Abstract
This 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.
Highlights
One of the most commonly used models in artificial intelligence is the Takagi–Sugeno (T-S) [25] fuzzy system
This paper proposes two methods for training Takagi–Sugeno (T-S) fuzzy systems using batch least squares (BLS) and particle swarm optimization (PSO)
The considered methods can be used for triangular or Gaussian membership functions in the antecedents and high-order polynomials in the consequents of fuzzy rules
Summary
One of the most commonly used models in artificial intelligence is the Takagi–Sugeno (T-S) [25] fuzzy system. Building T-S systems consists of two main tasks: structure identification and parameter estimation. The structure identification is mainly related with determining the number of fuzzy rules. The parameter estimation is related with determining the parameters of fuzzy sets and the coefficients of regression functions in the consequence part. These tasks can be achieved by various optimization techniques such as least squares [24, 26, 32], evolutionary algorithms [5, 8, 32] or particle swarm optimization. Particle swarm optimization (PSO) is a stochastic optimization method that was developed by Kennedy and Eberhart [9, 12]. The usefulness of the PSO in solving a wide range of optimization problems has been repeatedly confirmed [1, 13, 14, 22]
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