The basic philosophy of Functional Data Analysis (FDA) is to think of the observed data
functions as elements of a possibly infinite-dimensional function space. Most of the current
research topics on FDA focus on advancing theoretical tools and extending existing
multivariate techniques to accommodate the infinite-dimensional nature of data. This dissertation
reports contributions on both fronts, where a unifying inverse regression theory
for both the multivariate setting (Li 1991) and functional data from a Reproducing Kernel
Hilbert Space (RKHS) prospective is developed.
We proposed a functional multiple-index model which models a real response variable
as a function of a few predictor variables called indices. These indices are random
elements of the Hilbert space spanned by a second order stochastic process and they constitute
the so-called Effective Dimensional Reduction Space (EDRS). To conduct inference
on the EDRS, we discovered a fundamental result which reveals the geometrical association
between the EDRS and the RKHS of the process. Two inverse regression procedures,
a âÂÂslicingâ approach and a kernel approach, were introduced to estimate the counterpart of
the EDRS in the RKHS. Further the estimate of the EDRS was achieved via the transformation
from the RKHS to the original Hilbert space. To construct an asymptotic theory, we
introduced an isometric mapping from the empirical RKHS to the theoretical RKHS, which
can be used to measure the distance between the estimator and the target. Some general computational issues of FDA were discussed, which led to the smoothed versions of the
functional inverse regression methods. Simulation studies were performed to evaluate the
performance of the inference procedures and applications to biological and chemometrical
data analysis were illustrated.
| Identifer | oai:union.ndltd.org:tamu.edu/oai:repository.tamu.edu:1969.1/4203 |
| Date | 30 October 2006 |
| Creators | Ren, Haobo |
| Contributors | Hsing, Tailen |
| Publisher | Texas A&M University |
| Source Sets | Texas A and M University |
| Language | en_US |
| Detected Language | English |
| Type | Electronic Dissertation, text |
| Format | 616148 bytes, electronic, application/pdf, born digital |
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