On the Simulation of Polynomial NARMAX Models
For researchers in system identification, this work addresses a fundamental bias issue in NARMAX model simulation, though the proposed solutions are incremental improvements over existing methods.
The paper identifies that the common approach of setting noise contributions to zero in simulating polynomial NARMAX models yields biased responses, and that unbiased simulation generally requires infinite-order models. It proposes a Hermite polynomial-based representation for exact translation to simulation models (when possible) and a parameterized approximation to truncate infinite-order models to finite order.
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.