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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Costa, M.D., Goldberger, A.L., Peng, C.K.: Multiscale entropy analysis of physiologic time series. Phys. Rev. Lett. 89, 0621021–4 (2002)
Shannon, C.E.: A Mathematical Theory of Communication. Bell Syst. Tech. J. 27(3), 379–423 (1948)
Grassberger, P., Procaccia, I.: Estimation of the Kolmogorov entropy from a chaotic signal. Phys. Rev. A. 28, 2591–2593 (1983)
Eckmann, J.P., Ruelle, D.: Ergodic theory of chaos and strange attractors. Rev. Mod. Phys. 57, 617–656 (1985)
Pincus, S.M.: Approximate entropy as a measure of system complexity. Proc. Natl. Acad. Sci. 88, 2297–2301 (1991)
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)
Costa, M.D., Goldberger, A.L., Peng, C.K.: Multiscale entropy analysis of biological signals. Phys. Rev. E. 71, 021906 (2005)
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)
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)
Liu, T., Yao, W., Wu, M., Shi, Z., Wang, J., Ning, X.: Multiscale permutation entropy analysis of electrocardiogram. Physica A. 471, 492–498 (2017)
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.
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)
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)
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)
Costa, M., Peng, C.K., Goldberger, A.L., Hausdorff, J.M.: Multiscale entropy analysis of human gait dynamics. Physica A. 330, 53–60 (2003)
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)
Rizal, A., Hidayat, R., Nugroho, H.A.: Multiscale Hjorth descriptor for lung sound classification. International Conference on Science and Technology, 160008–1 (2015)
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)
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)
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)
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)
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)
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)
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)
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)
Xu, K., Wang, J.: Nonlinear multiscale coupling analysis of financial time series based on composite complexity synchronization. Nonlinear Dyn. 86, 441–458 (2016)
Lu, Y., Wang, J.: Nonlinear dynamical complexity of agent-based stochastic financial interacting epidemic system. Nonlinear Dyn. 86, 1823–1840 (2016)
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)
Fan, X., Li, S., Tian, L.: Complexity of carbon market from multiscale entropy analysis. Physica A. 452, 79–85 (2016)
Wang, J., Shang, P., Zhao, X., Xia, J.: Multiscale entropy analysis of traffic time series. Int. J. Mod. Phys. C. 24, 1350006 (2013)
Yin, Y., Shang, P.: Multivariate multiscale sample entropy of traffic time series. Nonlinear Dyn. 86, 479–488 (2016)
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)
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)
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)
Hu, M., Liang, H.: Adaptive multiscale entropy analysis of multivariate neural data. IEEE Trans. Biomed. Eng. 59(1), 12–15 (2012)
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)
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)
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)
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)
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)
Wang, J., Shang, P., Xia, J., Shi, W.: EMD based refined composite multiscale entropy analysis of complex signals. Physica A. 421, 583–593 (2015)
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)
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)
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)
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)
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)
Wu, Z., Huang, N.E.: Ensemble empirical mode decomposition: a noise-assisted data analysis method. Adv. Adapt. Data Anal. 1(1), 1–41 (2009)
Rehman, N., Mandic, D.P.: Multivariate Empirical Mode Decomposition. Proc. R. Soc. A. 466, 1291–1302 (2010)
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)
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)
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)
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)
Jiang, Y., Peng, C.K., Xu, Y.: Hierarchical entropy analysis for biological signals. J. Comput. Appl. Math. 236, 728–742 (2011)
Bandt, C., Pompe, B.: Permutation entropy—a natural complexity measure for time series. Phys. Rev. Lett. 88(17), 174102 (2002)
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)
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)
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)
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)
Zadeh, L.A.: Fuzzy sets. Inf. Control. 8, 338–353 (1965)
Chen, W., Zhuang, J., Yu, W., Wang, Z.: Measuring complexity using FuzzyEn, ApEn, and SampEn. Med. Eng. Phys. 31, 61–68 (2009)
Xie, H., He, W., Liu, H.: Measuring time series regularity using nonlinear similarity-based sample entropy. Phys. Lett. A. 372, 7140–7146 (2008)
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)
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)
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)
Ahmed, M.U., Mandic, D.P.: Multivariate multiscale entropy: a tool for complexity analysis of multichannel data. Phys. Rev. E. 84, 061918 (2011)
Ahmed, M.U., Mandic, D.P.: Multivariate multiscale entropy analysis. IEEE Signal Processing Letters. 19, 91–94 (2012)
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)
Sklar, A.: Random variables, joint distributions, and copulas. Kybernetica. 9, 449–460 (1973)
Nelsen, R.B.: An introduction to copulas. Springer, Berlin (2006)
Asai, M., McAleer, M., Yu, J.: Multivariate stochastic volatility: a review. Econ. Rev. 25, 145–175 (2006)
Aas, K., Czado, C., Frigessi, A., Bakken, H.: Pair-copula constructions of multiple dependence. Insur. Math. Econ. 44, 182–198 (2009)
Hu, M., Liang, H.: A copula approach to assessing Granger causality. Neuro. Image. 100, 125–124 (2014)
Elidan, G.: Copula Bayesian networks. Adv. Neural Inf. Proces. Syst. 23, 559–567 (2010)
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)
Schreiber, T.: Measuring Information Transfer. Phys. Rev. Lett. 85, 461–464 (2000)
Lungarella, M., Pitti, A., Kuniyoshi, Y.: Information transfer at multiple scales. Phys. Rev. E. 76, 056117 (2007)
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)
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)
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)
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)
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)
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)
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)
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)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this chapter
Cite this chapter
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
Download citation
DOI: https://doi.org/10.1007/978-3-319-58709-7_4
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-58708-0
Online ISBN: 978-3-319-58709-7
eBook Packages: Biomedical and Life SciencesBiomedical and Life Sciences (R0)