Title

Design and evaluation of radial basis function model for function approximation

Date of Award

1998

Degree Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Electrical Engineering and Computer Science

Advisor(s)

Amrit L. Goel

Second Advisor

Carlos Hartman

Keywords

Singular value decomposition, Radial basis function, Function approximation

Subject Categories

Computer Sciences

Abstract

The goal of function approximation is to construct a model which learns an input-output mapping from given data and performs good predictions on future inputs. Radial basis function (RBF) models, a particular class of neural networks, have recently become popular for function approximation because of their fast learning ability and good mathematical properties. The current algorithms for determining RBF model, however, tend to produce inconsistent designs due to their ad-hoc nature.

The main objective of this research is to develop a mathematical framework for the design and evaluation of RBF models for function approximation. In particular, we use singular value decomposition to investigate some key properties of the interpolation and design matrices which form the foundation of the SC algorithm proposed in this dissertation. This algorithm leads to a consistent approach for determining the RBF parameters, viz. number of basis functions, their widths, centers and weights.

It is shown that the number of basis functions can be determined by evaluating the interpolation matrix rank associated with a specified width. For this purpose, a new representational capability measure is introduced and its relationship to singular values is derived. The basis function centers for constructing the design matrix are determined by QR factorization with column pivoting of right singular vectors of the interpolation matrix. It is shown that the selected centers reflect the best compromise between structural stability and minimum residual. The weights are computed by the usual pseudo inverse method. The model which gives the minimum prediction error with respect to validation data is selected. A simulation study is also undertaken to investigate the bias-variance characteristics of the constructed RBF models. Some remarks about the SG algorithm from the perspective of best approximation are presented.

The SG algorithm proposed in this dissertation provides an objective and systematic design methodology due to its origins in the mathematical properties of the interpolation and design matrices associated with the RBF model. It also provides a rich analytical framework for performance and sensitivity analyses.

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