Kernel functions in Takagi-Sugeno-Kang fuzzy system with nonsingleton fuzzy input

Algorithms for supervised classification problems usually do not consider imprecise data, e.g., interval collections, histograms, list of values, fuzzy sets among others that represent observed data. Moreover, fuzzy set theory is a natural choice to model imprecision and kernel methods are the state of the art in learning machines. Previous works describe a link between both areas: the interaction between fuzzy rules of Takagi-Sugeno-Kang (TSK) fuzzy systems and singleton fuzzy inputs are equivalent to positive definite kernels (PDK). Current research in fuzzy systems shows that nonsingleton fuzzy sets can be used to model imprecise data. In this work, we study the relationship between positive definite kernels and TSK fuzzy systems with nonsingleton inputs. As a result, we define an extension of TSK fuzzy systems to deal with nonsingleton fuzzy input and we show that the interaction between fuzzy rules and nonsingleton fuzzy inputs induces a new class of PDK, the nonsingleton TSK kernel class, which are close related, but not equal, to Vapnik´s vicinal kernels. Finally, based on nonsingleton TSK kernels and distance substitution kernels we give a general procedure to formulate PDKs for interval data. Experiments conducted with interval datasets show better performance than the state of the art approaches.