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December  2015, 20(10): 3507-3524. doi: 10.3934/dcdsb.2015.20.3507

Entropy determination based on the ordinal structure of a dynamical system

Karsten Keller 1, , Sergiy Maksymenko 2, and  Inga Stolz 3,

1. 

Universität zu Lübeck, Institut für Mathematik, Ratzeburger Allee 160, 23562 Lübeck
2. 

Institute of Mathematics of NAS of Ukraine, Tereshchenkivs'ka str., 3, 01601 Kyiv
3. 

Universität zu Lübeck, Institut für Mathematik, Ratzeburger Allee 160, Lübeck, 23562, Germany

Received  December 2014 Revised  February 2015 Published  September 2015

The ordinal approach to evaluate time series due to innovative works of Bandt and Pompe has increasingly established itself among other techniques of nonlinear time series analysis. In this paper, we summarize and generalize the theory of determining the Kolmogorov-Sinai entropy of a measure-preserving dynamical system via increasing sequences of order generated partitions of the state space. Our main focus are measuring processes without information loss. Particularly, we consider the question of the minimal necessary number of measurements related to the properties of a given dynamical system.
Keywords: Kolmogorov-Sinai entropy, algebra reconstruction dimension., permutation entropy, ordinal time series analysis.
Mathematics Subject Classification: Primary: 37A35, 28D05; Secondary: 37M1.
Citation: Karsten Keller, Sergiy Maksymenko, Inga Stolz. Entropy determination based on the ordinal structure of a dynamical system. Discrete and Continuous Dynamical Systems - B, 2015, 20 (10) : 3507-3524. doi: 10.3934/dcdsb.2015.20.3507
References:
[1]

J. M. Amigó, K. Keller and V. A. Unakafova, Ordinal symbolic analysis and its application to biomedical recordings, Philosophical Transactions Royal Society A, 373 (2015), 20140091, 18pp.

[2]

J. M. Amigó, K. Keller and J. Kurths (eds.), Recent progess in symbolic dynamics and permutation complexity. Ten years of permutation entropy, The European Physical Journal Special Topics, 222 (2013), 241-598.

[3]

J. M. Amigó, The equality of Kolmogorov-Sinai entropy and metric permutation entropy generalized, Physica D: Nonlinear Phenomena, 241 (2012), 789-793. doi: 10.1016/j.physd.2012.01.004.

[4]

J. M. Amigó, Permutation Complexity in Dynamical Systems: Ordinal Patterns, Permutation Entropy and All That, Springer Series in Synergetics, Springer Complexity, Springer-Verlag, Berlin Heidelberg, 2010. doi: 10.1007/978-3-642-04084-9.

[5]

J. M. Amigó, M. B. Kennel and L. Kocarev, The permutation entropy rate equals the metric entropy rate for ergodic information sources and ergodic dynamical systems, Physica D: Nonlinear Phenomena, 210 (2005), 77-95. doi: 10.1016/j.physd.2005.07.006.

[6]

A. Antoniouk, K. Keller and S. Maksymenko, Kolmogorov-Sinai entropy via separation properties of order-generated sigma-algebras, Discrete and Continuous Dynamical Systems - A, 34 (2014), 1793-1809.

[7]

C. Bandt, G. Keller and B. Pompe, Entropy of interval maps via permutations, Nonlinearity, 15 (2002), 1595-1602. doi: 10.1088/0951-7715/15/5/312.

[8]

C. Bandt and B. Pompe, Permutation entropy: A natural complexity measure for time series, Physical Review Letters, 88 (2002), 174102.

[9]

M. Einsiedler, E. Lindenstrauss and T. Ward, Entropy in ergodic theory and homogeneous dynamics, unpublished, [access: July 02, 2014]. Available from: http://maths.dur.ac.uk/~tpcc68/entropy/welcome.html.

[10]

M. Einsiedler and T. Ward, Ergodic Theory With a View Towards Number Theory, Graduate Texts in Mathematics, Springer-Verlag, New York, 2010.

[11]

W. Hurewicz and H. Wallman, Dimension Theory, Princeton Mathematical Series, 4, Princeton University Press, Princeton, New Jersey, 1941.

[12]

A. S. Kechris, Classical Descriptive Set Theory, Graduate Texts in Mathematics, 156, Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4612-4190-4.

[13]

K. Keller, Permutations and the Kolmogorov-Sinai entropy, Discrete and Continuous Dynamical Systems - A, 32 (2012), 891-900. doi: 10.3934/dcds.2012.32.891.

