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Multiscale Entropy: Recent Advances

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Complexity and Nonlinearity in Cardiovascular Signals

Abstract

Multiscale entropy is a widely used metric for characterizing the complexity of physiological time series. The fundamental difference to classical entropy measures is it enables quantification of nonlinear dynamics underlying physiological processes over multiple time scales. The basic idea of multiscale entropy was initially developed in 2002 and has since witnessed considerable progress in methodological expansions along with growing applications. Here, we provide an overview of some recent developments in the theory, identify some methodological constraints of the originally introduced multiscale entropy analysis, and discuss some improvements that we, and others, have made regarding the definition of the time scales, its multivariate extension and improved methods for estimating the basic technique. Finally, the application of multiscale entropy to the analysis of cardiovascular data is summarized.

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References

  1. Costa, M.D., Goldberger, A.L., Peng, C.K.: Multiscale entropy analysis of physiologic time series. Phys. Rev. Lett. 89, 0621021–4 (2002)

    Article  Google Scholar 

  2. Shannon, C.E.: A Mathematical Theory of Communication. Bell Syst. Tech. J. 27(3), 379–423 (1948)

    Article  Google Scholar 

  3. Grassberger, P., Procaccia, I.: Estimation of the Kolmogorov entropy from a chaotic signal. Phys. Rev. A. 28, 2591–2593 (1983)

    Article  Google Scholar 

  4. Eckmann, J.P., Ruelle, D.: Ergodic theory of chaos and strange attractors. Rev. Mod. Phys. 57, 617–656 (1985)

    Article  CAS  Google Scholar 

  5. Pincus, S.M.: Approximate entropy as a measure of system complexity. Proc. Natl. Acad. Sci. 88, 2297–2301 (1991)

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  6. Richman, J.S., Moorman, J.R.: Physiological time-series analysis using approximate entropy and sample entropy. Amerian Journal of Physiology-Heart and Circulatory. Physiology. 278, H2039–H2049 (2000)

    CAS  Google Scholar 

  7. Costa, M.D., Goldberger, A.L., Peng, C.K.: Multiscale entropy analysis of biological signals. Phys. Rev. E. 71, 021906 (2005)

    Article  Google Scholar 

  8. Humeau-Heurtier, A., Wu, C.W., Wu, S.D., Mahe, G., Abraham, P.: Refined multiscale Hilbert–Huang spectral entropy and its application to central and peripheral cardiovascular data. IEEE Trans. Biomed. Eng. 63(11), 2405–2415 (2016)

    Article  PubMed  Google Scholar 

  9. Silva, L.E., Lataro, R.M., Castania, J.A., da Silva, C.A., Valencia, J.F., Murta Jr., L.O., Salgado, H.C., Fazan Jr., R., Porta, A.: Multiscale entropy analysis of heart rate variability in heart failure, hypertensive, and sinoaortic-denervated rats: classical and refined approaches. Am. J. Phys. Regul. Integr. Comp. Phys. 311(1), R150–R156 (2016)

    Google Scholar 

  10. Liu, T., Yao, W., Wu, M., Shi, Z., Wang, J., Ning, X.: Multiscale permutation entropy analysis of electrocardiogram. Physica A. 471, 492–498 (2017)

    Article  Google Scholar 

  11. Liu, Q., Chen, Y.F., Fan, S.Z., Abbod, M.F., Shieh, J.S.: EEG artifacts reduction by multivariate empirical mode decomposition and multiscale entropy for monitoring depth of anaesthesia during surgery. Med. Biol. Eng. Comput. (2016). Online, doi: 10.1007/s11517-016-1598-2.

  12. Shi, W., Shang, P., Ma, Y., Sun, S., Yeh, C.H.: A comparison study on stages of sleep: Quantifying multiscale complexity using higher moments on coarse-graining. Commun. Nonlinear Sci. Numer. Simul. 44, 292–303 (2017)

    Article  Google Scholar 

  13. Grandy, T.H., Garrett, D.D., Schmiedek, F., Werkle-Bergner, M.: On the estimation of brain signal entropy from sparse neuroimaging data. Sci. Rep. 6, 23073 (2016)

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  14. Kuo, P.C., Chen, Y.T., Chen, Y.S., Chen, L.F.: Decoding the perception of endogenous pain from resting-state MEG. NeuroImage. 144, 1–11 (2017)

    Article  PubMed  Google Scholar 

  15. Costa, M., Peng, C.K., Goldberger, A.L., Hausdorff, J.M.: Multiscale entropy analysis of human gait dynamics. Physica A. 330, 53–60 (2003)

