How to Determine the Corank and Noise Level of a System?

Abstract

The essence of identification of a linear system from inexact data is the determination of its corank, i.e. the number of linear relations of the system. Three approaches to the noisy realization problem using only data covariance matrices as inputs are described: 1.) the Frobenius-Kalman test, based on the work by Koopmans, Reiersøl and Kalman. The test determines if the corank is equal or greater than unity, 2.) the Reiethiøl tree search procedure. It determines the upperbound on the corank supportable by the inexact data; and 3.) Wilson's inequality (or Ledermann bound). This imposes an exact upperbound on the corank, if a Frisch diagonal noise matrix is presumed. This paper illustrates these findings with an empirical example drawn from the US economy to demonstrate their viability. It also presents for the first time the complete and exact algebraic solution for the Frisch (4, 2) scheme, i.e. for four variables and two relations, and provides thereby the definitive answer to this algebraic identification problem already formulated by Wileon and Worcester in 1934.