December  2015, 20(10): 3301-3343. doi: 10.3934/dcdsb.2015.20.3301

On entropy, entropy-like quantities, and applications

1. 

Universidad Miguel Hernández, Centro de Investigación Operativa, Avda. Universidad s/n, Elche (Alicante), 03202, Spain

2. 

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

3. 

Graduate School for Computing in Medicine and Life Science, Universität zu Lübeck, Ratzeburger Allee 160, 23562 Lübeck, Germany

Received  January 2015 Revised  March 2015 Published  September 2015

This is a review on entropy in various fields of mathematics and science. Its scope is to convey a unified vision of the classical entropies and some newer, related notions to a broad audience with an intermediate background in dynamical systems and ergodic theory. Due to the breadth and depth of the subject, we have opted for a compact exposition whose contents are a compromise between conceptual import and instrumental relevance. The intended technical level and the space limitation born furthermore upon the final selection of the topics, which cover the three items named in the title. Specifically, the first part is devoted to the avatars of entropy in the traditional contexts: many particle physics, information theory, and dynamical systems. This chronological order helps present the materials in a didactic manner. The axiomatic approach will be also considered at this stage to show that, quite remarkably, the essence of entropy can be encapsulated in a few basic properties. Inspired by the classical entropies, further akin quantities have been proposed in the course of time, mostly aimed at specific needs. A common denominator of those addressed in the second part of this review is their major impact on research. The final part shows that, along with its profound role in the theory, entropy has interesting practical applications beyond information theory and communications technology. For this sake we preferred examples from applied mathematics, although there are certainly nice applications in, say, physics, computer science and even social sciences. This review concludes with a representative list of references.
Citation: José M. Amigó, Karsten Keller, Valentina A. Unakafova. On entropy, entropy-like quantities, and applications. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3301-3343. doi: 10.3934/dcdsb.2015.20.3301
References:
[1]

S. Abe, Tsallis entropy: How unique?,, Contin. Mech. Thermodyn., 16 (2004), 237. doi: 10.1007/s00161-003-0153-1. Google Scholar

[2]

U. R. Acharya, O. Faust, N. Kannathal, T. Chua and S. Laxminarayan, Non-linear analysis of EEG signals at various sleep stages,, Comput. Meth. Prog. Bio., 80 (2005), 37. doi: 10.1016/j.cmpb.2005.06.011. Google Scholar

[3]

U. R. Acharya, K. P. Joseph, N. Kannathal, C. M. Lim and J. S. Suri, Heart rate variability: A review,, Advances in Cardiac Signal Processing, (2007), 121. doi: 10.1007/978-3-540-36675-1_5. Google Scholar

[4]

J. Aczél and Z. Daróczy, Charakterisierung der Entropien positiver Ordnung und der Shannonschen Entropie,, Acta Math. Acad. Sci. Hung., 14 (1963), 95. doi: 10.1007/BF01901932. Google Scholar

[5]

R. Adler, A. Konheim and M. McAndrew, Topological entropy,, Trans. Amer. Mat. Soc., 114 (1965), 309. doi: 10.1090/S0002-9947-1965-0175106-9. Google Scholar

[6]

R. L. Adler and B. Marcus, Topological entropy and the equivalence of dynamical systems,, Mem. Amer. Math. Soc., 20 (1979). doi: 10.1090/memo/0219. Google Scholar

[7]

V. Afraimovich, M. Courbage and L. Glebsky, Directional complexity and entropy for lift mappings,, Discr. Contin. Dyn. Syst. B, 20 (2015), 3385. Google Scholar

[8]

L. Alsedà, J. Llibre and M. Misiurewicz, Combinatorial Dynamics and Entropy in Dimension One,, World Scientific, (2000). doi: 10.1142/4205. Google Scholar

[9]

J. M. Amigó, L. Kocarev and J. Szczepanski, Order patterns and chaos,, Phys. Lett. A, 355 (2006), 27. Google Scholar

[10]

J. M. Amigó, S. Zambrano and M. A. F. Sanjuán, True and false forbidden patterns in deterministic and random dynamics,, Eur. Phys. Lett., 79 (2007). doi: 10.1209/0295-5075/79/50001. Google Scholar

[11]

J. M. Amigó and M. B. Kennel, Forbidden ordinal patterns in higher dimensional dynamics,, Physica D, 237 (2008), 2893. doi: 10.1016/j.physd.2008.05.003. Google Scholar

[12]

J. M. Amigó, S. Zambrano and M. A. F. Sanjuán, Combinatorial detection of determinism in noisy time series,, Europhys. Lett., 82 (2008). Google Scholar

[13]

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

[14]

J. M. Amigó, S. Zambrano and M. A. F. Sanjuán, Detecting determinism in time series with ordinal patterns: A comparative study,, Int. J. Bif. Chaos, 20 (2010), 2915. doi: 10.1142/S0218127410027453. Google Scholar

[15]

J. M. Amigó, R. Monetti, T. Aschenbrenner and W. Bunk, Transcripts: An algebraic approach to coupled time series,, Chaos, 22 (2012). doi: 10.1063/1.3673238. Google Scholar

[16]

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

[17]

J. M. Amigó and K. Keller, Permutation entropy: One concept, two approaches,, Eur. Phys. J. Special Topics, 222 (2013), 263. Google Scholar

[18]

J. M. Amigó, P. Kloeden and A. Giménez, Switching systems and entropy,, J. Diff. Eq. Appl., 19 (2013), 1872. doi: 10.1080/10236198.2013.788166. Google Scholar

[19]

J. M. Amigó, P. E. Kloeden and A. Giménez, Entropy increase in switching systems,, Entropy, 15 (2013), 2363. doi: 10.3390/e15062363. Google Scholar

[20]

J. M. Amigó, T. Aschenbrenner, W. Bunk and R. Monetti, Dimensional reduction of conditional algebraic multi-information via transcripts,, Inform. Sciences, 278 (2014), 298. doi: 10.1016/j.ins.2014.03.054. Google Scholar

[21]

J. M. Amigó and A. Giménez, A simplified algorithm for the topological entropy of multimodal maps,, Entropy, 16 (2014), 627. doi: 10.3390/e16020627. Google Scholar

[22]

J. M. Amigó, K. Keller and V. A. Unakafova, Ordinal symbolic analysis and its application to biomedical recordings},, Phil. Trans. R. Soc. A, 373 (2015). doi: 10.1098/rsta.2014.0091. Google Scholar

[23]

J. M. Amigó and A. Giménez, Formulas for the topological entropy of multimodal maps based on min-max symbols,, Discr. Contin. Dyn. Syst. B, 20 (2015), 3415. Google Scholar

[24]

R. G. Andrzejak, K. Lehnertz, F. Mormann, C. Rieke, P. David and C. E. Elger, Indications of nonlinear deterministic and finite-dimensional structures in time series of brain electrical activity: Dependence on recording region and brain state,, Phys. Rev. E, 64 (2001). doi: 10.1103/PhysRevE.64.061907. Google Scholar

[25]

A. Antoniouk, K. Keller and S. Maksymenko, Kolmogorov-Sinai entropy via separation properties of order-generated sigma-algebras,, Discr. Contin. Dyn. Syst. A, 34 (2014), 1793. Google Scholar

[26]

R. B. Ash, Information Theory,, Dover Publications, (1990). Google Scholar

[27]

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

[28]

E. Beadle, J. Schroeder, B Moran and S. Suvorova, An overview of Rényi Entropy and some potential applications,, in 42nd Asilomar Conference on Signals, (2008), 1698. doi: 10.1109/ACSSC.2008.5074715. Google Scholar

