ADAPTIVE SPLINES AND GENETIC ALGORITHMS FOR OPTIMAL STATISTICAL MODELING
Open Access
- Author:
- Pittman, Jennifer Lynn
- Graduate Program:
- Statistics
- Degree:
- Doctor of Philosophy
- Document Type:
- Dissertation
- Date of Defense:
- January 10, 2000
- Committee Members:
- Rich Caruana, Committee Member
Calyampudi Radhakrishna Rao, Committee Chair/Co-Chair
Mark Handcock, Committee Member
Zhen Luo, Committee Member
Soundar Rajan Tirupatikumara, Committee Member - Keywords:
- model selection
evolutionary computation
splines - Abstract:
- Many statistical analyses and applications require the capture of a relationship between two variables $X$ and $Y$ that is more complex than a simple linear relationship. One approach to solving this problem is to use a modeling technique to attempt to replace the complex and/or noisy relationship which the data represent by something simple yet reasonable which captures the nature of the dependence in the data. In the case where little is known about the underlying function which relates $X$ to $Y$, the modeling technique should be {it{flexible}} or {it{adaptive}}, i.e., able to handle a wide variety of functional shapes and behaviors. Nonparametric modeling is one such technique which has been successful in characterizing features of datasets that would not be obtainable by other means cite{hans}. Due in part to the increased availability of computational power, spatially adaptive smoothing methods involving regression splines have become a popular and rapidly developing class of nonparametric modeling techniques. Most of these methods are based on nonlinear optimization and/or stepwise selection of basis functions. Although computationally fast and spatially adaptive, stepwise knot selection is necessarily suboptimal while determining the best model over the space of adaptive splines is a very poorly behaved nonlinear optimization problem cite{wahba}. A possible alternative is to use a genetic algorithm to perform knot selection. Genetic algorithms are stochastic search methods which, under certain design conditions, have the capability or potential of converging to the global optimum of the evaluation function of an optimization problem cite{bmp1}. In other words, a genetic algorithm search is not biased from reaching the global optimum. Hence, given a variable selection criterion and a search space of possible knot locations, a genetic algorithm has the potential to find models that are more appropriate in comparison to models selected using nonlinear optimization or stepwise methods. In this work we explore the use of genetic algorithms for adaptive spline modeling in low dimensional settings. We introduce our work in Chapter ef{intr} and discuss genetic algorithms in Chapter ef{ga}. Chapters ef{paper2} and ef{paper3} cover work involving optimal modeling with linear splines only while Chapter ef{paperS} involves the optimal fitting of polynomial splines. Experimental results are compared to those of Dierckx cite{dckx}, Luo and Wahba cite{luo}, Donoho and Johnstone cite{don2}, Friedman cite{frie}, and Stone, Kooperberg, Hansen, and Troung cite{stone}, among others. Future research is focused on comparisons with the recent Bayesian spline methods (e.g., Denison, Mallick, and Smith cite{dms}), extensions to higher dimensions, and explorations into other areas of study such as neural networks and fractals.