Robust Function Approximation

Abstract

In developing a function approximator, we assume that the training data do not include outliers. But outliers may occur in many occasions and the detection of outliers is difficult especially for multi-dimensional data (cf. Section 8.1). In addition, if outliers are included, they affect the approximator’s performance. Outliers may be excluded by preprocessing, but in this chapter we consider excluding outliers while generating fuzzy rules. We focus our discussions on a robust training method for FACG, namely the function approximator based on Takagi-Sugeno type model with the center-ofgravity defuzzification. In FACG, the parameters of the liner equation that defines the output value of the fuzzy rule are determined by the least-squares method. Therefore, if the training data include outliers, the method fails to determine the parameter values correctly. To overcome this problem we use the least-median-of-squares method. Among the original training data set, we randomly select training data more than the number of parameters, and determine the parameter values using the least-squares method. We repeat this for a specified number of times and determine the parameters with the smallest median of squared errors. We compare the proposed method with the least-squares method and the conventional least-median-of-squares method using the data generated by the Mackey-Glass differential equation.