[29]

C. H. Bennet, Notes on Landauer's principle, reversible computation and Maxwell's demon,, Stud. Hist. Philos. M. P., 34 (2003), 501. doi: 10.1016/S1355-2198(03)00039-X. Google Scholar

[30]

G. D. Birkhoff, Proof of a recurrence theorem for strongly transitive systems,, PNAS, 17 (1931), 650. doi: 10.1073/pnas.17.12.650. Google Scholar

[31]

S. A. Borovkova, Estimation and Prediction for Nonlinear Time Series,, Ph.D thesis, (1998). Google Scholar

[32]

M. Boyle and D. Lind, Expansive subdynamics,, Trans. Amer. Math. Soc., 349 (1997), 55. doi: 10.1090/S0002-9947-97-01634-6. Google Scholar

[33]

R. Bowen, Entropy for group endomorphisms and homogeneous spaces,, Trans. Amer. Math. Soc., 153 (1971), 401. doi: 10.1090/S0002-9947-1971-0274707-X. Google Scholar

[34]

L. Bowen, A measure-conjugacy invariant for free group actions,, Ann. of Math., 171 (2010), 1387. doi: 10.4007/annals.2010.171.1387. Google Scholar

[35]

L. Bowen, Measure-conjugacy invariants for actions of countable sofic groups,, J. Amer. Math. Soc., 23 (2010), 217. doi: 10.1090/S0894-0347-09-00637-7. Google Scholar

[36]

H. W. Broer and F. Takens, Dynamical Systems and Chaos,, Applied Mathematical Sciences, (2011). doi: 10.1007/978-1-4419-6870-8. Google Scholar

[37]

M. Brin and A. Katok, On local entropy,, in Geometric dynamics (ed. J. Palis), (1007), 30. doi: 10.1007/BFb0061408. Google Scholar

[38]

A. A. Bruzzo, B. Gesierich, M. Santi, C. A. Tassinari, N. Birbaumer and G. Rubboli, Permutation entropy to detect vigilance changes and preictal states from scalp EEG in epileptic patients. A preliminary study,, Neurol. Sci., 29 (2008), 3. doi: 10.1007/s10072-008-0851-3. Google Scholar

[39]

N. Burioka, M. Miyata, G. Cornélissen, F. Halberg, T. Takeshima, D. T. Kaplan, H. Suyama, M. Endo, Y. Maegaki and T. Nomura, et. al, Approximate entropy in the electroencephalogram during wake and sleep,, Clin. EEG Neurosci., 36 (2005), 21. doi: 10.1177/155005940503600106. Google Scholar

[40]

C. Cafaro, W. M. Lord, J. Sun and E. M. Bollt, Causation entropy from symbolic representations of dynamical systems,, Chaos, 25 (2015). Google Scholar

[41]

J. S. Cánovas and A. Guillamón, Permutations and time series analysis,, Chaos, 19 (2009). doi: 10.1063/1.3238256. Google Scholar

[42]

Y. Cao, W.-W. Tung, J. B. Gao, V. A. Protopopescu and L. M. Hively, Detecting dynamical changes in time series using the permutation entropy,, Phys. Rev. E, 70 (2004). doi: 10.1103/PhysRevE.70.046217. Google Scholar

[43]

A. Capurro, L. Diambra, D. Lorenzo, O. Macadar, M. T. Martin, C. Mostaccio, A. Plastino, E. Rofman, M. E. Torres and J. Velluti, Tsallis entropy and cortical dynamics: The analysis of EEG signals,, Physica A, 257 (1998), 149. doi: 10.1016/S0378-4371(98)00137-X. Google Scholar

[44]

A. Capurro, L. Diambra, D. Lorenzo, O. Macadar, M. T. Martin, C. Mostaccio, A. Plastino, J. Perez, E. Rofman and M. E. Torres, Human brain dynamics: The analysis of EEG signals with Tsallis information measure,, Physica A, 265 (1999), 235. doi: 10.1016/S0378-4371(98)00471-3. Google Scholar

[45]

D. Carrasco-Olivera, R. Metzger and C. A. Morales, Topological entropy for set-valued maps,, Discr. Contin. Dyn. Syst. B, 20 (2015), 3461. Google Scholar

[46]

H. C. Choe, Computational Ergodic Theory,, Algorithms and Computation in Mathematics, (2005). Google Scholar

[47]

R. Clausius, The Mechanical Theory of Heat,, MacMillan and Co., (1865). Google Scholar

[48]

M. Courbage and B. Kamiński, Space-time directional Lyapunov exponents for cellular automata,, J. Statis. Phys., 124 (2006), 1499. doi: 10.1007/s10955-006-9172-1. Google Scholar

[49]

T. M. Cover and J. A. Thomas, Elements of Information Theory,, John Wiley & Sons, (2006). Google Scholar

[50]

B. Dai and B. Hu, Minimum conditional entropy clustering: A discriminative framework for clustering,, in JMLR: Workshop and Conference Proceedings, (2010), 47. Google Scholar

[51]

K. Denbigh, How subjective is entropy,, in Maxwell's Demon, (1990), 109. Google Scholar

[52]

M. Denker, Finite generators for ergodic, measure-preserving transformation,, Zeit. Wahr. ver. Geb., 29 (1974), 45. doi: 10.1007/BF00533186. Google Scholar

[53]

M. Denker, C. Grillenberger and K. Sigmund, Ergodic Theory on Compact Spaces,, Lecture Notes in Mathematics, (1976). Google Scholar

[54]

M. Denker and G. Keller, Rigorous statistical procedures for data from dynamical systems,, J. Stat. Phys., 44 (1986), 67. doi: 10.1007/BF01010905. Google Scholar

[55]

E. I. Dinaburg, The relation between topological entropy and metric entropy,, Soviet Math., 11 (1970), 13. Google Scholar

[56]

R. J. V. dos Santos, Generalization of Shannon's theorem for Tsallis entropy,, J. Math. Phys., 38 (1997), 4104. doi: 10.1063/1.532107. Google Scholar

[57]

J.-P. Eckmann and D. Ruelle, Ergodic theory of chaos and strange attractors,, Rev. Mod. Phys., 57 (1985), 617. doi: 10.1103/RevModPhys.57.617. Google Scholar

[58]

M. Einsiedler and T. Ward, Ergodic Theory with a View Towards Number Theory,, Graduate Texts in Mathematics, (2011). doi: 10.1007/978-0-85729-021-2. Google Scholar

[59]

D. K. Faddeev, On the notion of entropy of finite probability distributions,, Usp. Mat. Nauk (in Russian), 11 (1956), 227. Google Scholar

[60]

F. Falniowski, On the connections of the generalized entropies and Kolmogorov-Sinai entropies,, Entropy, 16 (2014), 3732. doi: 10.3390/e16073732. Google Scholar

[61]

B. Frank, B. Pompe, U. Schneider and D. Hoyer, Permutation entropy improves fetal behavioural state classification based on heart rate analysis from biomagnetic recordings in near term fetuses,, Med. Biol. Eng. Comput., 44 (2006), 179. doi: 10.1007/s11517-005-0015-z. Google Scholar

[62]

S. Furuichi, On uniqueness Theorems for Tsallis entropy and Tsallis relative entropy,, IEEE Trans. Inf. Theory, 51 (2005), 3638. doi: 10.1109/TIT.2005.855606. Google Scholar

[63]

L. G. Gamero, A. Plastino and M. E. Torres, Wavelet analysis and nonlinear dynamics in a nonextensive setting,, Physica A, 246 (1997), 487. doi: 10.1016/S0378-4371(97)00367-1. Google Scholar

