Separation of uncorrelated stationary time series using autocovariance matrices

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.

Averaging orthogonal projectors

Dimensionality is a major concern in analyzing large data sets. Some well known dimension reduction methods are for example principal component analysis (PCA), invariant coordinate selection (ICS), sliced inverse regression (SIR), sliced average variance estimate (SAVE), principal hessian directions (PHD) and inverse regression estimator (IRE). However, these methods are usually adequate of finding only certain types of structures or dependencies within the data. This calls the need to combine information coming from several different dimension reduction methods. We propose a generalization of the Crone and Crosby distance, a weighted distance that allows to combine subspaces of different dimensions. Some natural choices of weights are considered in detail. Based on the weighted distance metric we discuss the concept of averages of subspaces as well to combine various dimension reduction methods. The performance of the weighted distances and the combining approach is illustrated via simulations.