One-dimensional CCA and SVD, and their relationship to regression maps

Abstract

The canonical correlation analysis (CCA) and singular value decomposition (SVD) approaches for estimating a time series from a time-dependent vector and vice versa are investigated, and their relationship to multiple linear regression (MLR) and to regression maps is discussed. Earlier findings are reviewed and combined with new aspects to provide a systematic overview. It is shown that regression maps are proportional to canonical patterns and to singular vectors and that the estimate of a time-dependent vector from a time series does not depend on whether CCA, SVD, or component-wise regressions are used. When a time series is linearly estimated from a time-dependent vector, it is known that CCA is equivalent to MLR. It is demonstrated that an estimate for the time series based on a time expansion coefficient of the regression map that is calculated by orthogonal projection is identical to an SVD estimate, but different from the CCA and MLR estimate. The two approaches also lead to different correlations between the time series and the time expansion coefficient of its signal.