[64]

P. G. Gaspard and X. J. Wang, Noise, chaos, and $(\varepsilon,\tau)$-entropy per unit time,, Phys. Rep., 235 (1993), 291. doi: 10.1016/0370-1573(93)90012-3. Google Scholar

[65]

T. N. T. Goodman, Relating topological entropy and measure entropy,, Bull. London Math. Soc., 3 (1971), 176. doi: 10.1112/blms/3.2.176. Google Scholar

[66]

N. Gradojevic and R. Genccay, Financial applications of nonextensive entropy,, IEEE Signal Process. Mag., 28 (2011). Google Scholar

[67]

B. Graff, G. Graff and A. Kaczkowska, Entropy Measures of Heart Rate Variability for Short ECG Datasets in Patients with Congestive Heart Failure,, Acta Phys. Pol. B Proc. Suppl., 5 (2012), 153. Google Scholar

[68]

C. W. J. Granger, Investigating causal relations by econometric models and cross-spectral methods,, Econometrica, 37 (1969), 424. Google Scholar

[69]

P. Grassberger and I. Procaccia, Dimensions and entropies of strange attractors from a fluctuating dynamics approach,, Physica D, 13 (1984), 34. doi: 10.1016/0167-2789(84)90269-0. Google Scholar

[70]

P. Grassberger and I. Procaccia, Measuring the strangeness of strange attractors,, Physica D, 9 (1983), 189. doi: 10.1016/0167-2789(83)90298-1. Google Scholar

[71]

P. Grassberger and I. Procaccia, Characterization of strange attractors,, Phys. Rev. Lett., 50 (1983), 346. doi: 10.1103/PhysRevLett.50.346. Google Scholar

[72]

D. W. Hahs and S. D. Pethel, Distinguishing anticipation from causality: Anticipatory bias in the estimation of information flow,, Phys. Rev. Lett., 107 (2011). Google Scholar

[73]

R. Hanel and S. Thurner, A comprehensive classification of complex statistical systems and an axiomatic derivation of their entropy and equidistribution functions,, EPL, 93 (2011). Google Scholar

[74]

R. Hanel, S. Thurner and M. Gell-Mann, Generalized entropies and logarithms and their duality relations,, Proceed. Nat. Acad. Scie., 109 (2012), 19151. doi: 10.1073/pnas.1216885109. Google Scholar

[75]

B. Hasselblatt and A. Katok, Principal structures,, in Handbook of Dynamical Systems (eds. B. Hasselblatt and A. Katok), (2002), 1. doi: 10.1016/S1874-575X(02)80003-0. Google Scholar

[76]

J. Havrda and F. Charvát, Quantification method of classification processes. Concept of structural $\alpha$-entropy,, Kybernetika, 3 (1967), 30. Google Scholar

[77]

G. A. Hedlund, Endomorphisms and automorphisms of the shift dynamical system,, Math. Syst. Theory, 3 (1969), 320. doi: 10.1007/BF01691062. Google Scholar

[78]

H. G. E. Hentschel and I. Procaccia, The infinite number of generalized dimensions of fractals and strange attractors,, Physica D, 8 (1983), 435. doi: 10.1016/0167-2789(83)90235-X. Google Scholar

[79]

R. Hornero, D. Abasolo, J. Escuredo and C. Gomez, Nonlinear analysis of electroencephalogram and magnetoencephalogram recordings in patients with Alzheimer's disease,, Phil. Trans. R. Soc. A, 367 (2009), 317. doi: 10.1098/rsta.2008.0197. Google Scholar

[80]

V. M. Ilić, M. S. Stanković and E. H. Mulalić, Comments on "Generalization of Shannon-Khinchin axioms to nonextensive systems and the uniqueness theorem for the nonextensive entropy'',, IEEE Trans. Inf. Theory, 59 (2013), 6950. Google Scholar

[81]

E. T. Jaynes, Information theory and statistical mechanics,, Phys. Rev., 106 (1957), 620. doi: 10.1103/PhysRev.106.620. Google Scholar

[82]

P. Jizba and T. Arimitsu, The world according to Rényi: Thermodynamics of multifractal systems,, Ann. Phys., 312 (2004), 17. doi: 10.1016/j.aop.2004.01.002. Google Scholar

[83]

C. C. Jouny and G. K. Bergey, Characterization of early partial seizure onset: Frequency, complexity and entropy,, Clin. Neurophysiol., 123 (2012), 658. doi: 10.1016/j.clinph.2011.08.003. Google Scholar

[84]

N. Kannathal, M. L. Choo, U. R. Acharya and P. K. Sadasivan, Entropies for detection of epilepsy in EEG,, Comput. Meth. Prog. Bio., 80 (2005), 187. doi: 10.1016/j.cmpb.2005.06.012. Google Scholar

[85]

H. Kantz and T. Schreiber, Nonlinear Time Series Analysis,, Cambridge University Press, (2004). Google Scholar

[86]

A. Katok, Fifty years of entropy in dynamics: 1978-2007,, J. Mod. Dyn., 1 (2007), 545. doi: 10.3934/jmd.2007.1.545. Google Scholar

[87]

K. Keller and H. Lauffer, Symbolic analysis of high-dimensional time series,, Int. J. Bif. Chaos, 13 (2003), 2657. doi: 10.1142/S0218127403008168. Google Scholar

[88]

K. Keller, Permutations and the Kolmogorov-Sinai entropy,, Discr. Contin. Dyn. Syst. A, 32 (2012), 891. doi: 10.3934/dcds.2012.32.891. Google Scholar

[89]

K. Keller, A. M. Unakafov and V. A. Unakafova, Ordinal Patterns, Entropy, and EEG,, Entropy, 16 (2014), 6212. doi: 10.3390/e16126212. Google Scholar

[90]

K. Keller, S. Maksymenko and I. Stolz, Entropy determination based on the ordinal structure of a dynamical system,, Discr. Contin. Dyn. Syst. B, 20 (2015), 3507. Google Scholar

[91]

A. I. Khinchin, Mathematical Foundations of Information Theory,, Dover, (1957). Google Scholar

[92]

A. N. Kolmogorov, A new metric invariant of transitive dynamical systems and Lebesgue space endomorphisms,, Dokl. Acad. Sci. USSR, 119 (1958), 861. Google Scholar

[93]

S. Kolyada and L. Snoha, Topological entropy of nonautonomous dynamical systems,, Random Comput. Dynam., 4 (1996), 205. Google Scholar

[94]

Z. S. Kowalski, Finite generators of ergodic endomorphisms,, Colloq. Math., 49 (1984), 87. Google Scholar

[95]

Z. S. Kowalski, Minimal generators for aperiodic endomorphisms,, Commentat. Math. Univ. Carol., 36 (1995), 721. Google Scholar

[96]

W. Krieger, On entropy and generators of measure-preserving transformations,, Trans. Amer. Math. Soc., 149 (1970), 453. doi: 10.1090/S0002-9947-1970-0259068-3. Google Scholar

[97]

J. Kurths, A. Voss, P. Saparin, A. Witt, H. J. Kleiner and N. Wessel, Quantitative analysis of heart rate variability,, Chaos, 5 (1995), 88. doi: 10.1063/1.166090. Google Scholar

[98]

D. E. Lake, J. S. Richman, M. P. Griffin and J. R. Moorman, Sample entropy analysis of neonatal heart rate variability,, Am. J. Physiol.-Reg. I., 283 (2002), 789. doi: 10.1152/ajpregu.00069.2002. Google Scholar

[99]

