Fuzzy Integral for Rule Aggregation in Fuzzy Inference Systems
The fuzzy inference system (FIS) has been tuned and revamped many times over and applied to numerous domains. New and improved techniques have been presented for fuzzification, implication, rule composition and defuzzification, leaving one key component r
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Electrical and Computer Engineering, Mississippi State University, Starkville, USA [email protected], [email protected] 2 School of Computer Science, University of Nottingham, Nottingham, UK [email protected] 3 Electrical and Computer Engineering, Computer Science, Michigan Technological University, Houghton, USA [email protected] 4 Electrical and Computer Engineering, Computer Science, University of Missouri, Columbia, USA [email protected]
Abstract. The fuzzy inference system (FIS) has been tuned and revamped many times over and applied to numerous domains. New and improved techniques have been presented for fuzzification, implication, rule composition and defuzzification, leaving one key component relatively underrepresented, rule aggregation. Current FIS aggregation operators are relatively simple and have remained more-or-less unchanged over the years. For many problems, these simple aggregation operators produce intuitive, useful and meaningful results. However, there exists a wide class of problems for which quality aggregation requires nonadditivity and exploitation of interactions between rules. Herein, we show how the fuzzy integral, a parametric non-linear aggregation operator, can be used to fill this gap. Specifically, recent advancements in extensions of the fuzzy integral to “unrestricted” fuzzy sets, i.e., subnormal and nonconvex, makes this now possible. We explore the role of two extensions, the gFI and the NDFI, discuss when and where to apply these aggregations, and present efficient algorithms to approximate their solutions. Keywords: Fuzzy inference system · Choquet integral · Fuzzy integral · gFI · NDFI · Fuzzy measure
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Introduction
In Lofti Zadeh’s seminal 1965 paper on fuzzy set (FS) theory, a new philosophy was put forth to address uncertain data and/or information [1]. In 1973, Zadeh introduced fuzzy logic and he suggested that it might be a useful mechanism to model higher-level thought and reasoning in humans [2]. The first application selected was a steam engine and boiler control system and the rules were provided c Springer International Publishing Switzerland 2016 J.P. Carvalho et al. (Eds.): IPMU 2016, Part I, CCIS 610, pp. 78–90, 2016. DOI: 10.1007/978-3-319-40596-4 8
Fuzzy Integral for Rule Aggregation in Fuzzy Inference Systems
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by the system operators [3]. The Mamdani-Assilian fuzzy inference system (FIS) is built on top of Zadeh’s compositional rule of inference (CRI), a generalization of modus ponens, modus tollens, etc. The CRI is a way to calculate a FS-valued output based on crisp or FS-valued inputs and an implication function. Other well-known FISs that generalize the CRI are the Takagi-Sugeno-Kang (TSK) [4], Tsukamoto [5], and single input rule modules (SIRM) FISs [6,7]. Nearly all FISs consist of some subset of fuzzification, CRI, rule firing, rule combination and defuzzification. Of particular importance to this paper is the aggregation step in an FIS, which is responsible for combining the output of different rules and deriving a final compreh
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