Universal schemes for learning the best nonlinear predictor given the infinite past and side information

Authors:
P. Algoet
Affiliations:
Ysselmeerstraat 5, Ardooie, Belgium
Venue:
IEEE Transactions on Information Theory
Year:
2006

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Abstract

Let {Xt} be a real-valued time series. The best nonlinear predictor of X0 given the infinite past X-∞-1 in the least squares sense, is equal to the conditional mean E{X0|X-∞-1}. Previously, it has been shown that certain predictors based on growing segments of past observations converge to the best predictor given the infinite past whenever {Xt} is a stationary process with values in a bounded interval. The present paper deals with universal prediction schemes for stationary processes with finite mean. We also discuss universal schemes for learning the conditional mean E{X0|X -∞-1Y-∞-1Y0 } from past observations of a stationary pair process {(Xt , Yt)}, and schemes for learning the repression function m(y)=E{X|Y=y} from independent samples of (X, Y)