A. M. Law and W. D. Kelton, Simulation, Modeling, and Analysis,, McGraw-Hill, (2000). Google Scholar

[100]

F. Ledrappier and L.-S. Young, The metric entropy of diffeomorphisms. II. Relations betweenentropy, exponents and dimension,, Ann. Math., 122 (1985), 540. doi: 10.2307/1971329. Google Scholar

[101]

X. Li, G. Ouyang and D. A. Richards, Predictability analysis of absence seizures with permutation entropy,, Epilepsy Res., 77 (2007), 70. doi: 10.1016/j.eplepsyres.2007.08.002. Google Scholar

[102]

J. Li, J. Yan, X. Liu and G. Ouyang, Using permutation entropy to measure the changes in eeg signals during absence seizures,, Entropy, 16 (2014), 3049. doi: 10.3390/e16063049. Google Scholar

[103]

H. Li, K. Zhang and T. Jiang, Minimum entropy clustering and applications to gene expression analysis,, in Proc. IEEE Comput. Syst. Bioinform. Conf., (2004), 142. Google Scholar

[104]

Z. Liang, Y. Wang, X. Sun, D. Li, L. J. Voss, J. W. Sleigh, S. Hagihira and X. Li, EEG entropy measures in anesthesia,, Front. Comput. Neurosci., 9 (2015). doi: 10.3389/fncom.2015.00016. Google Scholar

[105]

D. Lind and B. Marcus, An Introduction to Symbolic Dynamics and Coding,, Cambridge University Press, (1995). doi: 10.1017/CBO9780511626302. Google Scholar

[106]

J. Llibre, Brief survey on the topological entropy,, Discr. Contin. Dyn. Syst. B, 20 (2015), 3363. Google Scholar

[107]

N. Mammone, F. La Foresta and F. C. Morabito, Automatic artifact rejection from multichannel scalp EEG by wavelet ICA,, IEEE Sens. J., 12 (2012), 533. doi: 10.1109/JSEN.2011.2115236. Google Scholar

[108]

M. Matilla-García, A non-parametric test for independence based on symbolic dynamics,, J. Econ. Dyn. Control, 31 (2007), 3889. doi: 10.1016/j.jedc.2007.01.018. Google Scholar

[109]

M. Matilla-García and M. Ruiz Marín, A non-parametric independence test using permutation entropy,, J. Econ., 144 (2008), 139. doi: 10.1016/j.jeconom.2007.12.005. Google Scholar

[110]

A. M. Mesón and F. Vericat, On the Kolmogorov-like generalization of Tsallis entropy, correlation entropies and multifractal analysis,, J. Math. Phys., 43 (2002), 904. doi: 10.1063/1.1429323. Google Scholar

[111]

R. Miles and T. Ward, Directional uniformities, periodic points, and entropy,, Discr. Contin. Dyn. Syst. B, 20 (2015), 3525. Google Scholar

[112]

J. Milnor, On the entropy geometry of cellular automata,, Complex Systems, 2 (1988), 357. Google Scholar

[113]

J. Milnor and W. Thurston, On iterated maps of the interval,, in Dynamical Systems (ed. J. C. Alexander), (1342), 465. doi: 10.1007/BFb0082847. Google Scholar

[114]

M. Misiurewicz and W. Szlenk, Entropy of piecewise monotone mappings,, Studia Math., 67 (1980), 45. Google Scholar

[115]

M. Misiurewicz, Permutations and topological entropy for interval maps,, Nonlinearity, 16 (2003), 971. doi: 10.1088/0951-7715/16/3/310. Google Scholar

[116]

R. Monetti, J. M. Amigó, T. Aschenbrenner and W. Bunk, Permutation complexity of interacting dynamical systems,, Eur. Phys. J. Special Topics, 222 (2013), 421. doi: 10.1140/epjst/e2013-01850-y. Google Scholar

[117]

R. Monetti, W. Bunk, T. Aschenbrenner, S. Springer and J. M. Amigó, Information directionality in coupled time series using transcripts,, Phys. Rev. E, 88 (2013). doi: 10.1103/PhysRevE.88.022911. Google Scholar

[118]

F. C. Morabito, D. Labate, F. La Foresta, A. Bramanti, G. Morabito and I. Palamara, Multivariate multi-scale permutation entropy for complexity analysis of Alzheimer's disease EEG,, Entropy, 14 (2012), 1186. doi: 10.3390/e14071186. Google Scholar

[119]

N. Nicolaou and J. Georgiou, The use of permutation entropy to characterize sleep electroencephalograms,, Clin. EEG Neurosci., 42 (2011), 24. doi: 10.1177/155005941104200107. Google Scholar

[120]

D. Ornstein, Two Bernoulli shifts with the same entropy are isomorphic,, Adv. Math., 4 (1970), 337. doi: 10.1016/0001-8708(70)90029-0. Google Scholar

[121]

D. Ornstein, Two Bernoulli shifts with infinite entropy are isomorphic,, Adv. Math., 5 (1970), 339. doi: 10.1016/0001-8708(70)90008-3. Google Scholar

[122]

D. Ornstein and B. Weiss, Entropy and isomorphism theorem for actions of amenable groups,, J. Analyse Math., 48 (1987), 1. doi: 10.1007/BF02790325. Google Scholar

[123]

D. Ornstein, Newton's law and coin tossing,, Notices Amer. Math. Soc., 60 (2013), 450. doi: 10.1090/noti974. Google Scholar

[124]

D. Ornstein and B. Weiss, Entropy is the only finitely observable invariant,, J. Mod. Dyn., 1 (2007), 93. Google Scholar

[125]

G. Ouyang, C. Dang, D. A. Richards and X. Li, Ordinal pattern based similarity analysis for EEG recordings,, Clin. Neurophysiol, 121 (2010), 694. doi: 10.1016/j.clinph.2009.12.030. Google Scholar

[126]

S. Y. Park and A. K. Bera, Maximum entropy autoregressive conditional heretoskedasticity model,, J. Econometrics, 150 (2009), 219. doi: 10.1016/j.jeconom.2008.12.014. Google Scholar

[127]

U. Parlitz, S. Berg, S. Luther, A. Schirdewan, J. Kurths and N. Wessel, Classifying cardiac biosignals using ordinal pattern statistics and symbolic dynamics,, Comput. Biol. Med., 42 (2012), 319. doi: 10.1016/j.compbiomed.2011.03.017. Google Scholar

[128]

W. Parry, Intrinsic Markov chains,, Trans. Amer. Math. Soc., 112 (1964), 55. doi: 10.1090/S0002-9947-1964-0161372-1. Google Scholar

[129]

Y. Pesin, Characteristic Lyapunov exponents and smooth ergodic theory,, Russian Math. Surveys, 32 (1977), 55. Google Scholar

[130]

S. M. Pincus, Approximate entropy as a measure of system complexity,, PNAS, 88 (1991), 2297. doi: 10.1073/pnas.88.6.2297. Google Scholar

[131]

S. M. Pincus and R. R. Viscarello, Approximate entropy: A regularity measure for fetal heart rate analysis,, Obstet. Gynecol., 79 (1992), 249. Google Scholar

[132]

S. M. Pincus, Approximate entropy as a measure of irregularity for psychiatric serial metrics,, Bipolar Disord., 8 (2006), 430. doi: 10.1111/j.1399-5618.2006.00375.x. Google Scholar

[133]

B. Pompe and J. Runge, Momentary information transfer as a coupling measure of time series,, Phys. Rev. E, 83 (2011). doi: 10.1103/PhysRevE.83.051122. Google Scholar

[134]

