This paper investigates new ways of inferring nonlinear dependence from measured data. The existence of unique linear and nonlinear sub-spaces which are structural invariants of general nonlinear mappings is established and necessary and sufficient conditions determining these sub-spaces are derived. The importance of these invariants in an identification context is that they provide a tractable framework for minimising the dimensionality of the nonlinear modelling task. Specifically, once the linear/nonlinear sub-spaces are known, by definition the explanatory variables may be transformed to form two disjoint sub-sets spanning, respectively, the linear and nonlinear sub-spaces. The nonlinear modelling task is confined to the latter sub-set, which will typically have a smaller number of elements than the original set of explanatory variables. Constructive algorithms are proposed for inferring the linear and nonlinear sub-spaces from noisy data.
- nonlinear identification
- dimensionality reduction
- Gaussian process priors
- nonlinear mapping
Leith, D. J., Leithead, W. E., & Murray-Smith, R. (2006). Inference of disjoint linear and nonlinear sub-domains of a nonlinear mapping. Automatica, 42(5), 849-858. https://doi.org/10.1016/j.automatica.2006.01.019