    Article  Google Scholar 

  16. Khalil, A., Humeau-Heurtier, A., Gascoin, L., Abraham, P., Mahe, G.: Aging effect on microcirculation: a multiscale entropy approach on laser speckle contrast images. Med. Phys. 43(7), 4008–4015 (2016)

    Article  CAS  PubMed  Google Scholar 

  17. Rizal, A., Hidayat, R., Nugroho, H.A.: Multiscale Hjorth descriptor for lung sound classification. International Conference on Science and Technology, 160008–1 (2015)

    Google Scholar 

  18. Ma, Y., Zhou, K., Fan, J., Sun, S.: Traditional Chinese medicine: potential approaches from modern dynamical complexity theories. Front. Med. 10(1), 28–32 (2016)

    Article  PubMed  Google Scholar 

  19. Li, Y., Yang, Y., Li, G., Xu, M., Huang, W.: A fault diagnosis scheme for planetary gearboxes using modified multi-scale symbolic dynamic entropy and mRMR feature selection. Mech. Syst. Signal Process. 91, 295–312 (2017)

    Article  Google Scholar 

  20. Aouabdi, S., Taibi, M., Bouras, S., Boutasseta, N.: Using multi-scale entropy and principal component analysis to monitor gears degradation via the motor current signature analysis. Mech. Syst. Signal Process. 90, 298–316 (2017)

    Article  Google Scholar 

  21. Zheng, J., Pan, H., Cheng, J.: Rolling bearing fault detection and diagnosis based on composite multiscale fuzzy entropy and ensemble support vector machines. Mech. Syst. Signal Process. 85, 746–759 (2017)

    Article  Google Scholar 

  22. Zhuang, L.X., Jin, N.D., Zhao, A., Gao, Z.K., Zhai, L.S., Tang, Y.: Nonlinear multi-scale dynamic stability of oil–gas–water three-phase flow in vertical upward pipe. Chem. Eng. J. 302, 595–608 (2016)

    Article  CAS  Google Scholar 

  23. Tang, Y., Zhao, A., Ren, Y.Y., Dou, F.X., Jin, N.D.: Gas–liquid two-phase flow structure in the multi-scale weighted complexity entropy causality plane. Physica A. 449, 324–335 (2016)

    Article  CAS  Google Scholar 

  24. Gao, Z.K., Yang, Y.X., Zhai, L.S., Ding, M.S., Jin, N.D.: Characterizing slug to churn flow transition by using multivariate pseudo Wigner distribution and multivariate multiscale entropy. Chem. Eng. J. 291, 74–81 (2016)

    Article  CAS  Google Scholar 

  25. Xia, J., Shang, P., Wang, J., Shi, W.: Classifying of financial time series based on multiscale entropy and multiscale time irreversibility. Physica A. 400(15), 151–158 (2014)

    Article  Google Scholar 

  26. Xu, K., Wang, J.: Nonlinear multiscale coupling analysis of financial time series based on composite complexity synchronization. Nonlinear Dyn. 86, 441–458 (2016)

    Article  Google Scholar 

  27. Lu, Y., Wang, J.: Nonlinear dynamical complexity of agent-based stochastic financial interacting epidemic system. Nonlinear Dyn. 86, 1823–1840 (2016)

    Article  Google Scholar 

  28. Hemakom, A., Chanwimalueang, T., Carrion, A., Aufegger, L., Constantinides, A.G., Mandic, D.P.: Financial stress through complexity science. IEEE J. Sel. Topics Signal Process. 10(6), 1112–1126 (2016)

    Article  Google Scholar 

  29. Fan, X., Li, S., Tian, L.: Complexity of carbon market from multiscale entropy analysis. Physica A. 452, 79–85 (2016)

    Article  Google Scholar 

  30. Wang, J., Shang, P., Zhao, X., Xia, J.: Multiscale entropy analysis of traffic time series. Int. J. Mod. Phys. C. 24, 1350006 (2013)

    Article  Google Scholar 

  31. Yin, Y., Shang, P.: Multivariate multiscale sample entropy of traffic time series. Nonlinear Dyn. 86, 479–488 (2016)

    Article  Google Scholar 

  32. Guzman-Vargas, L., Ramirez-Rojas, A., Angulo-Brown, F.: Multiscale entropy analysis of electroseismic time series. Nat. Hazards Earth Syst. Sci. 8, 855–860 (2008)

    Article  Google Scholar 

  33. Zeng, M., Zhang, S., Wang, E., Meng, Q.: Multiscale entropy analysis of the 3D near-surface wind field. World Congress on Intelligent Control and Automation, pp. 2797–2801, IEEE, Piscataway, NJ (2016)

