Problems of Control and Information Theory 8. (Budapest, 1979)

1979 / 1. szám - Greblicki, W.: Nonparametric system identification by orthogonal series

68 GREBLICKI: NONPARAMETUIC SYSTEM IDENTIFICATION BY ORTHOGONAL SERIES is minimized by the optimal model фо(х) = j yf(ylx) dy . (2) Throughout this paper all integrals are taken over either the space % or the space К or both of them, respectively. Let us assume that the input density f(x) is known, but the conditional output density f(y/x) is completely unknown. The identification problem is to estimate the optimal model having a learning sequence, i.e. a sample of independent observations (Xv Гх),.. . , (Xn, Yn) of the pair (X, Y). In this paper the problem is treated by orthogonal series. It is shown that the method is asymptotically optimal, i.e. when a number of observations tends to infinity, the sequence of models converges to the optimal model and the quality index converges to the minimal one. 3. Identification procedures Let be the space of all /^-measurable and square integrable models, and let {(pj(x)} (i = 0, 1, 2, . . .) be a complete system of orthonormal functions of L.,. Of course, the optimal model is not necessarily square integrable. If, however, it is an element of L2, where at = j Фи(х) cpi(x) dx = E {Y <pi{X)lf(X)} . All the coefficients of this expansion are estimated from the learning sequence according to the following formula П = n^2lYjcpi(Xj)lf(Xj)] i=i N(n) л Фп(х) = 2 <Pi(x) , i — O where {N(n)} is a sequence of numbers. We shall now give a theorem on the asymptotic optimality of this model. and as a model we take фо(х) = a'i <P‘(X) - 1=0

Next