Structured covariance estimation for state prediction

In this paper we propose a structurally constrained expectation-maximization (EM) algorithm for estimating noise covariances in state-space models, for the purpose of state prediction and control. More specifically, we generalize the problem of covariance estimation on the basis of given i.i.d sample sequence to the dynamic setting where the samples (i.e. state and observation noises) are observed only through the measurement data, or equivalently, drawn from the conditional distribution governed by the dynamic model. By applying the expectation maximization (EM) algorithm to the innovation model representation, we view the resulting ML covariance estimates as the conditional sample covariances, and augment the negative log-likelihood function with matrix norm penalty terms that enforce low-rank and low cardinality structure in the estimated covariances or their inverses. These constraints serve to reflect realistic problem structure expected from model knowledge, yet are still general and flexible enough to be broadly applicable. In addition, the new derivation of the EM algorithm based on the innovation representation gives the common sufficient statistic for both the process and observation noise covariances. This illustrates the coupling between the two covariance estimates, and in simulated cases, enables the calculation of an upper performance bound against which the EM estimates can be compared. The use of the innovation representation also provides a tractable connection to the existing techniques such as the Autocovariance Least Squares (ALS) algorithm. Numerical results comparing the constrained EM and the ALS algorithms are also provided, showing favorable performance for the EM covariance estimates.