Time Series Source Separation Using Dynamic Mode Decomposition
Arvind Prasadan, Raj Rao Nadakuditi · SIAM Journal on Applied Dynamical Systems · 2020
The dynamic mode decomposition (DMD) extracted dynamic modes are the nonorthogonal eigenvectors of the matrix that best approximates the one-step temporal evolution of the multivariate samples. In the context of dynamical system analysis, the extracted dynamic modes are a generalization of global stability modes. We apply DMD to a data matrix whose rows are linearly independent, additive mixtures of latent time series. We show that when the latent time series are uncorrelated at a lag of one time-step then, in the large sample limit, the recovered dynamic modes will approximate, up to a columnwise normalization, the columns of the mixing matrix. Thus, DMD is a time series blind source separation algorithm in disguise but is different from closely related second-order algorithms such as the second-order blind identification (SOBI) method and the algorithm for multiple unknown signals extraction (AMUSE). All can unmix mixed stationary, ergodic Gaussian time series in a way that kurtosis-based independent components analysis fundamentally cannot. We use our insights on single-lag DMD to develop a higher lag extension, analyze the finite sample performance with and without randomly missing data, and identify settings where the higher lag variant can outperform the conventional single-lag variant. We validate our results with numerical simulations and highlight how DMD can be used in changepoint detection.