An M-estimator for reduced-rank system identification

High-dimensional time-series data from a wide variety of domains, such as neuroscience, are being generated every day. Fitting statistical models to such data, to enable parameter estimation and time-series prediction, is an important computational primitive. Existing methods, however, are unable to cope with the high-dimensional nature of these data, due to both computational and statistical reasons. We mitigate both kinds of issues by proposing an M-estimator for Reduced-rank System IDentification (Mr. Sid). A combination of low-rank approximations, ℓ1 and ℓ2 penalties, and some numerical linear algebra tricks, yields an estimator that is computationally efficient and numerically stable. Simulations and real data examples demonstrate the usefulness of this approach in a variety of problems. In particular, we demonstrate that Mr. Sid can accurately estimate spatial filters, connectivity graphs, and time-courses from native resolution functional magnetic resonance imaging data. Mr. Sid therefore enables big time-series data to be analyzed using standard methods, readying the field for further generalizations including nonlinear and non-Gaussian state-space models.