Kronecker product formulation of the cumulant based realization of stochastic systems.

The author derives recursive equations and closed-form expressions relating the parameters of an autoregressive-moving-average (ARMA) model (which may be non-minimum-phase and/or noncausal) with the cumulants of its output, in response to excitation by a non-Gaussian i.i.d. process. Based on these relationships, cumulant-based stochastic realization algorithms are developed. The output noise can be colored Gaussian or i.i.d. non-Gaussian with unknown variance. When a state-space representation is adopted, the stochastic realization problem reduces to the realization of a appropriate Hankel matrix formed by cumulant statistics. Using a Kronecker product formulation, exact expressions are presented for identifying state-space quantities when output cumulants are provided, or for computing output cumulants when the state-space triple is known. Conditions for stationarity of a linear process, with respect to its cumulants, are also presented. If a transfer function approach is used, alternative formulations are given to cover the case of noncausal models.