    Google Scholar 

  34. Gopinath, S., Prince, P.R.: Multiscale and cross entropy analysis of auroral and polar cap indices during geomagnetic storms. Adv. Space Res. 57, 289–301 (2016)

    Article  Google Scholar 

  35. Hu, M., Liang, H.: Adaptive multiscale entropy analysis of multivariate neural data. IEEE Trans. Biomed. Eng. 59(1), 12–15 (2012)

    Article  PubMed  Google Scholar 

  36. Chen, W., Wang, Z., Xie, H., Yu, W.: Characterization of surface EMG signal based on fuzzy entropy. IEEE Trans. Neural Syst. Rehabil Eng. 15(2), 266–272 (2007)

    Article  PubMed  Google Scholar 

  37. Amoud, H., Snoussi, H., Hewson, D., Doussot, M., Duchece, J.: Intrinsic mode entropy for nonlinear discriminant analysis. IEEE Signal Process.Lett. 14(5), 297–300 (2007)

    Article  Google Scholar 

  38. Valencia, J.F., Porta, A., Vallverdu, M., Claria, F., Baranowski, R., Orlowska-Baranowska, E., Caminal, P.: Refined multiscale entropy: application to 24-h Holter recordings of heart period variability in healthy and aortic stenosis subjects. IEEE Trans. Biomed. Eng. 56, 2202–2213 (2009)

    Article  PubMed  Google Scholar 

  39. Wu, S.D., Wu, C.W., Lin, S.G., Wang, C.C., Lee, K.Y.: Time series analysis using composite multiscale entropy. Entropy. 15, 1069–1084 (2013)

    Article  Google Scholar 

  40. Wu, S.D., Wu, C.W., Lin, S.G., Lee, K.Y., Peng, C.K.: Analysis of complex time series using refined composite multiscale entropy. Phys. Rev. A. 378, 1369–1374 (2014)

    CAS  Google Scholar 

  41. Wang, J., Shang, P., Xia, J., Shi, W.: EMD based refined composite multiscale entropy analysis of complex signals. Physica A. 421, 583–593 (2015)

    Article  Google Scholar 

  42. Chang, Y.C., Wu, H.T., Chen, H.R., Liu, A.B., Yeh, J.J., Lo, M.T., Tsao, J.H., Tang, C.J., Tsai, I.T., Sun, C.K.: Application of a modified entropy computational method in assessing the complexity of pulse wave velocity signals in healthy and diabetic subjects. Entropy. 16, 4032–4043 (2014)

    Article  Google Scholar 

  43. Wu, S.D., Wu, C.W., Lee, K.Y., Lin, S.G.: Modified multiscale entropy for short-term time series analysis. Physica A. 392, 5865–5873 (2013)

    Article  Google Scholar 

  44. Costa, M.D., Goldberger, A.L.: Generalized multiscale entropy analysis: Application to quantifying the complex volatility of human heartbeat time series. Entropy. 17, 1197–1203 (2015)

    Article  PubMed  PubMed Central  Google Scholar 

  45. Huang, N.E., Wu, M.L., Long, S.R., Shen, S.S., Qu, W.D., Gloersen, P., Fan, K.L.: A Confidence Limit for the Empirical Mode Decomposition and Hilbert Spectral Analysis. Proc. R. Soc. A. 459(2037), 2317–2345 (2003)

    Article  Google Scholar 

  46. Hu, M., Liang, H.: Intrinsic mode entropy based on multivariate empirical mode decomposition and its application to neural data analysis. Cogn. Neurodyn. 5(3), 277–284 (2011)

    Article  PubMed  PubMed Central  Google Scholar 

  47. Wu, Z., Huang, N.E.: Ensemble empirical mode decomposition: a noise-assisted data analysis method. Adv. Adapt. Data Anal. 1(1), 1–41 (2009)

    Article  Google Scholar 

  48. Rehman, N., Mandic, D.P.: Multivariate Empirical Mode Decomposition. Proc. R. Soc. A. 466, 1291–1302 (2010)

    Article  Google Scholar 

  49. Hu, M., Liang, H.: Perceptual suppression revealed by adaptive multi-scale entropy analysis of local field potential in monkey visual cortex. Int. J. Neural Syst. 23(2), 1350005 (2013)

    Article  PubMed  Google Scholar 

  50. Manor, B., Lipsitz, L.A., Wayne, P.M., Peng, C.K., Li, L.: Complexity-based measures inform tai chi’s impact on standing postural control in older adults with peripheral neuropathy. BMC Complement Altern. Med. 13, 87 (2013)