[14]

K. Keller and M. Sinn, Kolmogorov-Sinai entropy from the ordinal viewpoint, Physica D: Nonlinear Phenomena, 239 (2010), 997-1000. doi: 10.1016/j.physd.2010.02.006.

[15]

K. Keller and M. Sinn, A standardized approach to the Kolmogorov-Sinai entropy, Nonlinearity, 22 (2009), 2417-2422. doi: 10.1088/0951-7715/22/10/006.

[16]

K. Keller, M. Sinn and J. Emonds, Time series from the ordinal viewpoint, Stochastics and Dynamics, 7 (2007), 247-272. doi: 10.1142/S0219493707002025.

[17]

K. Keller, A. M. Unakafov and V. A. Unakafova, On the relation of KS entropy and permutation entropy, Physica D: Nonlinear Phenomena, 241 (2012), 1477-1481. doi: 10.1016/j.physd.2012.05.010.

[18]

W. Krieger, On entropy and generators of measure-preserving transformations, Transactions of the American Mathematical Society, 149 (1970), 453-464. doi: 10.1090/S0002-9947-1970-0259068-3.

[19]

W. Parry, Generators and strong generators in ergodic theory, Bulletin of the American Mathematical Society, 72 (1966), 294-296. doi: 10.1090/S0002-9904-1966-11498-2.

[20]

M. Riedl, A. Müller and N. Wessel, Practical considerations of permutation entropy; a tutorial review, The European Physical Journal Special Topics, 222 (2013), 249-262.

[21]

V. A. Rohlin, Exact endomorphisms of a Lebesgue space, Izvestiya Akademii Nauk SSSR Seriya Matematicheskaya, 25 (1961), 499-530; English translation: American Mathematical Society Translations, 39 (1964), 1-36.

[22]

T. Sauer, J. A. Yorke and M. Casdagli, Embeddology, Journal of Statistical Physics, 65 (1991), 579-616. doi: 10.1007/BF01053745.

[23]

F. Takens, Detecting strange attractors in turbulence, in Dynamical Systems and Turbulence (eds. D. A. Rand and L. S. Young), Lecture Notes in Mathematics, 898, Springer-Verlag, Berlin New York, 1981, 366-381.

[24]

A. M. Unakafov and K. Keller, Conditional entropy of ordinal patterns, Physica D: Nonlinear Phenomena, 269 (2014), 94-102. doi: 10.1016/j.physd.2013.11.015.

[25]

V. A. Unakafova and K. Keller, Efficiently measuring complexity on the basis of real-world data, Entropy, 15 (2013), 4392-4415. doi: 10.3390/e15104392.

[26]

V. A. Unakafova, A. M. Unakafov and K. Keller, An approach to comparing Kolmogorov-Sinai and permutation entropy, The European Physical Journal Special Topics, 222 (2013), 353-361. doi: 10.1140/epjst/e2013-01846-7.

[27]

P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, 79, Springer-Verlag, New York, 2000. doi: 10.1007/978-1-4612-5775-2.

[28]

M. Zanin, L. Zunino, O. A. Rosso and D. Papo, Permutation entropy and its main biomedical and econophysics applications: A review, Entropy, 14 (2012), 1553-1577. doi: 10.3390/e14081553.

show all references

References:
[1]

J. M. Amigó, K. Keller and V. A. Unakafova, Ordinal symbolic analysis and its application to biomedical recordings, Philosophical Transactions Royal Society A, 373 (2015), 20140091, 18pp.

[2]

J. M. Amigó, K. Keller and J. Kurths (eds.), Recent progess in symbolic dynamics and permutation complexity. Ten years of permutation entropy, The European Physical Journal Special Topics, 222 (2013), 241-598.

[3]

J. M. Amigó, The equality of Kolmogorov-Sinai entropy and metric permutation entropy generalized, Physica D: Nonlinear Phenomena, 241 (2012), 789-793. doi: 10.1016/j.physd.2012.01.004.

[4]

J. M. Amigó, Permutation Complexity in Dynamical Systems: Ordinal Patterns, Permutation Entropy and All That, Springer Series in Synergetics, Springer Complexity, Springer-Verlag, Berlin Heidelberg, 2010. doi: 10.1007/978-3-642-04084-9.

[5]

J. M. Amigó, M. B. Kennel and L. Kocarev, The permutation entropy rate equals the metric entropy rate for ergodic information sources and ergodic dynamical systems, Physica D: Nonlinear Phenomena, 210 (2005), 77-95. doi: 10.1016/j.physd.2005.07.006.

[6]

A. Antoniouk, K. Keller and S. Maksymenko, Kolmogorov-Sinai entropy via separation properties of order-generated sigma-algebras, Discrete and Continuous Dynamical Systems - A, 34 (2014), 1793-1809.

