Globally Optimal Least-Squares ARMA Model Identification is an Eigenvalue Problem

Abstract

We show that globally optimal least-squares identification of autoregressive moving-average (ARMA) models is an eigenvalue problem (EP). The first order optimality conditions of this identification problem constitute a system of multivariate polynomial equations, in which most variables appear linearly. This system is basically a multiparameter eigenvalue problem (MEP), which we solve by iteratively building a so-called block Macaulay matrix, the null space of which is block multi-shift-invariant. The set of all stationary points of the optimization problem, i.e., the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> -tuples of eigenvalues and eigenvectors of the MEP, follows from a standard EP related to the multidimensional realization problem in that null space. At least one of these <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> -tuples corresponds to the global minimum of the original least-squares objective function. Contrary to existing heuristic techniques, this approach yields the globally optimal parameters of the ARMA model. We provide a numerical example to illustrate the new identification method.