A robust multiple regression model based on fuzzy random variables

Abstract

In the present paper, a novel robust multiple regression model with fuzzy intercepts and non-fuzzy regression coefficients was proposed. A two-stage robust procedure adopted with fuzzy random variables and $\alpha$-values of $LR$-fuzzy was also introduced to estimate the components of the model. Some common goodness-of-fit criteria were also used to evaluate the performance of the proposed method. The effectiveness of the proposed method was compared to some common fuzzy robust regression models through three numerical examples including a simulation study. The numerical results indicated the lower sensitivity of the proposed model to outliers and its higher precision compared to the other existing robust regression methods.

Introduction

Multiple regression analysis is the most basic and commonly used statistical technique to estimate the relationships between some independent predictor variables and a dependent response variable. Such techniques often rely on the least square errors. However, the least square estimations may misbehave when the data set involves some outliers. One approach to overcome this problem is to remove the influential observations from the least-square fit [1]. A modified approach, known as robust regression, employs a fitting criterion with no such susceptibility to these unusual data as much as the least squares. The most common general method of robust regression is M-estimation introduced by Huber [2]. It generally gives better accuracies compared to the least-square method as it utilizes a weighting mechanism to weigh down the influential observations. However, the traditional regression analysis often relies on exact information such as data or coefficients. Since its introduction by Tanaka et al. [3], the fuzzy regression methods have gained considerable attention in real-life applications. These methods can be categorized into two classes: the observations of the predictors can be either fuzzy numbers [4], [5], [6], [7], [8], [9], [10], [11], [12] or real value quantities [13], [14], [15], [16], [17].

Several outlier robust methods have been proposed based on fuzzy information [18], [19], [20], [21], [22], [23]. In this paper, a common and popular robust estimation method (called M-estimation [2]) was employed in cases where both predictors and responses are reported as random fuzzy quantities. For this purpose, a novel idea was proposed for the fuzzy multiple regression model. To do so, a two-stage procedure was proposed to estimate the fuzzy intercept and exact regression coefficients of the model at the presence of outliers. The performance of the developed method was then compared with several already-available fuzzy robust regression models in terms of some common goodness-of-fit criteria. For practical reasons, the proposed method was further evaluated through a simulated study and two applied examples. The numerical results indicated the sufficient accuracy of the proposed method (compared to others) at the presence of outliers in the data set.

The rest of this paper is organized as follows: Section 2 reviews some concepts including $\alpha$-values of fuzzy numbers and fuzzy random variables. The methodology is proposed in Section 3 to estimate the fuzzy intercept and non-fuzzy coefficients of a fuzzy multiple regression model with fuzzy predictors and fuzzy responses. Section 4 illustrates three numerical examples to assess the effectiveness and performance of the proposed method relative to the other fuzzy multiple regression methods in terms of some common performance measures. Finally, the main contributions of this paper are summarized in Section 5.