Multiobjective Optimization and Comparisonof Nonsingleton Type-1 and Singleton IntervalType-2 Fuzzy Logic Systems
Ana Bel ́en Cara, Christian Wagner
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Singleton interval type-2 fuzzy logic systems (FLSs)have been widely applied in several real-world applications, whereit was shown that the singleton interval type-2 FLSs outperformtheir singleton type-1 counterparts in applications with high un-certainty levels. However, one of the main criticisms of singletoninterval type-2 FLSs is the fact that they outperform singletontype-1 FLSs solely based on their use of extra degrees of freedom(extra parameters) and that type-1 FLSs with a sufficiently largenumber of parameters may provide the same performance as in-terval type-2 FLSs. In addition, most works on type-2 FLSs onlycompare their results with singleton type-1 FLSs but fail to con-sider nonsingleton type-1 systems. In this paper, we aim to directlyaddress and investigate this criticism. In order to do so, we willperform a comparative study between optimized singleton type-1,nonsingleton type-1, and singleton interval type-2 FLSs under thepresence of noise. We will also present a multiobjective evolution-ary algorithm (MOEA) for the optimization of singleton type-1,nonsingleton type-1, and singleton interval type-2 fuzzy systems forfunction approximation problems. The MOEA will aim to satisfytwo objectives to maximize the accuracy of the FLS and minimizethe number of rules in the FLS, thus improving its interpretability.Furthermore, we will present a methodology to obtain “optimal”consequents for the FLSs. Hence, this paper has two main con-tributions: First, it provides a common methodology to learn thethree types of FLSs (i.e., singleton type-1, nonsingleton type-1, andsingleton interval type-2 FLSs) from data samples. The secondcontribution is the creation of a common framework for the com-parison of type-1 and type-2 FLSs that allows us to address theaforementioned criticism. We provide details of a series of experi-ments and include statistical analysis showing that the type-2 FLSis able to handle higher levels of noise than its nonsingleton andsingleton type-1 counterparts.