QUARKS: Identification of large-scale Kronecker vector-autoregressive models

In this paper, we address the identification of two-dimensional (2-D) spatial-temporal dynamical systems described by the Vector-AutoRegressive (VAR) form. The coefficient-matrices of the VAR model are parametrized as sums of Kronecker products. When the number of terms in the sum is small compared to the size of the matrices, such a Kronecker representation efficiently models large-scale VAR models. Estimating the coefficient matrices in least-squares sense gives rise to a bilinear estimation problem that is tackled using an Alternating Least Squares (ALS) algorithm. Regularization or parameter constraints on the coefficient-matrices allows to induce temporal network properties, such as stability, as well as spatial properties, such as sparsity or Toeplitz structure. Convergence of the regularized ALS is proved using fixed-point theory. A numerical example demonstrates the advantages of the new modeling paradigm. It leads to comparable variance of the prediction error with the unstructured least-squares estimation of VAR models. However, the number of parameters grows only linearly with respect to the number of nodes in the 2-D sensor network instead of quadratically in the case of fully unstructured coefficient-matrices.