Direct identification of nonlinear structure using Gaussian process prior models

When inferring nonlinear dependence from measured data,the nonlinear nature of the relationship may be characterisedin terms of all the explanatory variables. However, this israrely the most parsimonious, or insightful, approach.Instead, it is usually much more useful to be able to exploitthe inherent nonlinear structure to characterise the nonlineardependence in terms of the least possible number of variables.In this paper a new way of inferring nonlinear structure frommeasured data is investigated. The measured data isinterpreted as providing information on a nonlinear map. Thespace containing the domain of the map is sub-divided intounique linear and nonlinear sub-spaces that are structuralinvariants. The most parsimonious representation of the mapis obtained by the restriction of the map to the nonlinear sub-space. A direct constructive algorithm based on Gaussianprocess prior models, defined using a novel covariancefunction, is proposed. The algorithm infers the linear andnonlinear sub-spaces from noisy data and provides a non-parametric model of the parsimonious map. Use of thealgorithm is illustrated by application to a Wiener-Hammerstein system