Deflation-based separation of uncorrelated stationary time series

In this paper we assume that the observed pp time series are linear combinations of pp latent uncorrelated weakly stationary time series. The problem is then to find an estimate for an unmixing matrix that transforms the observed time series back to uncorrelated time series. The so called SOBI (Second Order Blind Identification) estimate aims at a joint diagonalization of the covariance matrix and several autocovariance matrices with varying lags. In this paper, we propose a novel procedure that extracts the latent time series one by one. The limiting distribution of this deflation-based SOBI is found under general conditions, and we show how the results can be used for the comparison of estimates. The exact formula for the limiting covariance matrix of the deflation-based SOBI estimate is given for general multivariate MA(∞) processes. Finally, a whole family of estimates is proposed with the deflation-based SOBI as a special case, and the limiting properties of these estimates are found as well. The theory is widely illustrated by simulation studies.