Kernel-based methods for parameter estimation in multidimensional systems (Q2851166)

This Ph.D. thesis is concerned with kernel-based methods as a way to estimate continuous phenomena based on discrete and finite-dimensional measurements. It is shown how kernel regularization approaches can be used for parameter estimation of different multidimensional systems. The main aim of this dissertation is to develop extensions to the vector-valued case for three methods of parameter estimation. In this frame, the present Ph.D. thesis is divided into two parts. The first part consists of the first two chapters, while the second part consists of the last chapter. The full list of references is given at the end of the thesis. In the sequel, the subject of each one of the three chapters is briefly presented.NEWLINENEWLINE Chapter 1 (Learning vector-valued functions.) summarizes two basic concepts for estimating vector-valued functions based on a finite set of measurements. The first approach assumes that the unknown function is an element of a reproducing kernel Hilbert space (RKHS) of vector-valued functions, while the second assumes that the function is drawn from a vector-valued Gaussian process. In Chapter 2 (Kernel methods in multidimensional systems.) two applications, which serve as examples for the two methods of Chapter 1, are presented. In order to present the RKHS perspective a new approach for identification of ARX-Hammerstein models with multiple inputs and multiple outputs is developed. The second application of the chapter uses the kernel as a cross-covariance model in the context of Gaussian processes. In this frame, a new estimation criterion for simultaneous kriging of vector-valued random variables is suggested.NEWLINENEWLINE In the second part of this thesis (Chapter 3: A mathematical model for the glucose-insulin regulatory system.) a mathematical model that connects the dynamics of glucose and insulin concentration with the \(\beta\)-cell cycle is introduced. The interaction of glucose, insulin and the \(\beta\)-cell cycle is described by a system of ordinary differential equations. Initially, the motivation of the model, the general setup, its development as well as its mathematical analysis, which investigated the steady states of the model and their stability, are presented in detail. The model is used as a concrete application for a kernel based approximation method to solve systems of ordinary differential equations.NEWLINENEWLINE Afterwards two simulations of the model are presented. The first simulation describes the behavior of the glucose-insulin regulatory system in the physiological case, while the second one shows the adaption of \(\beta\)-cell mass under long term glucose infusion. Some of the results of this chapter are included in Gallenberger et al., Theor. Biol. and Med. Model. 9, 46\,ff (2012).NEWLINENEWLINE Summarizing, the results of this Ph D thesis are the extension of estimation methods to the case of vector-valued data. Furthermore, a new model for the glucose-insulin regulatory system is presented and is used as a concrete example for the use of a radial basis function (RBF) approximation network to estimate the solution of a system of differential equations.