J. Poza, R. Hornero, J. Escudero, A. Fernández and C. A. Sánchez, Regional analysis of spontaneous MEG rhythms in patients with Alzheimer's disease using spectral entropies,, Ann. Biomed. Eng., 36 (2008), 141. doi: 10.1007/s10439-007-9402-y. Google Scholar

[135]

J. C. Principe, Information Theoretic Learning. Renyi's Entropy and Kernel Perspectives,, Springer, (2010). doi: 10.1007/978-1-4419-1570-2. Google Scholar

[136]

A. Rényi, On measures of entropy and information,, in Proceedings of the 4th Berkeley Symposium on Mathematics, (1961), 547. Google Scholar

[137]

J. S. Richman and J. R. Moorman, Physiological time-series analysis using approximate entropy and sample entropy,, Am. J. Physiol.-Heart C., 278 (2000), 2039. Google Scholar

[138]

S. J. Roberts, R. Everson and I. Rezek, Minimum entropy data partitioning,, in Proceedings of International Conference on Artificial Neural Networks, (1999), 844. Google Scholar

[139]

E. A. Robinson and A. Şahin, Rank-one $\mathbbZ^d$ actions and directional entropy,, Ergod. Theor. Dynam. Syst., 31 (2011), 285. doi: 10.1017/S0143385709000911. Google Scholar

[140]

O. A. Rosso, M. T. Martin and A. Plastino, Brain electrical activity analysis using wavelet-based informational tools,, Physica A, 313 (2002), 587. doi: 10.1016/S0378-4371(02)00958-5. Google Scholar

[141]

D. Rudolph and B. Weiss, Entropy and mixing for amenable group actions,, Ann. of Math., 151 (2000), 1119. doi: 10.2307/121130. Google Scholar

[142]

D. Ruelle, Statistical mechanics on a compact set with $Z^{\nu}$ action satisfying expansiveness and specification,, Trans. Amer. Math. Soc., 187 (1973), 237. Google Scholar

[143]

J. Runge, J. Heitzig, N. Marwan and J. Kurths, Quantifying causal coupling strength: A lag-specific measure for multivariate time series related to transfer entropy,, Phys. Rev. E, 86 (2012). Google Scholar

[144]

X. San Liang, Unraveling the cause-effect relation between time series,, Phys. Rev. E, 90 (2014). Google Scholar

[145]

T. Schreiber, Measuring information transfer,, Phys. Rev. Lett., 85 (2000), 461. doi: 10.1103/PhysRevLett.85.461. Google Scholar

[146]

C. E. Shannon, A mathematical theory of communication,, Bell Syst. Tech. J., 27 (1948), 379. doi: 10.1002/j.1538-7305.1948.tb01338.x. Google Scholar

[147]

Y. G. Sinai, On the notion of entropy of dynamical systems,, Dokl. Acad. Sci. USSR, 125 (1959), 768. Google Scholar

[148]

Y. G. Sinai, Flows with finite entropy,, Dokl. Acad. Sci. USSR, 125 (1959), 1200. Google Scholar

[149]

Y. G. Sinai, Gibbs measures in ergodic theory,, Uspehi Mat. Nauk, 27 (1972), 21. Google Scholar

[150]

D. Smirnov, Spurious causalities with transfer entropy,, Phys. Rev. E, 87 (2013). Google Scholar

[151]

M. Smorodinsky, Ergodic Theory, Entropy,, Lectures Notes in Mathematics, (1971). Google Scholar

[152]

R. Sneddon, The Tsallis entropy of natural information,, Physica A, 386 (2007), 101. doi: 10.1016/j.physa.2007.05.065. Google Scholar

[153]

V. Srinivasan, C. Eswaran and N. Sriraam, Approximate entropy-based epileptic EEG detection using artificial neural networks,, IEEE T. Inf. Technol. B., 11 (2007), 288. doi: 10.1109/TITB.2006.884369. Google Scholar

[154]

H. Suyari, Generalization of Shannon-Khinchin axioms to nonextensive systems and the uniqueness theorem for the nonextensive entropy,, IEEE T. Inform. Theory, 50 (2004), 1783. doi: 10.1109/TIT.2004.831749. Google Scholar

[155]

F. Takens, Detecting strange attractors in turbulence,, in Dynamical Systems and Turbulence (eds. D. A. Rand and L. S. Young), (1981), 366. Google Scholar

[156]

F. Takens and E. Verbitskiy, Rényi entropies of aperiodic dynamical systems,, Israel J. Math., 127 (2002), 279. doi: 10.1007/BF02784535. Google Scholar

[157]

F. Takens and E. Verbitskiy, Generalized entropies: Rényi and correlation integral approach,, Nonlinearity, 11 (1998), 771. doi: 10.1088/0951-7715/11/4/001. Google Scholar

[158]

S. Tong, A. Bezerianos, A. Malhotra, Y. Zhu and N. Thakor, Parameterized entropy analysis of EEG following hypoxicischemic brain injury,, Phys. Lett. A, 314 (2003), 354. doi: 10.1016/S0375-9601(03)00949-6. Google Scholar

[159]

M. E. Torres and L. G. Gamero, Relative complexity changes in time series using information measures,, Physica A, 286 (2000), 457. doi: 10.1016/S0378-4371(00)00309-5. Google Scholar

[160]

B. Tóthmérész, Comparison of different methods for diversity ordering,, J. Veg. Sci., 6 (1995), 283. Google Scholar

[161]

C. Tsallis, Possible generalization of Boltzmann-Gibbs statistics,, J. Stat. Phys., 52 (1988), 479. doi: 10.1007/BF01016429. Google Scholar

[162]

C. Tsallis, The nonadditive entropy Sq and its applications in physics and elsewhere: Some remarks,, Entropy, 13 (2011), 1765. doi: 10.3390/e13101765. Google Scholar

[163]

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

[164]

V. A. Unakafova, Investigating Measures of Complexity for Dynamical Systems and for Time Series,, Ph.D thesis, (2015). Google Scholar

[165]

E. A. Verbitskiy, Generalized Entropies in Dynamical Systems,, Ph.D thesis, (2000). Google Scholar

[166]

A. Voss, S. Schulz, R. Schroeder, M. Baumert and P. Caminal, Methods derived from nonlinear dynamics for analysing heart rate variability,, Phil. Trans. R. Soc. A, 367 (2009), 277. doi: 10.1098/rsta.2008.0232. Google Scholar

[167]

P. Walters, An Introduction to Ergodic Theory,, Springer Verlag, (1982). Google Scholar

[168]

B. Weiss, Subshifts of finite type and sofic systems,, Monats. Math., 77 (1973), 462. doi: 10.1007/BF01295322. Google Scholar

[169]

B. Weiss, Entropy and actions of sofic groups,, Discr. Contin. Dyn. Syst. B, 20 (2015), 3375. Google Scholar

[170]

N. Wiener, The theory of prediction,, in Modern Mathematics for the Engineer (ed. E. F. Beckenbach), (1956). Google Scholar

[171]

A. D. Wissner-Gross and C. E. Freer, Causal entropic forces,, Phys. Rev. Lett., 110 (2013). doi: 10.1103/PhysRevLett.110.168702. Google Scholar

[172]

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. doi: 10.3390/e14081553. Google Scholar

[173]

D. Zhang, X. Jia, H. Ding, D. Ye and N. V. Thakor, Application of Tsallis entropy to EEG: Quantifying the presence of burst suppression after asphyxial cardiac arrest in rats,, IEEE Trans. Bio.-Med. Eng., 57 (2010), 867. Google Scholar

show all references

References:
[1]

