Statistical Methods in Prediction, Filtering, and Detection Problems

Abstract

Introduction. The closely related problems of prediction and filtering of random time series and of the detection of signals in noise from continuous observations over a finite time interval have received extensive treatment in recent scientific literature. The books by Grenander and Rosenblatt [5] and Laning and Battin [8] contain fairly adequate bibliographies on the subject. The primary reason for the present paper on this same subject is the exposition of a practical, direct method for the solution of a certain restricted class of these problems. An examination of the difficulties involved in the indirect methods for the solution of such problems by means of linear operators which are solutions of linear integral equations should convince the reader of the need for practical, direct methods (see Laning and Battin [81, Chapter 8). The methods of the present paper are not simple. Indeed they may be very tedious in nontrivial problems. The best that can be said is that application of these methods should lead to substantially shorter and easier calculations than does the integral equation method. In most of the existing literature on prediction, filtering, and detection problems that involve random time series, solutions for these problems have been dependent upon the solutions of certain linear integral equations or systems of such equations. Notable exceptions are found in the papers by Mann [9] and Reich and Swerling [12] where integral equations are not used in solving certain special cases of the filtering and detection problems. The methods used by Mann [9] and Mann and Moranda [10] will be simplified in the present paper and extended to cover somewhat more general stochastic processes than were considered by those authors. They treated two noise processes by similar but entirely separate statistical analyses. In the present paper it is shown that an important extension of their second case is reducible to the first case by a simple linear transformation. Thus separate treatments are unnecessary. The standard criterion of minimum error variance will be applied to prediction, filtering, and detection of the mean value function of a stationary stochastic process which has a spectral density function that is the reciprocal of a polynomial. This problem is distinct from the problems of prediction and filtering of a stochastic function considered by Wiener [15]. A criterion alternative to the minimum error variance will be considered in comparison with the latter.