Separation of uncorrelated stationary time series using autocovariance matrices

TL;DR

This paper analyzes the statistical properties of the symmetric SOBI method for blind source separation of stationary time series, comparing its efficiency to the deflation-based approach through theory, simulations, and real EEG data.

Contribution

It provides a rigorous statistical analysis of the symmetric SOBI estimator, including its limiting distribution and efficiency comparison with the deflation-based method.

Findings

Derived the limiting distribution of the symmetric SOBI estimator.

Compared asymptotic efficiencies of symmetric and deflation-based SOBI methods.

Validated theoretical results with simulations and EEG data analysis.

Abstract

Blind source separation (BSS) is a signal processing tool, which is widely used in various fields. Examples include biomedical signal separation, brain imaging and economic time series applications. In BSS, one assumes that the observed $p$ time series are linear combinations of $p$ latent uncorrelated weakly stationary time series. The aim is then to find an estimate for an unmixing matrix, which transforms the observed time series back to uncorrelated latent time series. In SOBI (Second Order Blind Identification) joint diagonalization of the covariance matrix and autocovariance matrices with several lags is used to estimate the unmixing matrix. The rows of an unmixing matrix can be derived either one by one (deflation-based approach) or simultaneously (symmetric approach). The latter of these approaches is well-known especially in signal processing literature, however, the rigorous analysis of its statistical properties has been missing so far. In this paper, we fill this gap and investigate the statistical properties of the symmetric SOBI estimate in detail and find its limiting distribution under general conditions. The asymptotical efficiencies of symmetric SOBI estimate are compared to those of recently introduced deflation-based SOBI estimate under general multivariate MA$(\infty)$ processes. The theory is illustrated by some finite-sample simulation studies as well as a real EEG data example.