Functional inverse regression and reproducing kernel Hilbert space

Functional Inverse Regression and Reproducing Kernel Hilbert Space. (August 2005) Haobo Ren, B.S., Peking University; M.S., Peking University Chair of Advisory Committee: Dr. Tailen Hsing The basic philosophy of Functional Data Analysis (FDA) is to think of the observed data functions as elements of a possibly infinite-dimensional fu nction space. Most of the current research topics on FDA focus on advancing theoretical t ools and extending existing multivariate techniques to accommodate the infinite-dimen sional nature of data. This dissertation reports contributions on both fronts, where a uni fying inverse regression theory for both the multivariate setting (Li 1991) and functional d ta 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 indice s. These indices are random elements of the Hilbert space spanned by a second order stoch a ic process and they constitute the so-called Effective Dimensional Reduction Spac e (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 re gr ssion 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 achieve d via the transformation from the RKHS to the original Hilbert space. To constru ct an asymptotic theory, we introduced an isometric mapping from the empirical RKHS to th e theoretical RKHS, which can be used to measure the distance between the estimator and the target. Some general