On the nonstationary covariance realization problem
Robert Kent Goodrich, Peter E. Caines · IEEE Transactions on Automatic Control · 1979
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. LetZ_{1}^{T}denote the random vector ofTdistinctp-component output values of the nonstationary output sample of a linear time-invariant stochastic system, and let the parameterized covariance matrix ofz_{1}^{T}be denoted by\Sigma_{T}(\theta)for\theta \in \Theta \subset R^{v}. We say that\thetais locally identifiable(T, N_{\theta})if the map\Sigma_{T}(\cdot): \theta \rightarrow R^{P} (p=pT(pT+ 1)/2)is one-to-one in the neighborhoodN_{\theta}of\theta. Among other results we show that under a nonstationarity condition\thetais locally identifiable(d+2, N_{\theta}), wheredis 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 ofzfrom\Sigma_{T}(\theta)in observable canonical form with state dimensionpd. 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.