An Extension of Wiener's Theory of Prediction

Abstract

The theory of prediction described in this paper is essentially an extension of Wiener's theory. It differs from the latter in the following respects. 1. The signal (message) component of the given time series is assumed to consist of two parts, (a) a non‐random function of time which is representable as a polynomial of degree not greater than a specified number n and about which no information other than n is available; and (b) a stationary random function of time which is described statistically by a given correlation function. (In Wiener's theory, the signal may not contain a non‐random part except when such a part is a known function of time.) 2. The impulsive response of the predictor or, in other words, the weighting function used in the process of prediction is required to vanish outside of a specified time interval 0≤t≤T. (In Wiener's theoryT is assumed to be infinite.) The theory developed in this paper is applicable to a broader and more practical class of problems than that covered in Wiener's theory. As in Wiener's theory, the determination of the optimum predictor reduces to the solution of an integral equation which, however, is a modified form of the Wiener‐Hopf equation. A simple method of solution of the equation is developed. This method can also be applied with advantage to the solution of the particular case considered by Wiener. The use of the theory is illustrated by several examples of practical interest.