Parameter estimation for Choquet fuzzy integral based on Takagi–Sugeno fuzzy model

Abstract

Currently, both Choquet fuzzy integral and Takagi–Sugeno fuzzy models are popular synthetic evaluation and fuzzy modeling tools. In this paper, we prove that Choquet fuzzy integral is a special version of Takagi–Suguno fuzzy model in the sense of structure, thus the learning algorithm of the latter is used to develop a parameter estimation procedure for the former. The parameter estimation procedure actually is performed in each ordinal subspace of input space, in which all input data have unique ordering of components. The proposed approach in this paper has been proven to possess better performance than the existing ones by not only theoretical analysis but also experiments.

Introduction

The traditional tool of aggregation for information fusion is the weighted average method that is basically a linear integral (Lebesgue integral). It is assumed that all attributes are non-interactive and, hence, their weighted effects are viewed as additive ones. This assumption is not realistic in many applications [1]. To overcome this problem, a new mathematical tool, fuzzy integral [1], [2], has been introduced with respect to the non-additive fuzzy measure in place of the traditional weighted average. Aggregation functions given by the Choquet fuzzy integral (CFI) and the Sugeno fuzzy integral play an important role in multicriteria decision-making. Some well-known aggregation techniques such as the weighted arithmetic mean and OWA aggregation operators are just special cases of them [3]. In this paper, we focus on CFI that is the most popular aggregating tool for information fusion at present.

Despite the success of CFI in real applications, the practical use of fuzzy measures could be difficult, because for n attributes, one has to identify 2ⁿ parameters in order to define a fuzzy measure. This identification step is very important since all the knowledge concerning these attributes is embedded into the fuzzy measure. In nature, there are essentially three approaches [3]: identification based on the semantics, identification based on learning data, and combining semantics and learning data. However, this first approach is far from being a well-established methodology since it involves a non-negligible amount of experience, and there exist the same defect on the third one. By contrast, the second approach has been addressed in many applications. Especially, the issue from a set of input–output data to determine the fuzzy measure of fuzzy integral was considered as an inverse problem on the synthetic evaluation, and some approaches for this purpose have been described in literature [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14]. Genetic algorithm (GA) [4], [5], [6], [7] is a popular choice in this category, and some gradient descend-based and least square (LS)-based approaches [8], [9], [10], [11] also have shown their success to a certain extent. Grabisch [8] presented a heuristic method called HLMS for identifying parameters of CFI in the insufficient information, where we will address it carefully later in this paper. Chiang [12] developed a learning algorithm for CFI parameters by means of fuzzy neurocomputation, in which a λ-fuzzy measure has been shown powerful feasibility by gradient-descent method. Chen et al. [13], [14] proposed an efficient approach for general fuzzy measure in CFI. Interesting enough, Sugeno [15] and Fujimoto [16] gave a necessary and sufficient condition for a CFI to be decomposable into a group of piecewise linear functions [17]. These work enlighten our research in different views.

Takagi–Sugeno (TS) fuzzy model is very popular additive fuzzy system [18]. It can represent arbitrary non-linear functions by using a few linear functions to certain extent, thus being widely researched and achieving many results [19]. In addition to specific characters of CFI, we shall verify that it is only the special version of TS fuzzy model in the sense of structure. This means that the methods used TS fuzzy model help solve these problems in CFI.

This paper is organized as follows. Section 2 introduces several major characteristics of the CFI and the TS fuzzy models. Then a new TS-based approach of parameter estimation for CFI is discussed in Section 3. Two experiments are used to verify the feasibility and efficiency of the new approach in Section 4, and conclusions are presented in Section 5.