Abstract
While ordinal techniques are commonplace in statistics, they have been introduced to time series fairly recently by Hallin and coauthors. Permutation entropy, an average of frequencies of order patterns, was suggested by Bandt and Pompe in 2002 and used by many authors as a complexity measure in physics, medicine, engineering, and economy. Here a modified version is introduced, the “distance to white noise.” For datasets with tens of thousands or even millions of values, which are becoming standard in many fields, it is possible to study order patterns separately, determine certain differences of their frequencies, and define corresponding autocorrelation type functions. In contrast to classical autocorrelation, these functions are invariant with respect to nonlinear monotonic transformations of the data. For order three patterns, a variance-analytic “Pythagoras formula” combines the different autocorrelation functions with our new version of permutation entropy. We demonstrate the use of such correlation type functions in sliding window analysis of biomedical and environmental data.
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References
Amigo, J.: Permutation Complexity in Dynamical Systems. Springer, Heidelberg (2010)
Amigo, J., Keller, K., Kurths, J. (eds.): Recent progress in symbolic dynamics and permutation entropy. Eur. Phys. J. Spec. Top. 222 (2013)
Aragoneses, A., Rubido, N., Tiana-Aisina, J., Torrent, M.C., Masoller, C.: Distinguishing signatures of determinism and stochasticity in spiking complex systems. Sci. Rep. 3, Article 1778 (2012)
Bandt, C.: Autocorrelation type functions for big and dirty data series (2014). http://arxiv.org/abs/1411.3904
Bandt, C., Pompe, B.: Permutation entropy: a natural complexity measure for time series. Phys. Rev. Lett. 88, 174102 (2002)
Bandt, C., Shiha, F.: Order patterns in time series. J. Time Ser. Anal. 28, 646–665 (2007)
California Environmental Protection Agency, Air Resources Board: www.arb.ca.gov/aqd/aqdcd/aqdcddld.htm (2014)
Ferguson, T.S., Genest, C., Hallin, M.: Kendall’s tau for serial dependence. Can. J. Stat. 28, 587–604 (2000)
Ferlazzo, E. et al.: Permutation entropy of scalp EEG: a tool to investigate epilepsies. Clin. Neurophysiol. 125, 13–20 (2014)
German Weather Service: www.dwd.de, Climate and Environment, Climate Data (2014)
Goldberger, A.L. et al.: PhysioBank, PhysioToolkit, and PhysioNet: components of a new research resource for complex physiologic signals. Circulation 101(23), e215–e220 (2000). Data at: http://www.physionet.org/physiobank/database/capslpdb (2014)
Hallin, M., Puri, M.L.: Aligned rank tests for linear models with autocorrelated error terms. J. Multivar. Anal. 50, 175–237 (1994)
Kuo, C.-E., Liang, S.-F.: Automatic stage scoring of single-channel sleep EEG based on multiscale permutation entropy. In: 2011 IEEE Biomedical Circuits and Systems Conference (BioCAS), pp. 448–451 (2011)
Lange, H., Rosso, O.A., Hauhs, M.: Ordinal pattern and statistical complexity analysis of daily stream flow time series. Eur. Phys. J. Spec. Top. 222, 535–552 (2013)
Nair, U., Krishna, B.M., Namboothiri, V.N.N., Nampoori, V.P.N.: Permutation entropy based real-time chatter detection using audio signal in turning process. Int. J. Adv. Manuf. Technol. 46, 61–68 (2010)
Nicolaou, N., Georgiou, J.: The use of permutation entropy to characterize sleep encephalograms. Clin. EEG Neurosci. 42, 24 (2011)
Ouyang, G., Dang, C., Richards, D.A., Li, X.: Ordinal pattern based similarity analysis for EEG recordings. Clin. Neurophysiol. 121, 694–703 (2010)
Soriano, M.C., Zunino, L., Rosso, O.A., Fischer, I., Mirasso, C.R.: Time scales of a chaotic semiconductor laser with optical feedback under the lens of a permutation information analysis. IEEE J. Quantum Electron. 47(2), 252–261 (2011)
Staniek, M., Lehnertz, K.: Symbolic transfer entropy. Phys. Rev. Lett. 100, 158101 (2008)
Terzano, M.G., et al.: Atlas, rules, and recording techniques for the scoring of cyclic alternating pattern (CAP) in human sleep. Sleep Med. 2(6), 537–553 (2001)
The National Water Level Observation Network: www.tidesandcurrents.noaa.gov/nwlon.html (2014)
Toomey, J.P., Kane, D.M.: Mapping the dynamical complexity of a semiconductor laser with optical feedback using permutation entropy. Opt. Express 22(2), 1713–1725 (2014)
Unakafov, A.M., Keller, K.: Conditional entropy of ordinal patterns. Physica D 269, 94–102 (2014)
Yan, R., Liu, Y., Gao, R.X.: Permutation entropy: a nonlinear statistical measure for status characterization of rotary machines. Mech. Syst. Signal Process. 29, 474–484 (2012)
Zanin, M., Zunino, L., Rosso, O.A., Papo, D.: Permutation entropy and its main biomedical and econophysics applications: a review. Entropy 14, 1553–1577 (2012)
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Bandt, C. (2016). Permutation Entropy and Order Patterns in Long Time Series. In: Rojas, I., Pomares, H. (eds) Time Series Analysis and Forecasting. Contributions to Statistics. Springer, Cham. https://doi.org/10.1007/978-3-319-28725-6_5
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DOI: https://doi.org/10.1007/978-3-319-28725-6_5
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