Development of functional principal components analysis and estimating the time-varying gene regulation network

Functional data analysis (FDA) addresses the analysis of information on curves or functions. Examples of such curves or functions include time-course gene expression measurements, the Electroencephalography (EEG) data motoring the brain activity, the emission rate of automobiles after acceleration and the growth curve of children on body fat percentage made over a growth time period. The primary interests for the underlying curves or functions varies in different fields. In this thesis, new methodology for constructing time-varying net- work based on functional observations is proposed. Several variations of Functional Principal Component Analysis (FPCA) are developed in the context of functional regression model. Lastly, the new use of FPCA are explored in terms of recovering trajectory functions and estimating derivatives.