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Weighted multiscale permutation entropy of financial time series

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Abstract

In this paper, we propose the weighted multiscale permutation entropy (WMPE) as a novel measure to quantify the complexity of nonlinear time series containing amplitude-coded information. WMPE is different from multiscale permutation entropy (MPE) in the sense that it suits better signals having considerable amplitude information and also succeed in accounting for the multiple time scales inherent in the financial systems. We first perform the MPE and WMPE methods on synthetic data showing the power of WMPE. Then, we apply the MPE and WMPE methods to the US and Chinese stock markets and make a comparison to discuss their differences and similarities between these different markets. The WMPE of each US and Chinese stock market points out the necessity of applying permutation entropy on multiple scales and reveals amplitude-coded information contained in the signals and immunity to degradation by noise when \(m = 5-7\). Some new conclusions are gotten by the new characteristics we detected in the comparison. WMPE method can distinguish ShangZheng and ShenCheng from these markets and imply that HSI is more similar to the US markets. WMPE moves the fluctuation range of entropy values of different market down differently reflecting more accurate complexity of different stock markets containing amplitude information. Compared WMPE of ShangZheng and ShenCheng with the WMPE of US markets and HSI, US stock market and HSI may have more amplitude information carried by the signals of stock market. Furthermore, compared with MPE, WMPE can reduce the standard deviation which ensures the results more robust. The conclusion that \(m = 7\) is the best embedding dimension to investigate the WMPE results can be drawn because the WMPE tends to be stable in a certain range and reflects the necessity of investigation on multiscale and advantages of adding different weight, and it can distinguish these markets while reducing the standard deviations of all the markets.

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Acknowledgments

The financial supports from the funds of the China National Science (61071142, 61371130), the Beijing National Science (4122059), and the National High Technology Research Development Program of China (863 Program) (2011AA110306) are gratefully acknowledged.

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Authors and Affiliations

  1. Department of Mathematics, School of Science, Beijing Jiaotong University, Beijing, 100044, People’s Republic of China

    Yi Yin & Pengjian Shang

Authors
  1. Yi Yin
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  2. Pengjian Shang
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Corresponding author

Correspondence to Pengjian Shang.

Appendix: MPE and WMPE results of US and Chinese stock markets on different time delay \(\tau \) from 2 to 5

Appendix: MPE and WMPE results of US and Chinese stock markets on different time delay \(\tau \) from 2 to 5

Figures 14, 15, 16 and 17 show the MPE and WMPE results of US markets: S&P500, DJI and NQCI on various embedding dimension \(m\) when time delay \(\tau =2-5\), while Figs. 18, 19, 20 and 21 show the MPE and WMPE results of Chinese markets: ShangZheng, ShenCheng and HSI on various embedding dimension \(m\) when time delay \(\tau =2-5\).

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MPE and WMPE results of US markets: S&P500, DJI and NQCI on various embedding dimension \(m\) and time delay \(\tau =2\)

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MPE and WMPE results of US markets: S&P500, DJI and NQCI on various embedding dimension \(m\) and time delay \(\tau =3\)

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MPE and WMPE results of US markets: S&P500, DJI and NQCI on various embedding dimension \(m\) and time delay \(\tau =4\)

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Fig. 17
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MPE and WMPE results of US markets: S&P500, DJI and NQCI on various embedding dimension \(m\) and time delay \(\tau =5\)

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Fig. 18
figure 18

MPE and WMPE results of Chinese markets: ShangZheng, ShenCheng and HSI on various embedding dimension \(m\) and time delay \(\tau =2\)

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Fig. 19
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MPE and WMPE results of Chinese markets: ShangZheng, ShenCheng and HSI on various embedding dimension \(m\) and time delay \(\tau =3\)

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Fig. 20
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MPE and WMPE results of Chinese markets: ShangZheng, ShenCheng and HSI on various embedding dimension \(m\) and time delay \(\tau =4\)

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Fig. 21
figure 21

MPE and WMPE results of Chinese markets: ShangZheng, ShenCheng and HSI on various embedding dimension \(m\) and time delay \(\tau =5\)

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Yin, Y., Shang, P. Weighted multiscale permutation entropy of financial time series. Nonlinear Dyn 78, 2921–2939 (2014). https://doi.org/10.1007/s11071-014-1636-2

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  • DOI: https://doi.org/10.1007/s11071-014-1636-2

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