基于Takenaka-Malmquist自适应时频分布的特征指标——来自上证和深证的数据

股票价格指数度量并反映了股票市场总体价格水平及其变动趋势，包含了丰富的市场信息，受到投资者和政策制定者的普遍关注.利用一定的数学方法对其进行分析和研究，挖掘股指的潜在价值，对加快资本市场治理，提升金融效率，促进国民经济的平稳快速发展具有十分重要的意义.本文利用基于Takenaka-Malmquist自适应傅里叶分解（简称自适应傅里叶分解或AFD）的时频分布，有效提取了股票价格指数的时频特征，分析股票市场的变动趋势.为满足自适应傅里叶分解的要求，首先利用H-P滤波算法对时间序列进行预处理，去除时间序列的趋势项，然后利用AFD算法处理周期项数据，在此基础上得到股票价格指数的时频分布，并进一步分析股指变动趋势.基于自适应傅里叶分解的算法可以有效提取股指在时频两域的信息，避免了单一域分析的缺陷，且比现有的小波分解方法具有更高的分辨率和准确度.为检验指标的有效性，本文利用上海证券交易所的上证综合指数（代码000001）和深圳证券交易所深证成份指数（代码399001）实证检验了指标的有效性，结果表明基于自适应傅里叶分解的时频分布提出的股市技术分析指标可以用于中短期股票市场的变动特征分析.

Characteristic index based on Takenaka-Malmquist adaptive time-frequency distribution — Data from Shanghai Composite Index and Shenzhen Component Index

The stock price index measures and reflects the overall price level of the stock market and its trend. It contains abundant market information and has been widely concerned by investors and policymakers. It is of great significance to analyze, study, and mine the potential value of the stock index by using some mathematical methods for accelerating capital market governance, improving financial efficiency, and promoting the steady and rapid development of the national economy. This paper proposes an adaptive Fourier decomposition (AFD) method based Takenaka-Malmquist approach to extract the time-frequency characteristics of stock price index and analyze the trend of stock market effectively. In order to satisfy the requirement of AFD, the H-P filtering algorithm is applied first to pre-process the time series and remove the trend term. Then the AFD algorithm is employed to process the periodic term data. Finally, the trend of the stock index is analyzed by the time-frequency distribution chart of the stock price index. The algorithm based on AFD can effectively extract the information of stock index in both time and frequency domains, which overcomes the drawback of single domain analysis. Meanwhile it has higher resolution and accuracy than the existing wavelet decomposition methods. The effectiveness of the proposed indicator is tested by using the Shanghai Composite Index (Stock code 000001) and Shenzhen Component Index (Stock code 399001), respectively. The results demonstrate that the technical analysis of the stock market index established by the time-frequency distribution based on AFD can be used to analyze the characteristics of short and medium-term stock market changes.