Jolliffe I. Principal Component Analysis (2ed., Springer, 2002)(518s)

12.4. PCA and Non-Independent Data—Some Additional Topics 337compared to a multivariate ARMA model involving all p of the originalseries.Wold (1994) suggests exponentially weighted moving principal componentsin the context of process control (see Section 13.7), and Diamantarasand Kung (1996, Section 3.5) advocate PCs based on weighted covariancematrices for multivariate time series, with weights decreasing exponentiallyfor less recent observations.Yet another rôle for PCs in the analysis of time series data is presentedby Doran (1976). In his paper, PCs are used to estimate the coefficients in aregression analysis of one time series variable on several others. The idea issimilar to that of PC regression (see Section 8.1), but is more complicatedas it involves the frequency domain. Consider the distributed lag model,which is a time series version of the standard regression model y = Xβ + ɛof equation (8.1.1), with the time series structure leading to correlationbetween elements of ɛ. There is a decomposition of the least squares estimatorof the regression coefficients β rather like equation (8.1.8) fromPC regression, except that the eigenvalues are replaced by ratios of spectraldensity estimates for the predictors (signal) and error (noise). Doran(1976) suggests using an estimate in which the terms corresponding tothe smallest values of this signal to noise ratio are omitted from the leastsquares decomposition.