Yin Yi, Shang Pengjian
Department of Mathematics, Beijing Jiaotong University, No. 3 Shangyuan Residence, Haidian District, Beijing, 100044, People's Republic of China.
Chaos. 2015 Mar;25(3):032101. doi: 10.1063/1.4913765.
The paper proposes the asymmetric multiscale cross-sample entropy (AMCSE) method and applies it to analyze the financial time series of US, Chinese, and European stock markets. The asynchronies of these time series in USA, China, and Europe all decrease (the correlations increase) with the increase in scale which declares that taking into account bigger time scale to study these financial time series is capable of revealing the intrinsic relations between these stock markets. Meanwhile, we find that there is a crossover between the upwards and the downwards in these AMCSE results, which indicates that when the scale reach a certain value, the asynchronies of the upwards and the downwards for these stock markets are equal and symmetric. But for the other scales, the asynchronies of the upwards and the downwards are different from each other indicating the necessity and importance of multiscale analysis for revealing the most comprehensive information of stock markets. The series with a positive trend have a higher decreasing pace on asynchrony than those with a negative trend, while the asynchrony between the series with a positive or negative trend is lower than that between the original series. Moreover, it is noticeable that there are some small abnormal rises at some abnormal scales. We find that the asynchronies are the highest at scales smaller than 2 when investigating the time series of stock markets with a negative trend. The existences of asymmetries declare the inaccuracy and weakness of multiscale cross-sample entropy, while by comparing the asymmetries of US, Chinese, and European markets, similar conclusions can be drawn and we acquire that the asymmetries of Chinese markets are the smallest and the asymmetries of European markets are the biggest. Thus, it is of great value and benefit to investigate the series with different trends using AMCSE method.
本文提出了非对称多尺度交叉样本熵（AMCSE）方法，并将其应用于分析美国、中国和欧洲股票市场的金融时间序列。随着尺度的增加，美国、中国和欧洲这些时间序列的异步性均降低（相关性增加），这表明考虑更大的时间尺度来研究这些金融时间序列能够揭示这些股票市场之间的内在关系。同时，我们发现这些AMCSE结果中存在上升和下降之间的交叉，这表明当尺度达到某个值时，这些股票市场上升和下降的异步性是相等且对称的。但对于其他尺度，上升和下降的异步性彼此不同，这表明多尺度分析对于揭示股票市场最全面信息的必要性和重要性。具有正趋势的序列在异步性上的下降速度比具有负趋势的序列更高，而具有正或负趋势的序列之间的异步性低于原始序列之间的异步性。此外，值得注意的是，在一些异常尺度处存在一些小的异常上升。我们发现，在研究具有负趋势的股票市场时间序列时，尺度小于2时异步性最高。非对称性的存在表明多尺度交叉样本熵的不准确和不足，而通过比较美国、中国和欧洲市场的非对称性，可以得出类似的结论，我们发现中国市场的非对称性最小，欧洲市场的非对称性最大。因此，使用AMCSE方法研究具有不同趋势的序列具有很大的价值和益处。
