Real-time fuzzy regression analysis: A convex hull approach

Abstract

In this study, we present an enhancement of fuzzy regression analysis with regard to its aspect of real-time processing. Let us recall that fuzzy regression generalizes the concept of classical (numeric) regression in the sense of bringing additional capabilities that allow the model to deal with fuzzy (granular) data. We show that a convex hull method provides a useful vehicle to reduce computing time, which becomes of particular relevance in case of real-time data analysis. Our objective is to develop an efficient real-time fuzzy regression analysis based on the use of convex hull, specifically a Beneath-Beyond algorithm. In this algorithm, the re-construction of convex hull edges depends on incoming vertices while a re-computing procedure can be realized in real-time. We demonstrate the use of the developed enhancement to application to unit performance assessment and air pollution data. An important role of convex hull is contrasted with the limitations of linear programming used in the “standard” regression.

Introduction

In real-world optimization problems, which we routinely encounter in engineering, management, economy, medicine, psychology, and other disciplines, it is quite common to handle large amounts of various types of data (Tanaka et al., 1982, Bohm and Kriegel, 2000, Watada et al., 2001, Olafsson et al., 2008). In particular, real-time data analysis becomes more important given the growing demand to support efficient managerial practices, which call for on-line (real-time) timely results of data analysis (Gould et al., 2008). The main intent in this setting is to reduce computing overhead in supplying the results in real-time.

Soft computing techniques such as fuzzy logic have become an important alternative to realize effective data analysis, especially for decision making process (Watada et al., 2001, He et al., 2007, Kumar and Ravi, 2007). On the other hand, regression analysis is a generic statistical tool to explore and describe dependencies among variables. When forming a synergy between these two essential modeling methodologies, we arrive at fuzzy regression, which takes full advantage of the strengths of the contributing technologies; cf. Wang and Tsaur (2000). Linear programming (LP) was used to determine the location of the centres and the spreads of the fuzzy coefficients (fuzzy numbers) of the fuzzy regression hyperplane by minimizing an objective function which takes into consideration the total spread of the outputs of the model treated as fuzzy numbers (Tanaka et al., 1982, Sakawa and Yano, 1992, Yao and Yu, 2006).

Let us recall that convex hull is a fundamental concept present in many applications encountered in pattern recognition, image processing, and statistics. Convex hull is defined as the smallest convex polygon located in a multidimensional data space which contains all point set (vertices) (Shapiro, 2004). In other words, convex hull corresponds to the intuitive notion of a “boundary” of a set of points and as such can be used to approximate a shape of any object of complex geometry.

In real-world problems such as those encountered present in economics, bio-computing or engineering, we are concerned with the massive data sets of high dimensionality to analyze in a limited time or even in real-time (Taylor, 2008). In addition to experimental evidence of numeric nature, some data can be described in a linguistic term, which immediately invokes the concept of fuzzy sets (Chang and Ayyub, 2001).

Related to the methods used for real-time application where statistical regression has been shown to be highly relevant, we can also observe some shortcomings, which arise when dealing with several characteristics of the data or making some simplifying yet not necessarily fully legitimate assumptions, cf. (Shapiro, 2004):

• Difficulties with a thorough verification of assumptions about data distributions,

• Vagueness present in the relationships between input and output variables,

• Ambiguity of events or non-Boolean degrees to which they occur, and

• Inaccuracy and distortion introduced by linearization.

In all these scenarios, the use of the “standard” regression might raise some hesitation. Here the use of fuzzy regression arises as a viable alternative. There are two general ways supporting the development process of fuzzy regression (Dom, 2007).

• Models where the relationships among the variables is inherently fuzzy, and

• Models where the input (independent) variables themselves are fuzzy.

There are some arguments that are worth highlighting with regard to fuzzy regression. As an example, Wang and Tsaur (2000) concluded that there is no proper interpretation of fuzzy regression interval. Besides that expert could still provide an interval of possible values but also indicate the probability of occurrence of each one of them. Obviously, this approach would require more information (Aznar and Guijarro, 2007, Guo and Tanaka, 2010).

Given the explanation presented above, a convex hull approach can help implement real-time fuzzy regression analysis by serving as an alternative optimization vehicle.

The main objective of this research is to enhance the implementation procedure of real-time fuzzy regression analysis with the use of the convex hull approach. In addition, the adaptation of selected algorithm of this approach, specifically the Beneath-Beyond algorithm, helps address the limitations of the generic implementation of fuzzy regression when applied to the analysis of real-time data. The unit performance assessment and air pollution evaluation offer two numeric examples, which exemplify the performance of the proposed approach.

The paper is organized as follows. Section 2 serves as a related literature review, which includes real-time data analysis, brief review of the fundamentals of fuzzy regression, outlines an LP approach, computational geometry and convex hull approach, and related research studies. Next, Section 3 presents the real-time fuzzy regression model realized with the use of the convex hull approach. Section 4 is devoted to empirical experiments. Finally, Section 5 presents concluding remarks.