Some properties of the parameterization of ARMA systems with unknown order

Abstract

The first section of the paper introduces known theory relating to the description of the set of all ARMA structures via the concept of order, n, and the coordinatisation of all structures, M( n), of given order. The coordinates most easily used are related to the state space representation and in the first section these are related to coordinates obtained from the ARMA representation. In the second section geometric and topological properties of M( n) are considered. For example, the closure of M( n) is just the union of all M( j), j ≤ n. Such questions are important in connection with order estimation when the true order, n 0, may be less than n and in connection with algorithm construction. Similarly the coordinates of a set of points in M( n), converging to a boundary point for a system of lower order, need not themselves converge. Such a boundary point will, in coordinates, map into an affine subspace of Euclidean space. This is shown and also it is shown how to construct coordinates that relate simply to this affine subspace.