Geometric Analysis for the Characterization of Nonstationary Time Series

Subspace methodologies, such as the Karhunen-Loève (KL) transform, are powerful geometric tools for the characterization of high-dimensional data sets. The KL transform, or the related singular value decomposition (SVD), maximizes the mean-square projection of the data ensemble on subspaces of reduced rank. Other interesting subspace approaches solve modified optimization problems and have received comparably less attention in the literature. Here we present two such methodologies: 1) signal fraction analysis (SFA), a method that optimizes the amount of signal retained when signals are superposed and 2) canonical correlation analysis (CCA), a method for contructing transformations that allow the comparison of two data sets. We compare these methods to the more widely employed SVD in the context of real data. We address the important and practical problem of whether two time-series are generated by the same process. As a specific example, the classification of noisy multivariate electroencephalogram (EEG) time-series data is considered.