One of the most versatile and powerful methods for the identification of nonlinear dynamical systems is the NARMAX (Nonlinear Auto-regressive Moving Average with exogenous inputs) approach. The model represents the current output of a system by a nonlinear regression on past inputs and outputs and can also incorporate a nonlinear noise model in the most general case. Although the NARMAX model is most often given a polynomial form, this is not a restriction of the method and other formulations have been proposed based on multi-layer perceptron neural networks or radial basis function networks for example. All of these forms of the NARMAX model allow the computation of Higher-order Frequency Response Functions (HFRFs) which encode the model in the frequency domain and allow a direct interpretation of how frequencies interact in the nonlinear system under study. In a recent paper, one of the authors discussed a NARX (no noise model) formulation based on Gaussian Process (GP) regression. The objective of the current paper is to provide the theory for the HFRFs corresponding to GP NARX. Examples will be given based on simulated data.
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