The objective of this dissertation is to develop a suitable statistical methodology
for functional data analysis. Modern advanced technology allows researchers to collect
samples as functional which means the ideal unit of samples is a curve. We consider
each functional observation as the resulting of a digitized recoding or a realization
from a stochastic process. Traditional statistical methodologies often fail to be applied
to this functional data set due to the high dimensionality.
Functional hypothesis testing is the main focus of my dissertation. We suggested
a testing procedure to determine the significance of two curves with order
restriction. This work was motivated by a case study involving high-dimensional
and high-frequency tidal volume traces from the New York State Psychiatric Institute
at Columbia University. The overall goal of the study was to create a model
of the clinical panic attack, as it occurs in panic disorder (PD), in normal human
subjects. We proposed a new dimension reduction technique by non-negative basis
matrix factorization (NBMF) and adapted a one-degree of freedom test in the context
of multivariate analysis. This is important because other dimension techniques, such
as principle component analysis (PCA), cannot be applied in this context due to the
order restriction.
Another area that we investigated was the estimation of functions with constrained
restrictions such as convexification and/or monotonicity, together with the development of computationally efficient algorithms to solve the constrained least
square problem. This study, too, has potential for applications in various fields.
For example, in economics the cost function of a perfectly competitive firm must be
increasing and convex, and the utility function of an economic agent must be increasing
and concave. We propose an estimation method for a monotone convex function
that consists of two sequential shape modification stages: (i) monotone regression
via solving a constrained least square problem and (ii) convexification of the monotone
regression estimate via solving an associated constrained uniform approximation
problem.
| Identifer | oai:union.ndltd.org:tamu.edu/oai:repository.tamu.edu:1969.1/ETD-TAMU-1626 |
| Date | 15 May 2009 |
| Creators | Lee, Sang Han |
| Contributors | Vannucci, Marina |
| Source Sets | Texas A and M University |
| Language | en_US |
| Detected Language | English |
| Type | Book, Thesis, Electronic Dissertation, text |
| Format | electronic, application/pdf, born digital |
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