Functional data analysis (FDA) is an active field of statistics, in which the primary subjects
in the study are curves. My dissertation consists of two innovative applications of
functional data analysis in biology. The data that motivated the research broadened the
scope of FDA and demanded new methodology. I develop new nonparametric methods to
make various estimations, and I focus on developing large sample theories for the proposed
estimators.
The first project is motivated from a colon carcinogenesis study, the goal of which is to
study the function of a protein (p27) in colon cancer development. In this study, a number
of colonic crypts (units) were sampled from each rat (subject) at random locations along
the colon, and then repeated measurements on the protein expression level were made on
each cell (subunit) within the selected crypts. In this problem, measurements within each
crypt can be viewed as a function, since the measurements can be indexed by the cell
locations. The functions from the same subject are spatially correlated along the colon,
and my goal is to estimate this correlation function using nonparametric methods. We use
this data set as an motivation and propose a kernel estimator of the correlation function
in a more general framework. We develop a pointwise asymptotic normal distribution
for the proposed estimator when the number of subjects is fixed and the number of units within each subject goes to infinity. Based on the asymptotic theory, we propose a weighted
block bootstrapping method for making inferences about the correlation function, where the
weights account for the inhomogeneity of the distribution of the unit locations. Simulation
studies are also provided to illustrate the numerical performance of the proposed method.
My second project is on a lipoprotein profile data, where the goal is to use lipoprotein
profile curves to predict the cholesterol level in human blood. Again, motivated by the data,
we consider a more general problem: the functional linear models (Ramsay and Silverman,
1997) with functional predictor and scalar response. There is literature developing different
methods for this model; however, there is little theory to support the methods. Therefore,
we focus more on the theoretical properties of this model. There are other contemporary
theoretical work on methods based on Principal Component Regression. Our work is different
in the sense that we base our method on roughness penalty approach and consider a
more realistic scenario that the functional predictor is observed only on discrete points. To
reduce the difficulty of the theoretical derivations, we restrict the functions with a periodic
boundary condition and develop an asymptotic convergence rate for this problem in Chapter
III. A more general result based on splines is a future research topic that I give some
discussion in Chapter IV.
| Identifer | oai:union.ndltd.org:tamu.edu/oai:repository.tamu.edu:1969.1/ETD-TAMU-1867 |
| Date | 02 June 2009 |
| Creators | Li, Yehua |
| Contributors | Carroll, Raymond J., Hsing, Tailen |
| 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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