Fuzzy Sets and Fuzzy Systems

Theoretical aspects are provided for inference based on α-cuts and generalized mean (α-GEMII). In order to clarify the basic properties of the inference, fuzzy tautological rules (FTRs) are focused on, which are composed by setting fuzzy sets in consequent parts identical to those in antecedent parts of initially given fuzzy rules. It is mathematically proved that the consequences deduced with FTRs are closer to given facts as the number of FTRs increases. The aspects provided in this paper are appropriate from axiomatic viewpoints and can contribute to interpretability in fuzzy systems constructed with α-GEMII. They are not obtained in conventional methods based on the compositional rule of inference. Simulations are performed by evaluating difference (mean square errors) between given facts and deduced consequences under the condition that convex and symmetric fuzzy sets are given as facts. Their results show that the difference becomes smaller as the number of FTRs increases. Thereby, it is confirmed that α-GEMII has an advantage in the interpretability with respect to FTRs over the conventional methods.

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Training High-Order Takagi-Sugeno Fuzzy Systems Using Batch Least Squares and Particle Swarm Optimization

International Journal of Fuzzy Systems ◽

10.1007/s40815-019-00747-2 ◽

2019 ◽

Vol 22 (1) ◽

pp. 22-34 ◽

Cited By ~ 5

Author(s):

Krzysztof Wiktorowicz ◽

Tomasz Krzeszowski

Keyword(s):

Particle Swarm Optimization ◽

Least Squares ◽

Fuzzy Sets ◽

Fuzzy Systems ◽

Piecewise Linear ◽

Particle Swarm ◽

Fuzzy Model ◽

Pso Algorithm ◽

Swarm Optimization ◽

Takagi Sugeno

AbstractThis paper proposes two methods for training Takagi–Sugeno (T-S) fuzzy systems using batch least squares (BLS) and particle swarm optimization (PSO). The T-S system is considered with triangular and Gaussian membership functions in the antecedents and higher-order polynomials in the consequents of fuzzy rules. In the first method, the BLS determines the polynomials in a system in which the fuzzy sets are known. In the second method, the PSO algorithm determines the fuzzy sets, whereas the BLS determines the polynomials. In this paper, the ridge regression is used to stabilize the solution when the problem is close to the singularity. Thanks to this, the proposed methods can be applied when the number of observations is less than the number of predictors. Moreover, the leave-one-out cross-validation is used to avoid overfitting and this way to choose the structure of a fuzzy model. A method of obtaining piecewise linear regression by means of the zero-order T-S system is also presented.

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