Non-parametric estimation of non-linearity in a cascade time-series systemby multiscale approximation

The paper addresses the problem of using multiscale approximation for the i dentification of non-linearities in Hammerstein systems. The exciting signa ls are random, stationary and white, with a bounded (unknown) probability d ensity function, and system outputs are corrupted by a zero-mean stationary random noise - white or coloured. The a priori information is poor. In par ticular, no parametric form of the non-linear characteristics is known in a dvance. To recover non-linearities, a class of non-parametric identificatio n algorithms is proposed and investigated. The algorithms use only input-ou tput measurements and are based on multiscale orthogonal approximations ass ociated with scaling functions of compact support. We establish the pointwi se weak consistency of such routines along with asymptotic rates of converg ence. In particular, local ability of the algorithms to discover non-linear characteristics in dependence on local smoothness of the identified non-li nearity, input density and the scaling function is examined. It is shown th at under mild requirements the routines attain optimal rate of convergence. The form and convergence of the algorithms are insensitive to correlation of the noise. (C) 2001 Elsevier Science B.V. All rights reserved.