Selecting inputs and measuring nonlinearity in system identification

The problems of selecting inputs and measuring system nonlinearity are stated in terms of statistical estimation theory. The natural solution which emerges consists in estimating and comparing Bayes risk (minimum mean square error) with the linear minimum risk. For any subset of inputs, these quantities are defined as the overall best performance achievable using a nonlinear and a linear model, respectively. A method based on kernel density estimation is presented to compute both quantities directly from a batch of input-output data, prior to any model choice. The relevance of a particular input subset can thus be naturally quantified by its corresponding minimum risk. As a consequence, superfluous input variables can, in principle, be detected and removed before the tedious task of model structure design. Furthermore, the method enables to determine a priori the improvement which can potentially be achieved by using a nonlinear model instead of a linear one. In practice, the method is limited by the required amount of data increasing exponentially with the number of inputs, which is characteristic of nonparametric estimation techniques.