# Kronecker Sum Decompostions of Space-Time Data

Kristjan Greenewald, Theodoros Tsiligkaridis, Alfred O. Hero III

**Abstract**

In this paper we consider the use of the space vs. time Kronecker product decomposition in the estimation of covariance matrices for spatio-temporal data. This decomposition imposes lower dimensional structure on the estimated covariance matrix, thus reducing the number of samples required for estimation. To allow a smooth tradeoff between the reduction in the number of parameters (to reduce estimation variance) and the accuracy of the covariance approximation (affecting estimation bias), we introduce a diagonally loaded modification of the sum-of-kronecker products representation in [Tsiligkaridis et al. 2013]. We derive an asymptotic Cramer-Rao bound (CRB) on the minimum attainable mean squared predictor coefficient estimation error for unbiased estimators of Kronecker structured covariance matrices. We illustrate the accuracy of the diagonally loaded Kronecker sum decomposition by applying it to the prediction of human activity video.

**MATLAB Code**