S. Abe, Tsallis entropy: How unique?,, Contin. Mech. Thermodyn., 16 (2004), 237. doi: 10.1007/s00161-003-0153-1. Google Scholar

[2]

U. R. Acharya, O. Faust, N. Kannathal, T. Chua and S. Laxminarayan, Non-linear analysis of EEG signals at various sleep stages,, Comput. Meth. Prog. Bio., 80 (2005), 37. doi: 10.1016/j.cmpb.2005.06.011. Google Scholar

[3]

U. R. Acharya, K. P. Joseph, N. Kannathal, C. M. Lim and J. S. Suri, Heart rate variability: A review,, Advances in Cardiac Signal Processing, (2007), 121. doi: 10.1007/978-3-540-36675-1_5. Google Scholar

[4]

J. Aczél and Z. Daróczy, Charakterisierung der Entropien positiver Ordnung und der Shannonschen Entropie,, Acta Math. Acad. Sci. Hung., 14 (1963), 95. doi: 10.1007/BF01901932. Google Scholar

[5]

R. Adler, A. Konheim and M. McAndrew, Topological entropy,, Trans. Amer. Mat. Soc., 114 (1965), 309. doi: 10.1090/S0002-9947-1965-0175106-9. Google Scholar

[6]

R. L. Adler and B. Marcus, Topological entropy and the equivalence of dynamical systems,, Mem. Amer. Math. Soc., 20 (1979). doi: 10.1090/memo/0219. Google Scholar

[7]

V. Afraimovich, M. Courbage and L. Glebsky, Directional complexity and entropy for lift mappings,, Discr. Contin. Dyn. Syst. B, 20 (2015), 3385. Google Scholar

[8]

L. Alsedà, J. Llibre and M. Misiurewicz, Combinatorial Dynamics and Entropy in Dimension One,, World Scientific, (2000). doi: 10.1142/4205. Google Scholar

[9]

J. M. Amigó, L. Kocarev and J. Szczepanski, Order patterns and chaos,, Phys. Lett. A, 355 (2006), 27. Google Scholar

[10]

J. M. Amigó, S. Zambrano and M. A. F. Sanjuán, True and false forbidden patterns in deterministic and random dynamics,, Eur. Phys. Lett., 79 (2007). doi: 10.1209/0295-5075/79/50001. Google Scholar

[11]

J. M. Amigó and M. B. Kennel, Forbidden ordinal patterns in higher dimensional dynamics,, Physica D, 237 (2008), 2893. doi: 10.1016/j.physd.2008.05.003. Google Scholar

[12]

J. M. Amigó, S. Zambrano and M. A. F. Sanjuán, Combinatorial detection of determinism in noisy time series,, Europhys. Lett., 82 (2008). Google Scholar

[13]

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

[14]

J. M. Amigó, S. Zambrano and M. A. F. Sanjuán, Detecting determinism in time series with ordinal patterns: A comparative study,, Int. J. Bif. Chaos, 20 (2010), 2915. doi: 10.1142/S0218127410027453. Google Scholar

[15]

J. M. Amigó, R. Monetti, T. Aschenbrenner and W. Bunk, Transcripts: An algebraic approach to coupled time series,, Chaos, 22 (2012). doi: 10.1063/1.3673238. Google Scholar

[16]

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

[17]

J. M. Amigó and K. Keller, Permutation entropy: One concept, two approaches,, Eur. Phys. J. Special Topics, 222 (2013), 263. Google Scholar

[18]

J. M. Amigó, P. Kloeden and A. Giménez, Switching systems and entropy,, J. Diff. Eq. Appl., 19 (2013), 1872. doi: 10.1080/10236198.2013.788166. Google Scholar

[19]

J. M. Amigó, P. E. Kloeden and A. Giménez, Entropy increase in switching systems,, Entropy, 15 (2013), 2363. doi: 10.3390/e15062363. Google Scholar

[20]

J. M. Amigó, T. Aschenbrenner, W. Bunk and R. Monetti, Dimensional reduction of conditional algebraic multi-information via transcripts,, Inform. Sciences, 278 (2014), 298. doi: 10.1016/j.ins.2014.03.054. Google Scholar

[21]

J. M. Amigó and A. Giménez, A simplified algorithm for the topological entropy of multimodal maps,, Entropy, 16 (2014), 627. doi: 10.3390/e16020627. Google Scholar

[22]

J. M. Amigó, K. Keller and V. A. Unakafova, Ordinal symbolic analysis and its application to biomedical recordings},, Phil. Trans. R. Soc. A, 373 (2015). doi: 10.1098/rsta.2014.0091. Google Scholar

[23]

J. M. Amigó and A. Giménez, Formulas for the topological entropy of multimodal maps based on min-max symbols,, Discr. Contin. Dyn. Syst. B, 20 (2015), 3415. Google Scholar

[24]

R. G. Andrzejak, K. Lehnertz, F. Mormann, C. Rieke, P. David and C. E. Elger, Indications of nonlinear deterministic and finite-dimensional structures in time series of brain electrical activity: Dependence on recording region and brain state,, Phys. Rev. E, 64 (2001). doi: 10.1103/PhysRevE.64.061907. Google Scholar

[25]

A. Antoniouk, K. Keller and S. Maksymenko, Kolmogorov-Sinai entropy via separation properties of order-generated sigma-algebras,, Discr. Contin. Dyn. Syst. A, 34 (2014), 1793. Google Scholar

[26]

R. B. Ash, Information Theory,, Dover Publications, (1990). Google Scholar

[27]

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

[28]

E. Beadle, J. Schroeder, B Moran and S. Suvorova, An overview of Rényi Entropy and some potential applications,, in 42nd Asilomar Conference on Signals, (2008), 1698. doi: 10.1109/ACSSC.2008.5074715. Google Scholar

[29]

C. H. Bennet, Notes on Landauer's principle, reversible computation and Maxwell's demon,, Stud. Hist. Philos. M. P., 34 (2003), 501. doi: 10.1016/S1355-2198(03)00039-X. Google Scholar

[30]

G. D. Birkhoff, Proof of a recurrence theorem for strongly transitive systems,, PNAS, 17 (1931), 650. doi: 10.1073/pnas.17.12.650. Google Scholar

[31]

S. A. Borovkova, Estimation and Prediction for Nonlinear Time Series,, Ph.D thesis, (1998). Google Scholar

[32]

M. Boyle and D. Lind, Expansive subdynamics,, Trans. Amer. Math. Soc., 349 (1997), 55. doi: 10.1090/S0002-9947-97-01634-6. Google Scholar

[33]

R. Bowen, Entropy for group endomorphisms and homogeneous spaces,, Trans. Amer. Math. Soc., 153 (1971), 401. doi: 10.1090/S0002-9947-1971-0274707-X. Google Scholar

[34]

L. Bowen, A measure-conjugacy invariant for free group actions,, Ann. of Math., 171 (2010), 1387. doi: 10.4007/annals.2010.171.1387. Google Scholar

[35]

L. Bowen, Measure-conjugacy invariants for actions of countable sofic groups,, J. Amer. Math. Soc., 23 (2010), 217. doi: 10.1090/S0894-0347-09-00637-7. Google Scholar

[36]

H. W. Broer and F. Takens, Dynamical Systems and Chaos,, Applied Mathematical Sciences, (2011). doi: 10.1007/978-1-4419-6870-8. Google Scholar

[37]

M. Brin and A. Katok, On local entropy,, in Geometric dynamics (ed. J. Palis), (1007), 30. doi: 10.1007/BFb0061408. Google Scholar

[38]

