  In this paper, we show that the common approach for simulation non-linear
stochastic models, commonly used in system identification, via setting the
noise contributions to zero results in a biased response. We also demonstrate
that to achieve unbiased simulation of finite order NARMAX models, in general,
we require infinite order simulation models. The main contributions of the
paper are two-fold. Firstly, an alternate representation of polynomial NARMAX
models, based on Hermite polynomials, is proposed. The proposed representation
provides a convenient way to translate a polynomial NARMAX model to a
corresponding simulation model by simply setting certain terms to zero. This
translation is exact when the simulation model can be written as an NFIR model.
Secondly, a parameterized approximation method is proposed to curtail infinite
order simulation models to a finite order. The proposed approximation can be
viewed as a trade-off between the conventional approach of setting noise
contributions to zero and the approach of incorporating the bias introduced by
higher-order moments of the noise distribution. Simulation studies are provided
to illustrate the utility of the proposed representation and approximation
method.
