Reduced rank linear regression and weighted low rank approximations

This paper addresses parameter estimation in reduced rank linear regressions. This estimation problem has applications in several subject areas including system identification, sensor array processing, econometrics and statistics. A new estimation procedure, based on instrumental variable principles, is derived and analyzed. The proposed method is designed to handle noise that is both spatially and temporally autocorrelated. An asymptotical analysis shows that the proposed method outperforms previous methods when the noise is temporally correlated and that it is asymptotically efficient otherwise. A numerical study indicates that the performance is significantly improved also for finite sample set sizes. In addition, the Cramer-Rao lower bound (CRB) on unbiased estimator covariance for the data model is derived. A statistical test for rank determination is also developed. An important step in the new algorithm is the weighted low rank approximation (WLRA). As the WLRA lacks a closed form solution in its general form, two new, noniterative and approximate solutions are derived, both of them asymptotically optimal when part of the estimation procedure proposed here. These methods are also interesting in their own right since the WLRA has several applications.