Uniform random parameter generation of stable minimum-phase real ARMA (p,q) processes

Abstract

An algorithm to randomly generate the parameters of stable invertible autoregressive moving average processes of order (p; q)-ARMA (p; q)-is presented. The AR and MA portions are independent of each other, and their respective parameters have jointly uniform distributions with support defined by stability and invertibility considerations. The uniform density insures that each possible model is equally likely. The algorithm uses the Levinson-Durbin recursion to guarantee the poles and zeros are inside the unit circle, thus avoiding coefficient resampling typical of "generate and test" methods. To initialize the Levinson-Durbin recursion for each model order, the reflection coefficients are generated using a rejection sampling technique.

An improved method for uniform simulation of stable minimum phase real ARMA (p,q) processes

“…We define the coefficients for the th order autoregressive process, for (where is the highest integer such that and is the indicator function of even integers), and to be the associated probability density on Then, Lemma 1 in [3] ensures that the following algorithm provides sample from…”

“…• For Sample where is the Durbin-Levinson recursion which allows us to transform the dimensional vector into the dimensional vector [3]. We use the notation to mean that is distributed according to…”

“…is not obviously straightforward, leading the authors in [3] to use rejection sampling. We now briefly recall the rejection sampling recursion used to sample from a pdf [4].…”

“…This method is not efficient, as the rejection rate may be high particularly for highdimensional models. Recently, Beadle and Djuríc proposed an interesting method relying on the Durbin-Levinson procedure [3]. This method circumvents high rejection rates, which may occur when using the rejection technique on a highdimensional parameter space, by replacing it with unidimensional rejection sampling procedures.…”

“…Finally, in Section V we present simulation results. As in [3], we only present the synthesis of the AR coefficients since the synthesis of the MA portion follows similarly and can be performed separately.…”