One-Dimensional CCA and SVD, and Their Relationship to Regression Maps

Martin Widmann Institute for Coastal Research, GKSS Research Centre, Geesthacht, Germany

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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.

The CCA–MLR and the SVD–regression map approaches are compared in an example where the January Arctic Oscillation index for the period 1948–2002 was estimated from extratropical Northern Hemispheric 850-hPa temperature. For CCA–MLR the leading principal components (PCs) of the temperature field were used as predictors, while for SVD the full field was employed. For more than seven retained PCs the skill in terms of correlations and mean squared error based on cross validation was for both approaches practically identical, but CCA–MLR showed a higher bias. For a smaller number of predictor PCs the SVD–regression map approach performed better. The discrepancy between the skill on the fitting data and on the independent data used for validation was in this example larger for the CCA–MLR approach.

Corresponding author address: Dr. Martin Widmann, Institute for Coastal Research, GKSS Research Centre, D 21502 Geesthacht, Germany. Email: widmann@gkss.de

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.

The CCA–MLR and the SVD–regression map approaches are compared in an example where the January Arctic Oscillation index for the period 1948–2002 was estimated from extratropical Northern Hemispheric 850-hPa temperature. For CCA–MLR the leading principal components (PCs) of the temperature field were used as predictors, while for SVD the full field was employed. For more than seven retained PCs the skill in terms of correlations and mean squared error based on cross validation was for both approaches practically identical, but CCA–MLR showed a higher bias. For a smaller number of predictor PCs the SVD–regression map approach performed better. The discrepancy between the skill on the fitting data and on the independent data used for validation was in this example larger for the CCA–MLR approach.

Corresponding author address: Dr. Martin Widmann, Institute for Coastal Research, GKSS Research Centre, D 21502 Geesthacht, Germany. Email: widmann@gkss.de

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