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Furthermore, it is proven elsewhere (Ref. 4) that, for a gaussian timerarying signal, the optimal (minimum mean square error) predictor is a linear predictor. Additionally, as a practical fact, most often all we know about the characterization of a given random process is its autocorrelation function. But there always exists a gaussian random process possessing the same autocorrelation function; we therefore might as well assume that the given rsndom process is itself gaussian. That is, the two processes are indistinguishable from the standpoint of the amount of knowledge postulated.
The detrended data sequence is assumed to have stationary statistics. Here 46)= 21, - rk gl 92 APPLIED OPTIMAL ESTIMATION represents the prediction of zk based on the model and on knowledge of the infinite number of z's prior to time tk. It is easy to see that LINEAR DYNAMIC SYSTEMS 93 This type of time series is called an autoregressive (AR) process. Note that the residual rk is the only portion of the measurement zk which cannot be predicted from previous measurements. I t is assumed that the coefficients of this difference equation have been chosen so that the linear system is stable, thus making the autoregressive process stationary.
IS the computed value of a quantity, x, based upon a set of measurements, 2 . An unbiased estimate is one whose expected value is the same as that of the quantity being estimated. A miflimmrm vanonce (unbiased) estimate has the property that its error variance is less than or equal to that of any other unbiased estimate. A consisrenr estimate IS one which converges to the true value of x, as the number of measurements increases. Thus, we shall look for unbiased, minimum variance, consistent estimators.