Abstract
One of the most versatile and powerful approaches to the identification of nonlinear dynamical systems is the NARMAX (Nonlinear Auto_regressive Moving Average with eXogenous inputs) method. 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 nonparametric machine learning paradigms, 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. Recently, a NARX (no noise model) formulation based on Gaussian Process (GP) regression has been developed. One advantage of the GP NARX form is that confidence intervals are a natural part of the prediction process. The objective of the current paper is to discuss the GP formulation and show how to compute the HFRFS corresponding to GP NARX. Examples will be given based on simulated data.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Leontaritis, I.J., Billings, S.A.: Input-output parametric models for nonlinear systems, part I: deterministic nonlinear systems. Int. J. Control 41, 303–328 (1985)
Leontaritis, I.J., Billings, S.A.: Input-output parametric models for nonlinear systems, part II: stochastic nonlinear systems. Int. J. Control 41, 329–344 (1985)
Billings, S.A.: Nonlinear System Identification: NARMAX, Methods in the Time, Frequency, and Spatio-Temporal Domains. Wiley, Hoboken (2013)
Billings, S.A., Jamaluddin, H.B., Chen, S.: Properties of neural networks with applications to modelling non-linear dynamical systems. Int. J. Control 55, 193–224 (1992)
Chen, S., Billings, S.A., Cowan, C.F.N., Grant, P.M.: Practical identification of NARMAX models using radial basis functions. Int. J. Control 52, 1327–1350 (1990)
Bishop, C.M.: Neural Networks for Pattern Recognition. Oxford University Press, New York (1998)
Schetzen, M.: The Volterra and Wiener Theories of Nonlinear Systems. John Wiley Interscience Publication, New York (1980).
Worden, K., Tomlinson, G.R.: Nonlinearity in Structural Dynamics: Detection, Modelling and Identification. Institute of Physics Press, Bristol (2001)
Bedrosian, E., Rice, S.O.: The output properties of Volterra systems driven by harmonic and Gaussian inputs. Proc. IEEE 59, 1688–1707 (1971)
Billings, S.A., Tsang, K.M.: Spectral analysis for nonlinear systems, part I: parametric non-linear spectral analysis. Mech. Syst. Signal Process. 3, 319–339 (1989)
Chance, J.E., Worden, K., Tomlinson, G.R.: Frequency domain analysis of NARX neural networks. J. Sound Vib. 213, 915–941 (1997)
Murray-Smith, R., Johansen, T. A., Shorten, R.: On transient dynamics, off-equilibrium behaviour and identification in blended multiple model structures. In: European Control Conference, Karlsruhe, BA-14 (1999)
Kocijan, J.: Dynamic GP models: an overview and recent developments. In: ASM12, Proc. 6th Int. Conf. Appl. Maths Sim. Mod., pp. 38–43 (2012)
Krige, D.G.: A Statistical Approach to Some Mine Valuations and Allied Problems at the Witwatersrand. Master’s Thesis, University of Witwatersrand (1951)
Neal, R.M.: Monte Carlo implementation of Gaussian process models for Bayesian regression and classification. Arxiv preprint physics/9701026, (1997)
MacKay, D.J.C.: Gaussian processes - a replacement for supervised neural networks. Lecture Notes for Tutorial at Int. Conf. Neural Inf. Proc. Sys. (1997)
Rasmussen, C.E., Williams, C.K.I.: Gaussian Processes for Machine Learning. MIT Press, Cambridge (2005)
Quinonero-Candelo, Q. Rasmussen, C.E.: A unifying view of sparse approximation Gaussian process regression. J. Mach. Learn. Res. 6, 1939–1959 (2005)
Snelson, E. Ghahramani, Z.: Sparse Gaussian processes using pseudo-inputs. In: AAdvances in Neural Information Processing Systems. MIT Press, Cambridge (2006)
Giraud, A.: Approximate Methods for Propagation of Uncertainty with Gaussian Process Models. PhD Thesis, University of Glasgow (2004)
Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P.: Numerical Recipes: The Art of Scientific Computing, 3rd edn. Cambridge University Press, New York (2007)
Worden, K., Hensman, J.J.: Parameter estimation and model selection for a class of hysteretic systems using Bayesian inference. Mech. Syst. Signal Process. 32, 153–169 (2011)
Volterra, V.: Theory of Functionals and of Integral and Integro-Differential Equations. Dover, New York (1959)
Palm, G., Poggio, T.: The Volterra representation and the Wiener expansion: validity and pitfalls. SIAM J. Appl. Math. 33, 195–216 (1997)
Acknowledgements
The authors would like to thank Dr James Hensman of the University of Sheffield Centre for Translational Neuroscience for a number of interesting and useful discussions.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer International Publishing Switzerland
About this paper
Cite this paper
Worden, K., Manson, G., Cross, E.J. (2014). On Gaussian Process NARX Models and Their Higher-Order Frequency Response Functions. In: Koziel, S., Leifsson, L., Yang, XS. (eds) Solving Computationally Expensive Engineering Problems. Springer Proceedings in Mathematics & Statistics, vol 97. Springer, Cham. https://doi.org/10.1007/978-3-319-08985-0_14
Download citation
DOI: https://doi.org/10.1007/978-3-319-08985-0_14
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-08984-3
Online ISBN: 978-3-319-08985-0
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)