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
Z_{1}^{T}
denote the random vector of
T
distinct
p
-component output values of the nonstationary output sample of a linear time-invariant stochastic system, and let the parameterized covariance matrix of
z_{1}^{T}
be denoted by
\Sigma_{T}(\theta)
for
\theta \in \Theta \subset R^{v}
. We say that
\theta
is 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 neighborhood
N_{\theta}
of
\theta
. Among other results we show that under a nonstationarity condition
\theta
is locally identifiable
(d+2, N_{\theta})
, where
d
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
z
from
\Sigma_{T}(\theta)
in observable canonical form with state dimension
pd
. 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.