Regression And Dimension Reduction For Functional Data

High dimensional data are becoming more and more common in the field of multi-variate data analysis. However, the high dimensionality is problematic due to increasingcomputational costs and to the curse of dimensionality. Functional data analysis assumethe observed data from a smooth curve, and have its distinctive superiority to deal withhigh dimensional data. As the development of collecting data online and nonparametrictechnology, functional data analysis is a hot area of modern statistics research, and hasreceived more and more attention in diferent fields of application, including medical,biological, Micro-array data, Chemistry and Survival analysis. Many classic statisticsmethod have been extended to functional data analysis.This paper consists four parts. Firstly, I propose a new canonical correlation-mixeddata canonical correlation analysis. Mixed data canonical correlation analysis studiesthe linear correlation between random vector and random function. Furthermore, I studyits theory properties.Secondly, functional data view the whole function as an observation, thus, the di-mension of the observation is infinity, therefore dimension reduction becomes necessaryfor functional data. In this paper, I mainly focus on functional dimension reduction. iproposed functional multi-index model which treat the real response variable as a func-tion of the called index. These indexes are random elements which are generated bysecond order stochastic process in Hilbert space. To infer the EDR space, this paperconsider three cases of functional multiple index models: the first case is that the re-sponse is scalar and the predictor is random curve; the second one is that the responseis binary and the predictor is random curve, and the last case is that the response isp-dimensional random vector and the predictor is random curve. Furthermore, I applythese methods to real data analysis including spectrum data, coal analysis and climateforecasting and so on. Thirdly, the most studied functional linear model is that the response is scalar vari-able and the predictor is functional data. I propose a novel method to estimate the re-gression parameter function, which is based on the functional sufcient dimension reduc-tion. A specific procedure for the estimation of the regression parameter function usingfunctional sufcient dimension reduction is proposed and compared with an establishedfunctional principal component regression approach, and I proposed a new method toestimate the EDR space without needed the inverse of the covariance operator.Lastly, the fourth part of this paper is to consider functional transformed model.Functional linear regression has been widely used to model the relationship betweena scalar response and functional predictors. If the original data do not satisfy the lin-ear assumption, an intuitive solution is to perform some transformation such that trans-formed data will be linearly related. The problem of finding such transformations hasbeen rather neglected in the development of functional data analysis tools. This paperconsider transformation on the response variable in functional linear regression and pro-pose a nonparametric transformation model in which this paper use spline functions toconstruct the transformation function. The functional regression coefcients are thenestimated by an innovative procedure called mixed data canonical correlation analysis(MDCCA). MDCCA is analogous to the canonical correlation analysis between twomultivariate samples, but is between a multivariate sample and a set of functional data.Here, this paper apply the MDCCA to the projection of the transformation function onthe B-spline space and the functional predictors. then show that our estimates agree withthe regularized functional least squares estimate for the transformation model subjectto a scale multiplication. The dimension of the space of spline transformations can bedetermined by a model selection principle. Typically, a very small number of B-splineknots is needed. Some general computational issues of functional data analysis werediscussed.