Modeling and Segmentation using Multiple Models

Sammanfattning: An established area within the system identification field is identification of linear models. In practice, sometimes the performance of such models is not satisfactory and non-linear models are needed. This thesis is devoted to modeling and identification of systems by means of multiple models. The models used in a bank of models are simple models, often linear, that combined manage to well describe an input-output relationship of a more complex system.The material is presented in two parts. The parts differ by the assumption on availability of information. In Part I it is assumed that the switching between the models in the model bank is guided by an unobserved stochastic process. We have chosen to model the switching by a Markov chain. The task is to model the output of a "jumping" system and estimate parameters in the models. That task is not difficult to solve if the points where the models switch are known. Since they are not, the complexity of the task increases severely. In the first part of the thesis we extend known solutions for the one-dimensional case to two dimensions. The resulting methods are sub-optimal. Sub-optimality iis the price payed for avoiding an otherwise overwhelming computational complexity of the estimation algorithms. The methods are applied to data from a laser range system. As a spin-off the problem of estimation of the number of hidden states of the Markov chain, given output data only, is investigated.The second part treats the case when the choice of a model from a model bank is governed by the input to the system considered. We are, thus, in a framework of function approximation, since an input-output relation can be viewed as a function mapping from the input of the system to the output. A recently introduced model class, the hinging hyperplane models, is presented. It turns out to be related to other model classes such as neural network models, regression trees, etc. A number of issues related to the new model class is discussed, e.g., parameterization, conditioning, etc. Finally, as a last contribution, a variant of hinging hyperplane models, so called smooth hinging hyperplane models, is presented.