On the Simulation of Polynomial NARMAX Models

TL;DR

This paper reveals that standard simulation methods for polynomial NARMAX models are biased unless infinite order models are used, and introduces a Hermite polynomial-based representation and approximation technique to improve simulation accuracy.

Contribution

It proposes a new Hermite polynomial-based representation of polynomial NARMAX models and a finite order approximation method for unbiased simulation.

Findings

Standard zero-noise simulation is biased for polynomial NARMAX models.

The Hermite polynomial representation simplifies model translation to simulation.

The approximation method balances bias and complexity in finite order simulations.

Abstract

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.