Method Of Functional Data Analysis And Its Application In Stock Index Analysis

With the rapid development of computer software and hardware, the scale of data that people can obtain and need to analyze is also growing, which makes the traditional methods of data analysis be limited. Moreover, in practical applications, many of the individuals (i.e., the sample) in the study are often characterized by continuity in its some indicators. That is, the discontinuity of the sample data is getting smaller and smaller. At this point, the use of functional data analysis (FDA) in the study of the individual indicators will be more reasonable. This is because that functional data analysis can not only analyze the high dimensional data, it also requires that the number and the positions of observation points are same to different observation objects.In China, the Shanghai Composite Index and Shenzhen Component Index are both the most important stock indices and cover the widest range, while the Gem index and the small and medium-sized board index are also two of the major indexes in numerous indexes. These four indexes are not only the most common ones referred in the stock market and studied by scholars. Therefore, we select the four indexes for analysis and discussion. In this paper, we firstly reviewed the development of functional data analysis. We found that the application of functional principal component analysis in stock index analysis can help readers understand the changing trend and rule of the stock index from another point of view, and this also expand the application fields of this method. Secondly, the theory of base function fitting and functional principal component analysis was described. In the base function fitting, we defined the FIC criterion based on the AIC criterion in time series theory, which is used to determine the number of base functions. Finally, we analyzed the closing value of the above four indexes of each trading day in 2014. In the instance, we fitted the four indexes respectively with three kinds of base functions, then the Fourier base functions was determined by comparisons. Then we got the fitness of each index function, and found that the relative errors of each one are all less than 5, which is a relatively small value. Then the functional principal components analysis was proceeded for the all index functions. The results indicated that the cumulative contribution rate of the first two principal components has been more than 99%, which means that the information loss is quite small. The score of two principal components indicated that the Shanghai Composite Index, the Shenzhen Component Index, and the Gem index were mainly impacted by the first principal component, and that the small and medium-sized board index was impacted by both the first and second principal component. Combined with the actual situation, the results can be concluded as follows:the first principal component reflected the self-protection factors to maintain the stability inside the system, while the second principal component should be regard as the reflection of comprehensive factors which impact on the mean stock index in macro environment.