[7]

C. Bandt, G. Keller and B. Pompe, Entropy of interval maps via permutations, Nonlinearity, 15 (2002), 1595-1602. doi: 10.1088/0951-7715/15/5/312.

[8]

C. Bandt and B. Pompe, Permutation entropy: A natural complexity measure for time series, Physical Review Letters, 88 (2002), 174102.

[9]

M. Einsiedler, E. Lindenstrauss and T. Ward, Entropy in ergodic theory and homogeneous dynamics, unpublished, [access: July 02, 2014]. Available from: http://maths.dur.ac.uk/~tpcc68/entropy/welcome.html.

[10]

M. Einsiedler and T. Ward, Ergodic Theory With a View Towards Number Theory, Graduate Texts in Mathematics, Springer-Verlag, New York, 2010.

[11]

W. Hurewicz and H. Wallman, Dimension Theory, Princeton Mathematical Series, 4, Princeton University Press, Princeton, New Jersey, 1941.

[12]

A. S. Kechris, Classical Descriptive Set Theory, Graduate Texts in Mathematics, 156, Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4612-4190-4.

[13]

K. Keller, Permutations and the Kolmogorov-Sinai entropy, Discrete and Continuous Dynamical Systems - A, 32 (2012), 891-900. doi: 10.3934/dcds.2012.32.891.

[14]

K. Keller and M. Sinn, Kolmogorov-Sinai entropy from the ordinal viewpoint, Physica D: Nonlinear Phenomena, 239 (2010), 997-1000. doi: 10.1016/j.physd.2010.02.006.

[15]

K. Keller and M. Sinn, A standardized approach to the Kolmogorov-Sinai entropy, Nonlinearity, 22 (2009), 2417-2422. doi: 10.1088/0951-7715/22/10/006.

[16]

K. Keller, M. Sinn and J. Emonds, Time series from the ordinal viewpoint, Stochastics and Dynamics, 7 (2007), 247-272. doi: 10.1142/S0219493707002025.

[17]

K. Keller, A. M. Unakafov and V. A. Unakafova, On the relation of KS entropy and permutation entropy, Physica D: Nonlinear Phenomena, 241 (2012), 1477-1481. doi: 10.1016/j.physd.2012.05.010.

[18]

W. Krieger, On entropy and generators of measure-preserving transformations, Transactions of the American Mathematical Society, 149 (1970), 453-464. doi: 10.1090/S0002-9947-1970-0259068-3.

[19]

W. Parry, Generators and strong generators in ergodic theory, Bulletin of the American Mathematical Society, 72 (1966), 294-296. doi: 10.1090/S0002-9904-1966-11498-2.

[20]

M. Riedl, A. Müller and N. Wessel, Practical considerations of permutation entropy; a tutorial review, The European Physical Journal Special Topics, 222 (2013), 249-262.

[21]

V. A. Rohlin, Exact endomorphisms of a Lebesgue space, Izvestiya Akademii Nauk SSSR Seriya Matematicheskaya, 25 (1961), 499-530; English translation: American Mathematical Society Translations, 39 (1964), 1-36.

[22]

T. Sauer, J. A. Yorke and M. Casdagli, Embeddology, Journal of Statistical Physics, 65 (1991), 579-616. doi: 10.1007/BF01053745.

[23]

F. Takens, Detecting strange attractors in turbulence, in Dynamical Systems and Turbulence (eds. D. A. Rand and L. S. Young), Lecture Notes in Mathematics, 898, Springer-Verlag, Berlin New York, 1981, 366-381.

[24]

A. M. Unakafov and K. Keller, Conditional entropy of ordinal patterns, Physica D: Nonlinear Phenomena, 269 (2014), 94-102. doi: 10.1016/j.physd.2013.11.015.

[25]

V. A. Unakafova and K. Keller, Efficiently measuring complexity on the basis of real-world data, Entropy, 15 (2013), 4392-4415. doi: 10.3390/e15104392.

[26]

V. A. Unakafova, A. M. Unakafov and K. Keller, An approach to comparing Kolmogorov-Sinai and permutation entropy, The European Physical Journal Special Topics, 222 (2013), 353-361. doi: 10.1140/epjst/e2013-01846-7.

[27]

P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, 79, Springer-Verlag, New York, 2000. doi: 10.1007/978-1-4612-5775-2.

[28]

M. Zanin, L. Zunino, O. A. Rosso and D. Papo, Permutation entropy and its main biomedical and econophysics applications: A review, Entropy, 14 (2012), 1553-1577. doi: 10.3390/e14081553.

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