A Fuzzy Convex Nonparametric Least Squares Method with Different Shape Constraints

The main drawback of the typical fuzzy least squares approach is that the resulting fuzzy regression model is linear, and the model's accuracy decreases with the increases in the magnitudes and the number of independent variables. Some nonparametric methods, such as the kernel regression method, have been proposed to overcome these drawbacks. In this paper, we derive a novel fuzzy nonparametric regression method. A convex nonparametric least squares approach (CNLS) is employed for the fuzzy regression models with crisp input fuzzy output data (Fuzzy-CNLS). Like Diamond's fuzzy least squares method, the fuzzy regression model is divided into three submodels, the Center, the Left endpoint, and the Right endpoint. We employ CNLS for each sub-model. Hence, the resulting Fuzzy-CNLS regression model consists of three sets of CNLS results (hyperplanes) for each sub-model, representing a fuzzy nonlinear regression model. One of the advantages of the original CNLS over ordinary least squares (OLS) is that the coefficient of determination of CNLS must be greater than that of OLS. Hence, the goodness of fit of Fuzzy-CNLS is better than the fuzzy least squares methods. On the other hand, CNLS can accommodate the concavity or convexity constraints for the regression functions following the concavity or convexity pattern, respectively. With these shape (concavity or convexity) constraints, it is considered that Fuzzy-CNLS is less sensitive to outliers. A similarity-distance measure is used to select the shape constraints and to evaluate the performance of Fuzzy-CNLS. An illustrative example and an application example are given. The numerical results show that Fuzzy-CNLS is better than Diamond's least squares method and fuzzy least absolute linear regression method in terms of the similarity measure.

Automated summary

This paper presents a novel fuzzy nonparametric regression method by dividing the fuzzy regression model into three submodels and using a convex nonparametric least squares approach for fitting each submodel. The resulting fuzzy non-linear regression model shows better goodness of fit compared to traditional fuzzy least squares methods and is less sensitive to outliers.