The Finite Memory Prediction of Covariance Stationary Time Series
H. Joseph Newton, Marcello Pagano · SIAM Journal on Scientific and Statistical Computing · 1983
An algorithm is presented for conveniently calculating h step ahead minimum mean square error linear predictors and prediction variances given a finite number of observations from a covariance stationary time series Y. It is shown that elements of the modified Cholesky decomposition of the covariance matrix of observations play the role in finite memory prediction that the coefficients in the infinite order moving average representation of Y play in infinite memory prediction. A by-product of the algorithm is the extension of Pagano’s result (J. Assoc. Comput. Mach., 23 (1976), pp. 310–316) on the convergence down the diagonal of the Cholesky factors of a banded Toeplitz matrix to a similar result for a general Toeplitz matrix. This result is applied to autoregressive-moving average time series. A numerical example illustrating the results of the paper is presented.