TSK-fuzzy modeling based on ϵ-insensitive learning

In this paper, new learning methods tolerant to imprecision are introduced and applied to fuzzy modeling based on the Takagi-Sugeno-Kang fuzzy system. The fuzzy modeling has an intrinsic inconsistency. It may perform thinking tolerant to imprecision, but learning methods are zero-tolerant to imprecision. The proposed methods make it possible to exclude this intrinsic inconsistency of a fuzzy modeling, where zero-tolerance learning is used to obtain fuzzy model tolerant to imprecision. These new methods can be called ε-insensitive learning or ε learning, where, in order to fit the fuzzy model to real data, the ε-insensitive loss function is used. This leads to a weighted or "fuzzified" version of Vapnik\´s support vector regression machine. This paper introduces two approaches to solving the ε-insensitive learning problem. The first approach leads to the quadratic programming problem with bound constraints and one linear equality constraint. The second approach leads to a problem of solving a system of linear inequalities. Two computationally efficient numerical methods for the ε-insensitive learning are proposed. The ε-insensitive learning leads to a model with the minimal Vapnik-Chervonenkis dimension, which results in an improved generalization ability of this model and its outliers robustness. Finally, numerical examples are given to demonstrate the validity of the introduced methods.