Robust training of radial basis networks
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ROBUST TRAINING OF RADIAL BASIS NETWORKS O. G. Rudenkoa† and O. O. Bezsonova‡
UDC 519.71
Abstract. Robust training of radial-basis networks under non-normally distributed noise is considered. The simulation results show that multistep projection training algorithms minimizing various forms of module criteria are rather efficient in this case. Keywords: neural network, robust training, nonlinear object, identification, basis function. INTRODUCTION Radial basis networks (RBSs) are widely used in problems of identification, control, pattern recognition, etc. [1–3]. The presence of rather efficient recursive algorithms of training these networks allows one to use them to solve the mentioned problems in real time [4–8]. In practice, one is forced to train a network and to further use it with allowance for existing measurement noises. The majority of existing training methods are based on the use of rigid and hardly tested conditions that are connected with the hypothesis of normality of the noise distribution law and are substantiated by references to the central limit theorem. As is well known [9], the normal law of probability distribution describes noises inherent in measurements performed under absolutely stable measurement conditions, and the Laplace distribution having longer “tails” describes noises arising under maximally unstable conditions. Accordingly, in the case of Gaussian noises, training algorithms are based on the least-squares method (LSM), and, in the case of noises distributed according to the Laplace law, they are based on the least-modules method (LMM). Both methods are optimal under their conditions, and solutions obtained with their help can be significantly different. Moreover, since these extreme cases are very seldom realized in practice, neither the Gauss law nor the Laplace law, as a rule, do not take place. In [10, 11], some types of classes of distributions are considered that occur in solving practical problems, namely, Ð1 is the class of nondegenerate distributions, Ð 2 is the class of distributions with bounded dispersions, Ð 5 is the class of finite distributions (the absolute noise value is bounded, and any information on the density of its distribution is absent), Ð 3 , Ð 4 , and Ð 6 are, respectively, the classes of approximately normal, approximately uniform, and approximately finite distributions described by the Tukey–Huber model [12–14] (1) p( x ) = (1- e ) p 0 ( x ) + eq( x ) , where p 0 ( x ) is the density of the corresponding basic distribution, q( x ) is the density of (arbitrary) garbling distribution, and e Î[ 0, 1] is the parameter characterizing the distortion level of the basic distribution. For all these classes, the least favorable distributions are found, i.e., those minimizing Fisher information. In particular, the minimum of Fisher information for the class P1 yields the Laplace distribution p * ( x ) = L( 0, sx ) , for the class P2 yields the Gauss distribution p * ( x ) = N ( 0, s 2 ) , and, for the classes of finite distributions, the least favorable density
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