We discuss several aspects of the mathematical foundations of the nonlinear black-box
identification problem. We shall see that the quality of the identification procedure is always
a result of a certain trade-off between the expressive power of the model we try to identify
(the larger the number of parameters used to describe the model, the more flexible is the
approximation), and the stochastic error (which is proportional to the number of parameters).
A consequence of this trade-off is the simple fact that a good approximation technique can …

The basic idea behind the large family of non-linear black box models is described. It is
shown how neural networks, wavelet networks, hinging hyperplanes, kernel methods, fuzzy
models, etc all fit to a common framework of basis expansion, using a single “mother
function”. Most differences relate to how this signle function is expanded tohigher regressor
dimensions. The estimation theory, algorithmic aspects and applications to dynamical
systems are also covered.