Singular-value decomposition approach to time series modelling

Abstract

In various signal processing applications, as exemplified by spectral analysis, deconvolution and adaptive filtering, the parameters of a linear recursive model are to be selected so that the model is `most' representative of a given set of time series observations. For many of these applications, the parameters are known to satisfy a theoretical recursive relationship involving the time series' autocorrelation lags. Conceptually, one may then use this recursive relationship, with appropriate autocorrelation lag estimates substituted, to effect estimates for the operator's parameters. A procedure for carrying out this parameter estimation is given which makes use of the singular-value decomposition (SVD) of an extended-order autocorrelation matrix associated with the given time series. Unlike other SVD modelling methods, however, the approach developed does not require a full-order SVD determination. Only a small subset of the matrix's singular values and associated characteristic vectors need be computed. This feature can significantly alleviate an otherwise overwhelming computational burden that is necessitated when generating a full-order SVD. Furthermore, the modelling performance of this new method has been found empirically to excel that of a near maximum-likelihood SVD method as well as several other more traditional modelling methods.