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Guide to Radial Basis Functions for Model Building

A radial basis function has the form

where x is a n-dimensional vector, is an n-dimensional vector called the center of the radial basis function, ||.|| denotes Euclidean distance, and is a univariate function, defined for positive input values, that we shall refer to as the profile function.

The model is built up as a linear combination of N radial basis functions with N distinct centers. Given an input vector x, the output of the RBF network is the activity vector given by

where is the weight associated with the jth radial basis function, centered at , and . The output approximates a target set of values denoted by y.

A variety of radial basis functions are available in MBC, each characterized by the form of . All of the radial basis functions also have an associated width parameter , which is related to the spread of the function around its center. Selecting the box in the model setup provides a default setting for the width. The default width is the average over the centers of the distance of each center to its nearest neighbor. This is a heuristic given in Hassoun (see References) for Gaussians, but it is only a rough guide that provides a starting point for the width selection algorithm.

Another parameter associated with the radial basis functions is the regularization parameter . This (usually small) positive parameter is used in most of the fitting algorithms. The parameter penalizes large weights, which tends to produce smoother approximations of y and to reduce the tendency of the network to overfit (that is, to fit the target values y well, but to have poor predictive capability).

The following sections explain the different parameters for the radial basis functions available in the Model-Based Calibration Toolbox product, and how to use them for modeling.

Radial Basis Functions

Types of Radial Basis Functions

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