    Article  PubMed  PubMed Central  Google Scholar 

  51. Wayne, P.M., Gow, B.J., Costa, M.D., Peng, C.K., Lipsitz, L.A., Hausdorff, J.M., Davis, R.B., Walsh, J.N., Lough, M., Novak, V., Yeh, G.Y., Ahn, A.C., Macklin, E.A., Manor, B.: Complexity-based measures inform effects of tai chi training on standing postural control: cross-sectional and randomized trial studies. PLoS One. 9(12), e114731 (2014)

    Article  PubMed  PubMed Central  Google Scholar 

  52. Zhou, D., Zhou, J., Chen, H., Manor, B., Lin, J., Zhang, J.: Effects of transcranial direct current stimulation (tDCS) on multiscale complexity of dual-task postural control in older adults. Exp. Brain Res. 233(8), 2401–2409 (2015)

    Article  PubMed  PubMed Central  Google Scholar 

  53. Jiang, Y., Peng, C.K., Xu, Y.: Hierarchical entropy analysis for biological signals. J. Comput. Appl. Math. 236, 728–742 (2011)

    Article  Google Scholar 

  54. Bandt, C., Pompe, B.: Permutation entropy—a natural complexity measure for time series. Phys. Rev. Lett. 88(17), 174102 (2002)

    Article  PubMed  Google Scholar 

  55. Wu, S.D., Wu, P.H., Wu, C.W., Ding, J.J., Wang, C.C.: Bearing fault diagnosis based on multiscale permutation entropy and support vector machine. Entropy. 14, 1343–1356 (2012)

    Article  Google Scholar 

  56. Lo, M.T., Chang, Y.C., Lin, C., Young, H.W., Lin, Y.H., Ho, Y.L., Peng, C.K., Hu, K.: Outlier-resilient complexity analysis of heartbeat dynamics. Sci. Rep. 6(5), 8836 (2015)

    Article  Google Scholar 

  57. Humeau-Heurtier, A., Baumert, M., Mahé, G., Abraham, P.: Multiscale compression entropy of microvascular blood flow signals: comparison of results from laser speckle contrast and laser Doppler flowmetry data in healthy subjects. Entropy. 16, 5777–5795 (2014)

    Article  Google Scholar 

  58. Baumert, M., Baier, V., Haueisen, J., Wessel, N., Meyerfeldt, U., Schirdewan, A., Voss, A.: Forecasting of life threatening arrhythmias using the compression entropy of heart rate. Methods Inf. Med. 43(2), 202–206 (2004)

    CAS  PubMed  Google Scholar 

  59. Zadeh, L.A.: Fuzzy sets. Inf. Control. 8, 338–353 (1965)

    Article  Google Scholar 

  60. Chen, W., Zhuang, J., Yu, W., Wang, Z.: Measuring complexity using FuzzyEn, ApEn, and SampEn. Med. Eng. Phys. 31, 61–68 (2009)

    Article  PubMed  Google Scholar 

  61. Xie, H., He, W., Liu, H.: Measuring time series regularity using nonlinear similarity-based sample entropy. Phys. Lett. A. 372, 7140–7146 (2008)

    Article  CAS  Google Scholar 

  62. Xie, H., Zheng, Y., Guo, J., Chen, X.: Cross-fuzzy entropy: A new method to test pattern synchrony of bivariate time series. Inf. Sci. 180, 1715–1724 (2010)

    Article  Google Scholar 

  63. Zhang, L., Xiong, G., Liu, H., Zou, H., Guo, W.: Applying improved multi-scale entropy and support vector machines for bearing health condition identification. Proc. Inst. Mech. Eng. Part C. 224, 1315–1325 (2010)

    Article  Google Scholar 

  64. Xiong, G.L., Zhang, L., Liu, H.S., Zou, H.J., Guo, W.Z.: A comparative study on ApEn, SampEn and their fuzzy counterparts in a multiscale framework for feature extraction. J. Zhejiang Univ. Sci. A. 11, 270–279 (2010)

    Article  Google Scholar 

  65. Ahmed, M.U., Mandic, D.P.: Multivariate multiscale entropy: a tool for complexity analysis of multichannel data. Phys. Rev. E. 84, 061918 (2011)

    Article  Google Scholar 

  66. Ahmed, M.U., Mandic, D.P.: Multivariate multiscale entropy analysis. IEEE Signal Processing Letters. 19, 91–94 (2012)

    Article  Google Scholar 

  67. Poczos, B., Kirshner, S., Szepesvari, C.: REGO: rank-based Estimation of Renyi Information Using Euclidean Graph Optimization. Proceedings of the 13th International Conference on AI and Statistics, JMLR Workshop and Conference Proceedings, vol. 9, pp. 605–612, MIT Press, Cambridge, MA (2010)

