On the nonstationary covariance realization problem

This paper contains an algebraic result in system identifiability which is fundamental to the results of [1] concerning the maximum likelihood identification of the parameters of linear time-invariant systems from nonstationary cross sectional data. Let denote the random vector of distinct -component output values of the nonstationary output sample of a linear time-invariant stochastic system, and let the parameterized covariance matrix of be denoted by for . We say that is locally identifiable if the map is one-to-one in the neighborhood of . Among other results we show that under a nonstationarity condition is locally identifiable , where is the degree of the minimal polynomial of the state transition matrix of the system. This is established by explicitly constructing a wide-sense state space stochastic realization of from in observable canonical form with state dimension . The intimate connections between these results and the standard results [13]-[15] concerning the (wide-sense) realization of stationary processes from their covariance matrices are described.