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 non-linear and non-Gaussian state-space models.