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Description:
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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 â  slicingâ  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 . |