Topics in functional data analysis with biological applications

Functional data analysis (FDA) is an active field of statistics, in which the primary subjectsin the study are curves. My dissertation consists of two innovative applications offunctional data analysis in biology. The data that motivated the research broadened thescope of FDA and demanded new methodology. I develop new nonparametric methods tomake various estimations, and I focus on developing large sample theories for the proposedestimators.The first project is motivated from a colon carcinogenesis study, the goal of which is tostudy the function of a protein (p27) in colon cancer development. In this study, a numberof colonic crypts (units) were sampled from each rat (subject) at random locations alongthe colon, and then repeated measurements on the protein expression level were made oneach cell (subunit) within the selected crypts. In this problem, measurements within eachcrypt can be viewed as a function, since the measurements can be indexed by the celllocations. The functions from the same subject are spatially correlated along the colon,and my goal is to estimate this correlation function using nonparametric methods. We usethis data set as an motivation and propose a kernel estimator of the correlation functionin a more general framework. We develop a pointwise asymptotic normal distributionfor the proposed estimator when the number of subjects is fixed and the number of units within each subject goes to infinity. Based on the asymptotic theory, we propose a weightedblock bootstrapping method for making inferences about the correlation function, where theweights account for the inhomogeneity of the distribution of the unit locations. Simulationstudies are also provided to illustrate the numerical performance of the proposed method.My second project is on a lipoprotein profile data, where the goal is to use lipoproteinprofile curves to predict the cholesterol level in human blood. Again, motivated by the data,we consider a more general problem: the functional linear models (Ramsay and Silverman,1997) with functional predictor and scalar response. There is literature developing differentmethods for this model; however, there is little theory to support the methods. Therefore,we focus more on the theoretical properties of this model. There are other contemporarytheoretical work on methods based on Principal Component Regression. Our work is differentin the sense that we base our method on roughness penalty approach and consider amore realistic scenario that the functional predictor is observed only on discrete points. Toreduce the difficulty of the theoretical derivations, we restrict the functions with a periodicboundary condition and develop an asymptotic convergence rate for this problem in ChapterIII. A more general result based on splines is a future research topic that I give somediscussion in Chapter IV.