Neurocomputing. 2001. Vol. 28, Nos. 1-3 pp. 165-175 HYBRID TRAINING OF RADIAL BASIS FUNCTION NETWORKS IN A PARTITIONING CONTEXT OF CLASSIFICATION Latifa OUKHELLOU, Patrice AKNIN French National Institute for Transport and Safety Research. INRETS-LTN, 2 av. Malleret-Joinville. 94114 Arcueil Cedex, France Tel : 33 1 47 40 73 37. Email : aknin@inrets.fr Abstract : The design of radial basis function networks (RBF) is rather complex because of the great number of parameters that must be adjusted : positioning and number of kernels, choice of the distance type and centre widths, weight values. This article details these points in the framework of classification tasks with a partitioning approach : the global K-class problem is split into K 2-class sub-problems. An adaptation of the Orthogonal Least Square method is presented in order to select the centres of each sub-classifier in connection with a particular stopping criterion based on the addition of a random centre. Moreover, different choices of distance and centre widths are compared and illustrated by a 4-class problem in the Non Destructive Evaluation domain. Keywords : Radial Basis Function, Selection, Classification, Orthogonal Least Square, Mahalanobis Distance Acknowledgement : this research is supported by the French Research Ministry within the framework of PREDIT Program (Research, Development and Innovation in Transport Systems). 1 1. Introduction Neural networks are parsimonious approximators and they can achieve classification tasks by estimating the posterior probabilities of class membership [1]. In multilayer perceptron networks, made up of scalar product neurons, the discrimination frontiers between classes combine several hyperplans of the input space. In the RBF networks, made up of distance neurons, the classes are delimited by junctions of several ellipsoids. In this way, the RBF networks give more localized responses which show interesting performances, especially in supervised mode with outspread training data base [2]. After the introduction of the formalism of Gaussian RBF, a partitioning approach of the global K-class problem is presented. Then, a hybrid training of the RBF networks is tackled ; the choice of the position and number of centres is presented in the linear regression context, associated with different stopping criteria. Four couples (distance type - centre width) are treated and compared in terms of classification performances. All these particular points are illustrated by a Non Destructive Evaluation application which consists of the classification of eddy current signatures of metro rail defects into four classes. 2. Formalism The output S of a single Gaussian RBF neuron is defined by the following vectorized equation : S = ! X " C A ( ) = exp " X " C A 2 2# 2 $ % & & ’ ( ) ) 1 () X is the input vector, C the centre of the neuron, ! the activation function and β the centre width. Figure 1 : model of a single distance neuron The distance operator is considered in its generalized form :