases where some elements of the model are represented by fuzzy data. Since the introduction of the fuzzy regression a
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