An Introduction to Theoretical Properties of Functional Principal Component Analysis

The term “functional data” refers to data where each observation is a curve, a surface, or a hypersurface, as opposed to a point or a finite-dimensional vector. Functional data are intrinsically infinite dimensional and measurements on the same curve display high correlation, making assumptions of classical multivariate models invalid. An alternative approach, functional principal components analysis (FPCA), is used in this area as an important data analysis tool. However, presently there are very few academic reviews that summarize the known theoretical properties of FPCA. The purpose of this thesis is to provide a summary of some theoretical properties of FPCA when used in functional data exploratory analysis and functional linear regression. Practical issues in implementing FPCA and further topics in functional data analysis are also discussed, however, the emphasis is given to asymptotics and consistency results, their proofs and implications. Acknowledgements I would like to thank my teachers, Professors Peter Hall, Konstatin Borovkov, and Richard Huggins, for their guidance and editing assistance. Ngoc M. Tran, June 2008.