A. A. Bruzzo, B. Gesierich, M. Santi, C. A. Tassinari, N. Birbaumer and G. Rubboli, Permutation entropy to detect vigilance changes and preictal states from scalp EEG in epileptic patients. A preliminary study,, Neurol. Sci., 29 (2008), 3. doi: 10.1007/s10072-008-0851-3. Google Scholar

[39]

N. Burioka, M. Miyata, G. Cornélissen, F. Halberg, T. Takeshima, D. T. Kaplan, H. Suyama, M. Endo, Y. Maegaki and T. Nomura, et. al, Approximate entropy in the electroencephalogram during wake and sleep,, Clin. EEG Neurosci., 36 (2005), 21. doi: 10.1177/155005940503600106. Google Scholar

[40]

C. Cafaro, W. M. Lord, J. Sun and E. M. Bollt, Causation entropy from symbolic representations of dynamical systems,, Chaos, 25 (2015). Google Scholar

[41]

J. S. Cánovas and A. Guillamón, Permutations and time series analysis,, Chaos, 19 (2009). doi: 10.1063/1.3238256. Google Scholar

[42]

Y. Cao, W.-W. Tung, J. B. Gao, V. A. Protopopescu and L. M. Hively, Detecting dynamical changes in time series using the permutation entropy,, Phys. Rev. E, 70 (2004). doi: 10.1103/PhysRevE.70.046217. Google Scholar

[43]

A. Capurro, L. Diambra, D. Lorenzo, O. Macadar, M. T. Martin, C. Mostaccio, A. Plastino, E. Rofman, M. E. Torres and J. Velluti, Tsallis entropy and cortical dynamics: The analysis of EEG signals,, Physica A, 257 (1998), 149. doi: 10.1016/S0378-4371(98)00137-X. Google Scholar

[44]

A. Capurro, L. Diambra, D. Lorenzo, O. Macadar, M. T. Martin, C. Mostaccio, A. Plastino, J. Perez, E. Rofman and M. E. Torres, Human brain dynamics: The analysis of EEG signals with Tsallis information measure,, Physica A, 265 (1999), 235. doi: 10.1016/S0378-4371(98)00471-3. Google Scholar

[45]

D. Carrasco-Olivera, R. Metzger and C. A. Morales, Topological entropy for set-valued maps,, Discr. Contin. Dyn. Syst. B, 20 (2015), 3461. Google Scholar

[46]

H. C. Choe, Computational Ergodic Theory,, Algorithms and Computation in Mathematics, (2005). Google Scholar

[47]

R. Clausius, The Mechanical Theory of Heat,, MacMillan and Co., (1865). Google Scholar

[48]

M. Courbage and B. Kamiński, Space-time directional Lyapunov exponents for cellular automata,, J. Statis. Phys., 124 (2006), 1499. doi: 10.1007/s10955-006-9172-1. Google Scholar

[49]

T. M. Cover and J. A. Thomas, Elements of Information Theory,, John Wiley & Sons, (2006). Google Scholar

[50]

B. Dai and B. Hu, Minimum conditional entropy clustering: A discriminative framework for clustering,, in JMLR: Workshop and Conference Proceedings, (2010), 47. Google Scholar

[51]

K. Denbigh, How subjective is entropy,, in Maxwell's Demon, (1990), 109. Google Scholar

[52]

M. Denker, Finite generators for ergodic, measure-preserving transformation,, Zeit. Wahr. ver. Geb., 29 (1974), 45. doi: 10.1007/BF00533186. Google Scholar

[53]

M. Denker, C. Grillenberger and K. Sigmund, Ergodic Theory on Compact Spaces,, Lecture Notes in Mathematics, (1976). Google Scholar

[54]

M. Denker and G. Keller, Rigorous statistical procedures for data from dynamical systems,, J. Stat. Phys., 44 (1986), 67. doi: 10.1007/BF01010905. Google Scholar

[55]

E. I. Dinaburg, The relation between topological entropy and metric entropy,, Soviet Math., 11 (1970), 13. Google Scholar

[56]

R. J. V. dos Santos, Generalization of Shannon's theorem for Tsallis entropy,, J. Math. Phys., 38 (1997), 4104. doi: 10.1063/1.532107. Google Scholar

[57]

J.-P. Eckmann and D. Ruelle, Ergodic theory of chaos and strange attractors,, Rev. Mod. Phys., 57 (1985), 617. doi: 10.1103/RevModPhys.57.617. Google Scholar

[58]

M. Einsiedler and T. Ward, Ergodic Theory with a View Towards Number Theory,, Graduate Texts in Mathematics, (2011). doi: 10.1007/978-0-85729-021-2. Google Scholar

[59]

D. K. Faddeev, On the notion of entropy of finite probability distributions,, Usp. Mat. Nauk (in Russian), 11 (1956), 227. Google Scholar

[60]

F. Falniowski, On the connections of the generalized entropies and Kolmogorov-Sinai entropies,, Entropy, 16 (2014), 3732. doi: 10.3390/e16073732. Google Scholar

[61]

B. Frank, B. Pompe, U. Schneider and D. Hoyer, Permutation entropy improves fetal behavioural state classification based on heart rate analysis from biomagnetic recordings in near term fetuses,, Med. Biol. Eng. Comput., 44 (2006), 179. doi: 10.1007/s11517-005-0015-z. Google Scholar

[62]

S. Furuichi, On uniqueness Theorems for Tsallis entropy and Tsallis relative entropy,, IEEE Trans. Inf. Theory, 51 (2005), 3638. doi: 10.1109/TIT.2005.855606. Google Scholar

[63]

L. G. Gamero, A. Plastino and M. E. Torres, Wavelet analysis and nonlinear dynamics in a nonextensive setting,, Physica A, 246 (1997), 487. doi: 10.1016/S0378-4371(97)00367-1. Google Scholar

[64]

P. G. Gaspard and X. J. Wang, Noise, chaos, and $(\varepsilon,\tau)$-entropy per unit time,, Phys. Rep., 235 (1993), 291. doi: 10.1016/0370-1573(93)90012-3. Google Scholar

[65]

T. N. T. Goodman, Relating topological entropy and measure entropy,, Bull. London Math. Soc., 3 (1971), 176. doi: 10.1112/blms/3.2.176. Google Scholar

[66]

N. Gradojevic and R. Genccay, Financial applications of nonextensive entropy,, IEEE Signal Process. Mag., 28 (2011). Google Scholar

[67]

B. Graff, G. Graff and A. Kaczkowska, Entropy Measures of Heart Rate Variability for Short ECG Datasets in Patients with Congestive Heart Failure,, Acta Phys. Pol. B Proc. Suppl., 5 (2012), 153. Google Scholar

[68]

C. W. J. Granger, Investigating causal relations by econometric models and cross-spectral methods,, Econometrica, 37 (1969), 424. Google Scholar

[69]

P. Grassberger and I. Procaccia, Dimensions and entropies of strange attractors from a fluctuating dynamics approach,, Physica D, 13 (1984), 34. doi: 10.1016/0167-2789(84)90269-0. Google Scholar

[70]

P. Grassberger and I. Procaccia, Measuring the strangeness of strange attractors,, Physica D, 9 (1983), 189. doi: 10.1016/0167-2789(83)90298-1. Google Scholar

[71]

P. Grassberger and I. Procaccia, Characterization of strange attractors,, Phys. Rev. Lett., 50 (1983), 346. doi: 10.1103/PhysRevLett.50.346. Google Scholar

[72]

D. W. Hahs and S. D. Pethel, Distinguishing anticipation from causality: Anticipatory bias in the estimation of information flow,, Phys. Rev. Lett., 107 (2011). Google Scholar

[73]

