Functional inverse regression and reproducing kernel Hilbert space

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Title: Functional inverse regression and reproducing kernel Hilbert space
Author: Ren, Haobo
Abstract: 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 ? ? ? ? ? ?slicing ? ? ? ? ? ? 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 .
URI: http : / /hdl .handle .net /1969 .1 /4203
Date: 2006-10-30

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Functional inverse regression and reproducing kernel Hilbert space. Available electronically from http : / /hdl .handle .net /1969 .1 /4203 .

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