Functional Data Analysis Method And Its Application

The basic philosophy of functional data analysis (FDA) is to think of the observed data as an element of an infinite dimensional functional space. Over the past decade, with the rapid development of science and technology, FDA plays a more and more important role in modern scientific research. Many research fields encounter the statistical problems over functional data analysis, such as Psychology, Medical diagnosis, Weather forecast, Economics, Children growth analysis, and Life science, etc. Comparing with classical multivariate statistical methods and tools, FDA is still in development stage, and has very broad application future prospects. So, there are large number of problems on FDA needed to research.In this dissertation, we focus on some statistical analysis methods and its ap-plications on FDA. Firstly, we analyze a dataset of children growth survey with sparsity via one of sparse functional principal components analysis, named Princi-pal Analysis by Conditional Estimation (PACE) and proposed by Yao et al (2005). One may draw some interesting conclusions by analyzing real data.Secondly, we extend Fraiman-Muniz depth (Fraiman and Muniz,2001) to define some new functional depth functions. Based on these new depth functions, we propose a statistic as the functional counterpart of multivariate Liu-Singh depth-based rank sum statistic. To evaluate performance of our proposed statistic, we select functional analysis of variance and depth-based Wilcoxon rank sum test as the references and do apply them into analyze some simulation studies and real datasets.Recently, many Chinese authors begin to analyze China economic data by FDA, such as Yan Mingyi (2007a,2007b), Jin Liurui (2008) and so forth. We also try to divide Chinese (mainland) economic zones by functional clustering analysis method based on functional principal components analysis proposed by Chiou and Li (2007). The clustering result show that this method is feasible and could provide a reference for making policy.We propose a robust functional principal components analysis method based on functional depth functions mentionabove. Furthermore, one may construct a new discriminant analysis method so that one can remove the negative effect caused by some outliers in observed sample and obtain robust estimation for some unknown principal components. To evaluate its performance, one select some band depth-based functional classification methods, which is proposed by Lopez-Pintado and Romo (2006b), as the references. Simulation study shows that our new method not only give better error rate than the one based on references, but also save lots of computer running time. So it is worth to spread this new method into many applied fields in practice.