Abstract
The main purpose of this paper is to explore the principle components of Shanghai stock exchange 50 index by means of functional principal component analysis (FPCA). Functional data analysis (FDA) deals with random variables (or process) with realizations in the smooth functional space. One of the most popular FDA techniques is functional principal component analysis, which was introduced for the statistical analysis of a set of financial time series from an explorative point of view. FPCA is the functional analogue of the well-known dimension reduction technique in the multivariate statistical analysis, searching for linear transformations of the random vector with the maximal variance. In this paper, we studied the monthly return volatility of Shanghai stock exchange 50 index (SSE50). Using FPCA to reduce dimension to a finite level, we extracted the most significant components of the data and some relevant statistical features of such related datasets. The calculated results show that regarding the samples as random functions is rational. Compared with the ordinary principle component analysis, FPCA can solve the problem of different dimensions in the samples. And FPCA is a convenient approach to extract the main variance factors.
Highlights
In the present study of data analysis we have learned, the data we research is either cross-sectional data or panel data
Treating stock price series as random function in a space spanned by finite dimensional functional bases, we intensively explore methods of functional data analysis, especially functional principal component analysis
functional principal component analysis (FPCA) attempts to find the dominant modes of variation around an overall trend function and is a key technique in functional data analysis
Summary
In the present study of data analysis we have learned, the data we research is either cross-sectional data or panel data. We often meet with such data which has functional characteristics. Functional data is multivariate data with an ordering on the dimensions [1]. The typical dataset of this sort consists of time series and cross-sectional data, such as the time series of stock price, and some datasets even may take on curves or images. Advances in data collection and storage have tremendously increased the presence of such functional data, whose graphical representations are curves, images, or shapes. The theoretical and practical developments in functional data analysis are mainly from the last four decades, due to the rapid development of computer recording and storing facilities. As a new area of statistics, functional data analysis extends existing methodologies and theories from the fields of data analysis, generalized linear models, multivariate data analysis, nonparametric statistics, and many others. There were several impressive attempts to analyze functional dataset such as Ramsay et al [2,3,4,5], who proposed some new concepts and methods in the field of FDA
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