The Study On The Methods Of Functional Data Analysis And Their Application

In functional data analysis, data are analyzed from a functional angle with viewing the data as an entirety, which is shown as a smooth curve or continuous functions. Compared with the traditional analyzing methods, functional data analysis has its own advantages, it relies relatively on less assumed conditions and looser structural limit, does not require that different observed objects have the same observing point and times, not only can be used to analyze infinite dimensional data, but also can be used to analyze non-functional data. An important characteristic of functional data analysis is that derivative curves or differential curves can be found and used to mine more important information about the data.The research in this field has just began and there are still many problems needing us to pay more attention and efforts. What's more, this analysis method is seldom used in economic data analysis. This paper introduces systematically the method for functional data analysis and then conducts a research into the method and its application in economic data analysis.The thesis is entitled A Study on the Methods of Functional Data Analysis and Their Application, probes systematically into the methods of functional data analysis, including functional linear models, functional principle components analysis, functional cluster analysis, functional canonical correlation and discriminant analysis, and then apply the methods to analyze economic data.The creative points of this thesis can be viewed in the following aspects:(1) The thesis puts forward the functional linear model based on regression parameter functions. Compared with three basic functional linear models, the method improves the computability of the functional linear models, which can be used to not only analyze and forecast economic phenomenon from a dynamic angle in a traditional way, but also discover deeper information behind dynamic laws such as the speed and the acceleration of the developing and changing of economic phenomenon in continuous point of view, which presents the practical applicability of functional data analysis. (2) The thesis puts forward a new cluster method which measuring similarity based on milestones and taking both sequence values and shapes into account. The method overcomes the disadvantage of the cluster methods based on sequence values ignoring the trajectory shapes and of the cluster methods based on sequence shapes neglecting sequence values which involve information about similarities and trends between curves. The cluster method has practical significance in the cluster analysis of high-frequency financial time series.(3) The thesis studies on the application of functional data analysis in economic data analysis. The writer applies functional regression into investigating the developing and changing trends of China's urban expenditure tendency; functional principle components analysis into analyzing the main varying modes of the government expenditure by main item; functional cluster analysis into investigating the clustering problem of high-frequency financial time series; functional canonical correlation analysis into analyzing covarying modes between per capita annual income and living expenditure of urban households.(4) The thesis finds some economic laws, which are not easily identifiable using traditional analysis methods, in the process of applying functional data analysis into economic data. For example, the variation of marginal per capita annual expenditure lags that of income of urban households; the increase of per capita annual living expenditure may spur the increase of income in the regions where per capita annual income of urban households is higher, the increase of income may spur the increase of living expenditure in the regions where per capita annual income of urban households is middling or lower etc.