Kernel Functions in Takagi-Sugeno-Kang Fuzzy System with Nonsingleton Fuzzy Input
Résumé
Algorithms for supervised classification problems usually does not consider imprecise data, e.g., observed data whose samples can be represented by a collection of intervals, histograms, list of values, fuzzy sets among others. Fuzzy theory is a naturally choice for imprecise data, and state of the art techniques such a kernel methods is a natural choice for supervised classification problems. In this context, under some assumptions Takagi-Sugeno-Kang (TSK) fuzzy system rules are equivalent to positive definite kernels (PDK), such relationship was given considering only singleton fuzzy sets to model crisp input. However, imprecise data are better modeled by membership functions of nonsingleton fuzzy sets, but, the relationship among nonsingleton fuzzy input, TSK fuzzy systems and PDK's is unknown. In this work, we will study such relashionship, for that, we formulate an extension of TSK fuzzy systems to deal with nonsingleton fuzzy input and then we show that a new class of PDK's are derived from it. We give three examples of nonsingleton TSK kernels which are close related to Vapnik's vicinal kernels. Also, based on those TSK induced kernels and in the concept of distance substitution kernels, we formulate two PDK's for interval data. Potential applications for the proposed kernels are pattern recognition problems with imprecise data. Experiments conducted with interval datasets show better performances that state of art approaches
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