Date of Award

2017

Document Type

Open Access Dissertation

Department

Statistics

Sub-Department

College of Arts and Sciences

First Advisor

David B. Hitchcock

Abstract

In many subjects such as psychology, geography, physiology or behavioral science, researchers collect and analyze non-traditional data, i.e., data that do not consist of a set of scalar or vector observations, but rather a set of sequential observations measured over a fine grid on a continuous domain, such as time, space, etc. Because the underlying functional structure of the individual datum is of interest, Ramsay and Dalzell (1991) named the collection of topics involving analyzing these functional observations functional data analysis (FDA). Topics in functional data analysis include data smoothing, data registration, regression analysis with functional responses, cluster analysis on functional data, etc. Among these topics, data smoothing and data registration serve as preliminary steps that allow for more reliable statistical inference afterwards. In this dissertation, we include three research projects on functional data smoothing and its effects on functional data applications. In particular, Chapter 2 mainly presents a unified Bayesian approach that borrows the idea of time warping to represent functional curves of various shapes. Based on a comparison with the method of B-splines developed by de Boor (2001) and some other methods that are well known for its broad applications in curve fitting, our method is proved to adapt more flexibly to highly irregular curves. Then, Chapter 3 discusses subsequent regression and clustering methods for functional data, and investigates the accuracy of functional regression prediction as well as clustering results as measured by either traditional in-sample and out-of-sample sum of squares or the Rand index. It is showed that using our Bayesian smoothing method on the raw curves prior to carrying out the corresponding applications provides very competitive statistical inference and analytic results in most scenarios compared to using other standard smoothing methods prior to the applications. Lastly, notice that one restriction for our method in Chapter 2 is that it can only be applied to functional curves that are observed on a fine grid of time points. Hence, in Chapter 4, we extend the idea of our transformed basis smoothing method in Chapter 2 to the sparse functional data scenario. We show via simulations and analysis that the proposed method gives a very good approximation of the overall pattern as well as the individual trends for the data with the cluster of sparsely observed curves.

Share

COinS