Linear minimax estimation for random vectors with parametric uncertainty
Eilyan Y. Bitar, Enrique Baeyens, Alan B. Packard, Kameshwar R. Poolla · 2010
In this paper, we take a minimax approach to the problem of computing a worst-case linear mean squared error (MSE) estimate of X given Y , where X and Y are jointly distributed random vectors with parametric uncertainty in their distribution. We consider two uncertainty models, PAand PB. Model PArepresents X and Y as jointly Gaussian whose covariance matrix Λ belongs to the convex hull of a set of m known covariance matrices. Model PBcharacterizes X and Y as jointly distributed according to a Gaussian mixture model with m known zero-mean components, but unknown component weights. We show: (a) the linear minimax estimator computed under model PAis identical to that computed under model PBwhen the vertices of the uncertain covariance set in PAare the same as the component covariances in model PB, and (b) the problem of computing the linear minimax estimator under either model reduces to a semidefinite program (SDP). We also consider the dynamic situation where x(t) and y(t) evolve according to a discrete-time LTI state space model driven by white noise, the statistics of which is modeled by PAand PBas before. We derive a recursive linear minimax filter for x(t) given y(t).