Functional Data Analysis And Its Application In Securities Investment

Data smoothing is a perpetual topic in mathematical statistics. In practice we usually use regression to smooth data, including linear regression and nonlinear regression. This thesis will introduce a new solution to data analysis - Functional Data Analysis (FDA), which is based on the assumption that the discrete data come from a function. FDA selects a basis function system to linearly expand the function we consider, so the function has all the good properties the basis functions possess. We often use Least Squares Method to estimate the parameters of the model in practice. However, it has its limitations, especially the one of over-smoothing. This thesis introduces a roughness penalty and a roughness matrix to the Least Squares Criteria to avoid over-smoothing. After that, we develop the multivariate statistical theory, including Linear Model, Principle Component Analysis(PCA), Canonical Correlation Analysis(CCA) and Analysis of Variance(ANOVA), in order to put these theories into functional data. In Appendix, we give some functions and classes of Matlab involved in FDA.In Chapter 3, we put the theory of FDA into tow case studies. Shanghai Composite Index is an important index for Chinese stock market, and we apply the Least Squares Method with roughness penalty to smooth its historical data, using spline functions. Besides, Securities Investment Fund is an important financial tool, and agio-rate is an unique index of Close-end Fund, which is influenced directly by its closing rate and net asset value. Firstly we apply Functional PCA to the data of agio-rate in order to explore its variation pattern, and then we focus on Variable Standardization Method and Functional CCA to uncover its inner relationship with the opening rate.