On the Theory of Prediction of Nonstationary Stochastic Processes

We consider the following problem of prediction: During a finite time interval T the real valued function S(t)+N(t) is observed, in which S(t) is a signal and N(t) is a linearly superimposed noise disturbance. The problem is to predict the value of a given linear functional of S(t), the predictor formula having certain preassigned ``optimum properties'' among a certain class of predictors. In the case in which the mean value of S(t) is known, the random components of S(t) and N(t) are strictly stationary, and the time interval T is infinite, a complete solution to this problem has been given by N. Wiener. (In the case of discrete time series, the solution was given by A. Kolmogoroff.) This theory has been extended by L. Zadeh and J. Ragazzini [J. Appl. Phys. 21, 645 (1950] to the case in which T is a finite time interval and the mean value of S(t) is unknown but is restricted to be a polynomial in time. We extend the above theories to the case in which the random components of both S(t) and N(t) are nonst...