(Dr. Nader Sadegh, advisor)
"Nonparametric Statistical Modeling of High-Dimensional Nonlinear Dynamical Systems: A Projection Pursuit Solution"
Over the years, a number of approaches have been developed to model nonlinear systems. However, all suffer to some extent from what has been dubbed the “curse of dimensionality” when applied to high-dimensional systems. This research proposes a novel projection pursuit solution to this problem with wide-ranging applicability in an effort to alleviate some of these data sparsity issues.
The main theoretical results of this work include the theoretical foundations of the dimension reduction technique employed, as well as a theorem that shows that any high-dimensional L2 function can be decomposed into a finite number of single dimensional, mutually orthogonal functions.
Furthermore, this proposal applies the derived theory toward the construction of an innovative projection pursuit architecture capable of modeling the response of many high-dimensional systems with greater accuracy than current methods. Included in this proposal is a discussion of the choice of basis functions to be used in the network as well as a description of the Levenberg-Marquardt optimization scheme utilized to find the parameters that enable the network to best approximate the response surface. Additionally, preliminary work on a second suitable architecture (a projection pursuit approach with a genetic algorithm optimization) will also be presented.