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Dynamical complexity of short and noisy time series

Compression-Complexity vs. Shannon entropy

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Abstract

Shannon entropy has been extensively used for characterizing complexity of time series arising from chaotic dynamical systems and stochastic processes such as Markov chains. However, for short and noisy time series, Shannon entropy performs poorly. Complexity measures which are based on lossless compression algorithms are a good substitute in such scenarios. We evaluate the performance of two such Compression-Complexity Measures namely Lempel-Ziv complexity (LZ) and Effort-To-Compress (ETC) on short time series from chaotic dynamical systems in the presence of noise. Both LZ and ETC outperform Shannon entropy (H) in accurately characterizing the dynamical complexity of such systems. For very short binary sequences (which arise in neuroscience applications), ETC has higher number of distinct complexity values than LZ and H, thus enabling a finer resolution. For two-state ergodic Markov chains, we empirically show that ETC converges to a steady state value faster than LZ. Compression-Complexity measures are promising for applications which involve short and noisy time series.

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References

  1. C.E. Shannon, Bell Syst. Tech. J. 27, 379 (1948)

    Article  Google Scholar 

  2. M. Borowska, Studies in Logic, Grammar and Rhetoric 43, 21 (2015)

    Google Scholar 

  3. A. Porta, S. Guzzetti, N. Montano, R. Furlan, M. Pagani, A. Malliani, S. Cerutti, IEEE Trans. Biomed. Eng. 48, 1282 (2001)

    Article  Google Scholar 

  4. A. Li, Y. Pan, IEEE Trans. Inf. Theory 62, 3290 (2016)

    Article  Google Scholar 

  5. Z. Bar-Yossef, T.S. Jayram, R. Kumar, D. Sivakumar, in Proceedings of the 43rd Annual IEEE Symposium on Foundations of Computer Science, 2002 (IEEE, 2002), p. 209

  6. S. Vinga, Briefings in bioinformatics, bbt068 (2013)

  7. B.J. Strait, T.G. Dewey, Biophys. J. 71, 148 (1996)

    Article  ADS  Google Scholar 

  8. A. Golan, E. Maasoumi, Econometric Rev. 27, 317 (2008)

    Article  MathSciNet  Google Scholar 

  9. R. Zhou, R. Cai, G. Tong, Entropy 15, 4909 (2013)

    Article  ADS  MathSciNet  Google Scholar 

  10. P. Fiedor, Phys. Rev. E 89, 052801 (2014)

    Article  ADS  Google Scholar 

  11. C. Tsallis, Chaos, Solitons & Fractals 13, 371 (2002)

    Article  ADS  MathSciNet  Google Scholar 

  12. W. Gersch, D.M. Eddy, E. Dong Jr, Comp. Biomed. Res. 3, 385 (1970)

    Article  Google Scholar 

  13. D. Coast, R.M. Stern, G.G. Cano, S. Briller, et al., IEEE Trans. Biomed. Eng. 37, 826 (1990)

    Article  Google Scholar 

  14. W. Gersch, P. Lilly, E. Dong, Comp. Biomed. Res. 8, 370 (1975)

    Article  Google Scholar 

  15. S.-T. Pan, Y.-H. Wu, Y.-L. Kung, H.-C. Chen, in Proceedings of the 14th ACIS International Conference on Software Engineering, Artificial Intelligence, Networking and Parallel/Distributed Computing (SNPD), p. 586 (2013)

  16. M.S. Waterman, Mathematical Methods for DNA Sequences (CRC Press Inc., 1989)

  17. T.-J. Wu, Y.-C. Hsieh, L.-A. Li, Biometrics 57, 441 (2001)

    Article  MathSciNet  Google Scholar 

  18. I. Sergienko, A. Gupal, A. Ostrovsky, Cybernetics Syst. Anal. 48, 369 (2012)

    Article  Google Scholar 

  19. L. Narlikar, N. Mehta, S. Galande, M. Arjunwadkar, Nucl. Acids Res. 41, 1416 (2013)

    Article  Google Scholar 

  20. A. Varga, R. Moore, in Proceedings of the International Conference on Acoustics, Speech and Signal (ICASSP), p. 845 (1990)

