Canonical Correlation Analysis of Time Series and the Use of an Information Criterion

Abstract

This chapter starts with a brief introductory review of some of the recent developments of time-series analysis. One of the most established procedures of time-series analysis is the method of estimation of power spectrum through windowed sample covariance sequence. Except for the special situations where the orders are specified in advance, any method of the autoregressive model fitting is not well-defined as a method of estimation of the covariance sequence in the absence of the description of the rules for the determination of the order. Because the covariance sequence determines the power spectrum, once an appropriate rule for the order determination is given, the autoregressive-model fitting procedure automatically provides an estimate of the power spectrum. A solution to the problem of order determination of an auto-regressive model was obtained in 1969 by using the concept of final prediction error (FPE), which is defined as the one-step ahead prediction error variance when the least squares estimates of the autoregressive coefficients are used for prediction. The concept of the one-step ahead of prediction error variance had difficulty in extending the minimum FPE procedure to the multivariate situation because of the non-uniqueness of the measure of variance in the multivariate situation. A solution was found with the aid of the Gaussian model and the concept of maximum likelihood, which suggested the use of the generalized variance of the one-step ahead of prediction error.