Continuum Power CCA: A Unified Approach for Isolating Coupled Modes

Various multivariate statistical methods exist for analyzing covariance and isolating linear relationships between datasets. The most popular linear methods are based on singular value decomposition (SVD) and include canonical correlation analysis (CCA), maximum covariance analysis (MCA), and redundancy analysis (RDA). In this study, continuum power CCA (CPCCA) is introduced as one extension of continuum power regression for isolating pairs of coupled patterns whose temporal variation maximizes the squared covariance between partially whitened variables. Similar to the whitening transformation, the partial whitening transformation acts to decorrelate individual variables but only to a partial degree with the added benefit of preconditioning sample covariance matrices prior to inversion, providing a more accurate estimate of the population covariance. CPCCA is a unified approach in the sense that the full range of solutions bridges CCA, MCA, RDA, and principal component regression (PCR). Recommended CPCCA solutions include a regularization for CCA, a variance bias correction for MCA, and a regularization for RDA. Applied to synthetic data samples, such solutions yield relatively higher skill in isolating known coupled modes embedded in noise. Provided with some crude prior expectation of the signal-to-noise ratio, the use of asymmetric CPCCA solutions may be justifiable and beneficial. An objective parameter choice is offered for regularization with CPCCA based on the covariance estimate of O. Ledoit and M. Wolf, and the results are quite robust. CPCCA is encouraged for a range of applications.