  21. B.H. Juang, L.R. Rabiner, Technometrics 33, 251 (1991)

    Article  MathSciNet  Google Scholar 

  22. H. Veisi, H. Sameti, Speech Commun. 55, 205 (2013)

    Article  Google Scholar 

  23. R.P. Rao, N. Yadav, M.N. Vahia, H. Joglekar, R. Adhikari, I. Mahadevan, Proc. Nat. Acad. Sci. 106, 13685 (2009)

    Article  ADS  Google Scholar 

  24. R.P. Rao, IEEE Comp. 43, 76 (2010)

    Article  Google Scholar 

  25. G.A. Fink, Markov Models for Pattern Recognition: From Theory to Applications (Springer Science & Business Media, 2014)

  26. G.V. Cormack, R. Horspool, Comp. J. 30, 541 (1987)

    Article  Google Scholar 

  27. H.S. Wang, N. Moayeri, IEEE Trans. Veh. Technol. 44, 163 (1995)

    Article  Google Scholar 

  28. H. Zhou, J. Bruck, IEEE Trans. Inf. Theory 58, 2490 (2012)

    Article  Google Scholar 

  29. M. Svoboda, L. Lukas, in Proceedings of 30th International Conference Mathematical Methods in Economics (Silesian University, School of Business Administration, Karviná, 2012), p. 848

  30. F.O. Mettle, E.N.B. Quaye, R.A. Laryea, SpringerPlus 3, 1 (2014)

    Article  Google Scholar 

  31. K.T. Alligood, T.D. Sauer, J.A. Yorke, Chaos (Springer, 1997)

  32. R. Cilibrasi, Statistical inference through data compression, Ph.D. Thesis, University of Amsterdam, 2007

  33. A. Lempel, J. Ziv, IEEE Trans. Inf. Theory 22, 75 (1976)

    Article  Google Scholar 

  34. J. Ziv, A. Lempel, IEEE Trans. Inf. Theory 23, 337 (1977)

    Article  Google Scholar 

  35. M. Aboy, R. Hornero, D. Abásolo, D. Álvarez, IEEE Trans. Biomed. Eng. 53, 2282 (2006)

    Article  Google Scholar 

  36. J. Hu, J. Gao, J.C. Principe, IEEE Trans. Biomed. Eng. 53, 2606 (2006)

    Article  Google Scholar 

  37. J.M. Amigó, J. Szczepański, E. Wajnryb, M.V. Sanchez-Vives, Neural Comput. 16, 717 (2004)

    Article  Google Scholar 

  38. S. Zozor, P. Ravier, O. Buttelli, Physica A 345, 285 (2005)

    Article  ADS  Google Scholar 

  39. S. Shinkai, Y. Aizawa, Prog. Theory Phys. 116, 503 (2006)

    Article  ADS  Google Scholar 

  40. H.H. Otu, K. Sayood, Bioinformatics 19, 2122 (2003)

    Article  Google Scholar 

  41. V.D. Gusev, L.A. Nemytikova, N.A. Chuzhanova, Bioinformatics 15, 994 (1999)

    Article  Google Scholar 

  42. S. Azhar, G.J. Badros, A. Glodjo, M.-Y. Kao, J.H. Reif, in Proceedings of the Conference on Data Compression, 1994 (DCC’94) (IEEE, 1994), p. 72

  43. R. Giglio, R. Matsushita, S. Da Silva, Econ. Bull. 7, 1 (2008)

    Google Scholar 

  44. N. Nagaraj, K. Balasubramanian, S. Dey, Eur. Phys. J. Special Topics 222, 847 (2013)

    Article  ADS  Google Scholar 

  45. W. Ebeling, M.A. Jiménez-Montaño, Math. Biosci. 52, 53 (1980)

    Article  Google Scholar 

  46. K. Balasubramanian, N. Nagaraj, PeerJ 4, e2755 (2016)

    Article  Google Scholar 

  47. M. Talebinejad, G. Tsoulfas, S. Musallam, in Proc. Canadian Med. Biol. Engg. (2011)

  48. T.M. Cover, J.A. Thomas, Elements of Information Theory (John Wiley & Sons, 2012)

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Correspondence to Nithin Nagaraj or Karthi Balasubramanian.

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Nagaraj, N., Balasubramanian, K. Dynamical complexity of short and noisy time series. Eur. Phys. J. Spec. Top. 226, 2191–2204 (2017). https://doi.org/10.1140/epjst/e2016-60397-x

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  • DOI: https://doi.org/10.1140/epjst/e2016-60397-x

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