A Weighted Least Squares Fuzzy Regression for Crisp Input-Fuzzy Output Data

Abstract

Weighted regression approach is one of the popular problems in robust regression analysis. Recently, robust fuzzy regression models have proven to be alternative approaches to fuzzy regression models attempting to identify, down-weight and/or ignore unusual points (outliers). This paper proposes a new robust fuzzy regression modeling technique known as weighted least squares (LS) fuzzy regression to construct a model for crisp input-fuzzy output data. We introduce a new weighted objective function to overcome the disadvantages of the ordinary LS approach in the presence of outliers. We derive and describe an iterative reweighted algorithm for minimization of the objective function. The algorithm is presented to approximate the weighted estimators of the fuzzy regression by solving the weighted optimization problem. The proposed algorithm decreases the affect of outliers on the model fit attempting to identify/down-weight them. To this end, experiments on datasets with different numbers of outliers are performed. The accuracy of our approach in a real setting is also tested on establishing a predictive model for evaluation of suspended load based on a real world dataset in hydrology engineering. The numerical results show that in the presence of unusual points the proposed weighted fit tracks the main body of the data considerably better than the ordinary LS fuzzy regression fit both in terms of the selected performance criteria and in terms of identifying and down weighting unusual data (outliers). The results of the numerical examples show that this approach has the capability to examine the behavior of value changes in the goodness-of-fit criteria of the fuzzy regression models when the downweighted observations are omitted.