    Google Scholar 

  68. Sklar, A.: Random variables, joint distributions, and copulas. Kybernetica. 9, 449–460 (1973)

    Google Scholar 

  69. Nelsen, R.B.: An introduction to copulas. Springer, Berlin (2006)

    Google Scholar 

  70. Asai, M., McAleer, M., Yu, J.: Multivariate stochastic volatility: a review. Econ. Rev. 25, 145–175 (2006)

    Article  Google Scholar 

  71. Aas, K., Czado, C., Frigessi, A., Bakken, H.: Pair-copula constructions of multiple dependence. Insur. Math. Econ. 44, 182–198 (2009)

    Article  Google Scholar 

  72. Hu, M., Liang, H.: A copula approach to assessing Granger causality. Neuro. Image. 100, 125–124 (2014)

    PubMed  Google Scholar 

  73. Elidan, G.: Copula Bayesian networks. Adv. Neural Inf. Proces. Syst. 23, 559–567 (2010)

    Google Scholar 

  74. Hu, M., Clark, K., Gong, X., Noudoost, B., Li, M., Moore, T., Liang, H.: Copula regression analysis of simultaneously recorded frontal eye field and inferotemporal spiking activity during object-based working memory. J. Neurosci. 35, 8745–8757 (2015)

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  75. Schreiber, T.: Measuring Information Transfer. Phys. Rev. Lett. 85, 461–464 (2000)

    Article  CAS  PubMed  Google Scholar 

  76. Lungarella, M., Pitti, A., Kuniyoshi, Y.: Information transfer at multiple scales. Phys. Rev. E. 76, 056117 (2007)

    Article  Google Scholar 

  77. Costa, M.D., Peng, C.K., Goldberger, A.L.: Multiscale analysis of heart rate dynamics: entropy and time irreversibility measures. Cardiovasc. Eng. 8(2), 88–93 (2008)

    Article  PubMed  PubMed Central  Google Scholar 

  78. Nardelli, M., Valenza, G., Cristea, I.A., Gentili, C., Cotet, C., David, D., Lanata, A., Scilingo, E.P.: Characterization of behavioral activation in non-pathological subjects through heart rate variability monovariate and multivariate multiscale entropy analysis. The 8th conference of the European study group on cardiovascular oscillations, pp. 135–136, IEEE, Piscataway, NJ (2014)

    Google Scholar 

  79. Cornforth, D., Herbert, F.J., Tarvainen, M.: A comparison of nonlinear measures for the detection of cardiac autonomic neuropathy from heart rate variability. Entropy. 17, 1425–1440 (2015)

    Article  Google Scholar 

  80. Pan, W.Y., Su, M.C., Wu, H.T., Lin, M.C., Tsai, I.T., Sun, C.K.: Multiscale entropy analysis of heart rate variability for assessing the severity of sleep disordered breathing. Entropy. 17, 231–243 (2015)

    Article  Google Scholar 

  81. Bari, V., Marchi, A., Maria, B.D., Girardengo, G., George Jr., A.L., Brink, P.A., Cerutti, S., Crotti, L., Schwartz, P.J., Porta, A.: Low-pass filtering approach via empirical mode decomposition improves short-scale entropy-based complexity estimation of QT interval variability in long QT syndrome type 1 patients. Entropy. 16, 4839–4854 (2014)

    Article  Google Scholar 

  82. Valenza, G., Nardelli, M., Bertschy, G., Lanata, A., Scilingo, E.P.: Mood states modulate complexity in heartbeat dynamics: a multiscale entropy analysis. Europhys. Lett. 107(1), 18003 (2014)

    Article  Google Scholar 

  83. Valenza, G., Citi, L., Scilingo, E., Barbieri, R.: Inhomogeneous point-process entropy: An instantaneous measure of complexity in discrete systems. Phys. Rev. E. 89, 052803 (2014)

    Article  Google Scholar 

  84. Valenza, G., Citi, L., Scilingo, E., Barbieri, R.: Point-process nonlinear models with Laguerre and Volterra expansions: instantaneous assessment of heartbeat dynamics. IEEE Trans. Signal Process. 61, 2914 (2013)

    Article  Google Scholar 

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Hu, M., Liang, H. (2017). Multiscale Entropy: Recent Advances. In: Barbieri, R., Scilingo, E., Valenza, G. (eds) Complexity and Nonlinearity in Cardiovascular Signals. Springer, Cham. https://doi.org/10.1007/978-3-319-58709-7_4

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