An overview of adaptive linear minimum mean square error predictor performance

In discrete time prediction applications, the minimum mean square error performance criterion is often coupled with the implementation constraint of a linear predictor structure, resulting in a linear minimum mean square error (LMMSE) predictor. The LMMSE predictor thus realized is a simple linear combination of a finite number of observations immediately preceding the data point to be predicted, where the optimal coefficients which weight these observations are functions of the mean and covariance of the random process on which the predictor is to operate. Unfortunately, in many applications, these moments are not known a priori. In such applications, adaptive LMMSE predictors, the coefficients of which are "adapted to" or "learned from" a finite number of previous data observations in such a manner as to continually seek to minimize the mean square prediction error, are employed. This paper will review the historical investigations into the performance of adaptive LMMSE predictors, beginning with the results of Davisson (1964, 1965, and 1966) followed by the later work of Akaike (1969, 1970, and 1971), who rediscovered and extended the work of Davisson. An analysis of the assumptions upon which these investigations were predicated, as well as a comparison of the results which followed, will be presented. The paper will conclude with a delineation of areas which require further investigation.