Time Series Source Separation using Dynamic Mode Decomposition

TL;DR

This paper demonstrates that Dynamic Mode Decomposition (DMD) can be used as a blind source separation method for multivariate time series, especially when latent sources are uncorrelated at lag one, and extends it to higher lags with practical advantages.

Contribution

The paper reveals DMD's capability as a blind source separation algorithm for time series and introduces a higher-lag extension with performance analysis and applications.

Findings

DMD approximates the mixing matrix for uncorrelated latent sources.

Higher-lag DMD can outperform single-lag DMD in certain settings.

DMD can be applied to change point detection in time series.

Abstract

The Dynamic Mode Decomposition (DMD) extracted dynamic modes are the non-orthogonal 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 column-wise 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 (ICA) 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 change point detection.