Linear minimax estimation for random vectors with parametric uncertainty
KAUST Grant Number025478
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AbstractIn 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, PA and PB. Model PA represents X and Y as jointly Gaussian whose covariance matrix Λ belongs to the convex hull of a set of m known covariance matrices. Model PB characterizes 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 PA is identical to that computed under model PB when the vertices of the uncertain covariance set in PA are 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 PA and PB as before. We derive a recursive linear minimax filter for x(t) given y(t).
CitationBitar E, Baeyens E, Packard A, Poolla K (2010) Linear minimax estimation for random vectors with parametric uncertainty. Proceedings of the 2010 American Control Conference. Available: http://dx.doi.org/10.1109/ACC.2010.5531063.
SponsorsSupported in part by OOF991-KAUST US LIMITED underaward number 025478, the UC Discovery Grant ele07-10283 underthe IMPACT program, NASA Langley NRA NNH077ZEA001N,and NSF under Grant EECS-0925337.