In all of the preceding chapters, we have examined the identification problem for block-oriented systems of various forms, that are characterized by a one-dimensional input process. In numerous applications, we confront the problem of identifying a system that has multiple inputs and multiple interconnecting signals. The theory and practical algorithms for identification of multivariate linear systems have been thoroughly examined in the literature [332]. On the other hand, the theory of identification of multivariate nonlinear systems has been far less explored. This is mainly due to the mathematical and computational difficulties appearing in multivariate problems. In this chapter, we examine some selected multivariate nonlinear models that are natural generalizations of the previously introduced block-oriented connections. An apparent curse of dimensionality that takes place in high-dimensional estimation problems forces us to focus on low-dimensional counterparts of the classical block-oriented structures. In particular, we examine a class of additive models, which provides a parsimonious representation for multivariate systems. Indeed, we show that the additive systems provide simple and interpretable structures, which also give a reasonable trade-off between the systematic modeling error and the estimation error of an identification algorithm. The theory of finding an optimal additive model is examined.

Multivariate nonparametric regression

As in all of the previous chapters, we will make use of the notion of a regression function.