摘要:

We consider the following problem of prediction: During a finite time intervalTthe real valued functionS(t)+N(t) is observed, in whichS(t) is a signal andN(t) is a linearly superimposed noise disturbance. The problem is to predict the value of a given linear functional ofS(t), the predictor formula having certain preassigned ``optimum properties'' among a certain class of predictors. In the case in which the mean value ofS(t) is known, the random components ofS(t) andN(t) are strictly stationary, and the time intervalTis 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 whichTis a finite time interval and the mean value ofS(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 bothS(t) andN(t) are nonstationary in time and merely possess finite continuous covariance and cross‐covariance functions. The analytical tools of probability theory which we use are those developed independently by M. Loe`ve and K. Karhunen in their studies of a class of stochastic processes usually termed processes of second order. Apparently these techniques are practically unknown outside of certain mathematical circles. In the opinion of the author these techniques are extremely powerful in the analysis of transient random phenomena in linear systems.Finally we give an exposition of a method of prediction by the theory of conditional probabilities. This method is applicable when the form of the joint probability distribution of signal and noise is known and hence is applicable to many practical problems, since this distribution is often Gaussian. In this particular case the predictor formula given by the theory of conditional probabilities is identical with the usual linear predictor formula given by the theory of least squares. In case the signal is a Markoff process, the method of conditional probabilities yields a much simpler prediction formula than the usual method of least squares.

ISSN:0021-8979    

DOI:10.1063/1.1702343

出版商:AIP    

年代:1952

数据来源: AIP

ISSN:0021-8979    

DOI:10.1063/1.1702345

出版商:AIP    

年代:1952

数据来源: AIP

ISSN:0021-8979    

DOI:10.1063/1.1702347

出版商:AIP    

年代:1952

数据来源: AIP

ISSN:0021-8979    

DOI:10.1063/1.1702349

出版商:AIP    

年代:1952

数据来源: AIP

ISSN:0021-8979    

DOI:10.1063/1.1702350

出版商:AIP    

年代:1952

数据来源: AIP

ISSN:0021-8979    

DOI:10.1063/1.1702351

出版商:AIP    

年代:1952

数据来源: AIP

ISSN:0021-8979    

DOI:10.1063/1.1702352

出版商:AIP    

年代:1952

数据来源: AIP