Mathematical realization formulas that specify a Markovian model for a stochastic process from the covariance sequence of the process have interested researchers for some time. Such formulas construct the model so that its output covariance sequence is equal to the given sequence; the resulting model may therefore be called an exact realization of the given covariance sequence.

Let {y_k} be a stationary, scalar, and Gaussian stochastic process with zero mean. Further assume that {y_k} is a finite-dimensional process of order p so that it can be realized by a Markovian model of the form

x_k+1 = Fx_k + gu_k, - E[x_kx^T_k+j] = P&dgr;_k - j

y_k = hx_k + u_k, Var(u_k) = &rgr;

where F is a p × p matrix and &dgr; is the Kronecker delta function. The output covariance sequence of the model is given by

r_j = E[y_ky_k+j], j = 0,1,2, . . . ,L.

In this chapter the author demonstrates a method of constructing a state-space model for a stochastic process from a finite number of estimated covariance lags. He suggests a two-phase approach: first obtain a high-order model that exactly matches the estimated covariance sequence, and then use balanced model reduction techniques to obtain a lower-order model that approximates the sequence.

Vaccaro goes on to show that one can obtain the balanced models from a realization algorithm that uses an infinite covariance sequence; then he introduces scaling ideas that allow one to obtain the balanced realization from finite covariance sequences. He does not, however, develop details such as relating the finite factorizations to infinite factorizations and balancing the scaled realization.