R. Hanel and S. Thurner, A comprehensive classification of complex statistical systems and an axiomatic derivation of their entropy and equidistribution functions,, EPL, 93 (2011). Google Scholar

[74]

R. Hanel, S. Thurner and M. Gell-Mann, Generalized entropies and logarithms and their duality relations,, Proceed. Nat. Acad. Scie., 109 (2012), 19151. doi: 10.1073/pnas.1216885109. Google Scholar

[75]

B. Hasselblatt and A. Katok, Principal structures,, in Handbook of Dynamical Systems (eds. B. Hasselblatt and A. Katok), (2002), 1. doi: 10.1016/S1874-575X(02)80003-0. Google Scholar

[76]

J. Havrda and F. Charvát, Quantification method of classification processes. Concept of structural $\alpha$-entropy,, Kybernetika, 3 (1967), 30. Google Scholar

[77]

G. A. Hedlund, Endomorphisms and automorphisms of the shift dynamical system,, Math. Syst. Theory, 3 (1969), 320. doi: 10.1007/BF01691062. Google Scholar

[78]

H. G. E. Hentschel and I. Procaccia, The infinite number of generalized dimensions of fractals and strange attractors,, Physica D, 8 (1983), 435. doi: 10.1016/0167-2789(83)90235-X. Google Scholar

[79]

R. Hornero, D. Abasolo, J. Escuredo and C. Gomez, Nonlinear analysis of electroencephalogram and magnetoencephalogram recordings in patients with Alzheimer's disease,, Phil. Trans. R. Soc. A, 367 (2009), 317. doi: 10.1098/rsta.2008.0197. Google Scholar

[80]

V. M. Ilić, M. S. Stanković and E. H. Mulalić, Comments on "Generalization of Shannon-Khinchin axioms to nonextensive systems and the uniqueness theorem for the nonextensive entropy'',, IEEE Trans. Inf. Theory, 59 (2013), 6950. Google Scholar

[81]

E. T. Jaynes, Information theory and statistical mechanics,, Phys. Rev., 106 (1957), 620. doi: 10.1103/PhysRev.106.620. Google Scholar

[82]

P. Jizba and T. Arimitsu, The world according to Rényi: Thermodynamics of multifractal systems,, Ann. Phys., 312 (2004), 17. doi: 10.1016/j.aop.2004.01.002. Google Scholar

[83]

C. C. Jouny and G. K. Bergey, Characterization of early partial seizure onset: Frequency, complexity and entropy,, Clin. Neurophysiol., 123 (2012), 658. doi: 10.1016/j.clinph.2011.08.003. Google Scholar

[84]

N. Kannathal, M. L. Choo, U. R. Acharya and P. K. Sadasivan, Entropies for detection of epilepsy in EEG,, Comput. Meth. Prog. Bio., 80 (2005), 187. doi: 10.1016/j.cmpb.2005.06.012. Google Scholar

[85]

H. Kantz and T. Schreiber, Nonlinear Time Series Analysis,, Cambridge University Press, (2004). Google Scholar

[86]

A. Katok, Fifty years of entropy in dynamics: 1978-2007,, J. Mod. Dyn., 1 (2007), 545. doi: 10.3934/jmd.2007.1.545. Google Scholar

[87]

K. Keller and H. Lauffer, Symbolic analysis of high-dimensional time series,, Int. J. Bif. Chaos, 13 (2003), 2657. doi: 10.1142/S0218127403008168. Google Scholar

[88]

K. Keller, Permutations and the Kolmogorov-Sinai entropy,, Discr. Contin. Dyn. Syst. A, 32 (2012), 891. doi: 10.3934/dcds.2012.32.891. Google Scholar

[89]

K. Keller, A. M. Unakafov and V. A. Unakafova, Ordinal Patterns, Entropy, and EEG,, Entropy, 16 (2014), 6212. doi: 10.3390/e16126212. Google Scholar

[90]

K. Keller, S. Maksymenko and I. Stolz, Entropy determination based on the ordinal structure of a dynamical system,, Discr. Contin. Dyn. Syst. B, 20 (2015), 3507. Google Scholar

[91]

A. I. Khinchin, Mathematical Foundations of Information Theory,, Dover, (1957). Google Scholar

[92]

A. N. Kolmogorov, A new metric invariant of transitive dynamical systems and Lebesgue space endomorphisms,, Dokl. Acad. Sci. USSR, 119 (1958), 861. Google Scholar

[93]

S. Kolyada and L. Snoha, Topological entropy of nonautonomous dynamical systems,, Random Comput. Dynam., 4 (1996), 205. Google Scholar

[94]

Z. S. Kowalski, Finite generators of ergodic endomorphisms,, Colloq. Math., 49 (1984), 87. Google Scholar

[95]

Z. S. Kowalski, Minimal generators for aperiodic endomorphisms,, Commentat. Math. Univ. Carol., 36 (1995), 721. Google Scholar

[96]

W. Krieger, On entropy and generators of measure-preserving transformations,, Trans. Amer. Math. Soc., 149 (1970), 453. doi: 10.1090/S0002-9947-1970-0259068-3. Google Scholar

[97]

J. Kurths, A. Voss, P. Saparin, A. Witt, H. J. Kleiner and N. Wessel, Quantitative analysis of heart rate variability,, Chaos, 5 (1995), 88. doi: 10.1063/1.166090. Google Scholar

[98]

D. E. Lake, J. S. Richman, M. P. Griffin and J. R. Moorman, Sample entropy analysis of neonatal heart rate variability,, Am. J. Physiol.-Reg. I., 283 (2002), 789. doi: 10.1152/ajpregu.00069.2002. Google Scholar

[99]

A. M. Law and W. D. Kelton, Simulation, Modeling, and Analysis,, McGraw-Hill, (2000). Google Scholar

[100]

F. Ledrappier and L.-S. Young, The metric entropy of diffeomorphisms. II. Relations betweenentropy, exponents and dimension,, Ann. Math., 122 (1985), 540. doi: 10.2307/1971329. Google Scholar

[101]

X. Li, G. Ouyang and D. A. Richards, Predictability analysis of absence seizures with permutation entropy,, Epilepsy Res., 77 (2007), 70. doi: 10.1016/j.eplepsyres.2007.08.002. Google Scholar

[102]

J. Li, J. Yan, X. Liu and G. Ouyang, Using permutation entropy to measure the changes in eeg signals during absence seizures,, Entropy, 16 (2014), 3049. doi: 10.3390/e16063049. Google Scholar

[103]

H. Li, K. Zhang and T. Jiang, Minimum entropy clustering and applications to gene expression analysis,, in Proc. IEEE Comput. Syst. Bioinform. Conf., (2004), 142. Google Scholar

[104]

Z. Liang, Y. Wang, X. Sun, D. Li, L. J. Voss, J. W. Sleigh, S. Hagihira and X. Li, EEG entropy measures in anesthesia,, Front. Comput. Neurosci., 9 (2015). doi: 10.3389/fncom.2015.00016. Google Scholar

[105]

D. Lind and B. Marcus, An Introduction to Symbolic Dynamics and Coding,, Cambridge University Press, (1995). doi: 10.1017/CBO9780511626302. Google Scholar

[106]

J. Llibre, Brief survey on the topological entropy,, Discr. Contin. Dyn. Syst. B, 20 (2015), 3363. Google Scholar

[107]

N. Mammone, F. La Foresta and F. C. Morabito, Automatic artifact rejection from multichannel scalp EEG by wavelet ICA,, IEEE Sens. J., 12 (2012), 533. doi: 10.1109/JSEN.2011.2115236. Google Scholar

[108]

M. Matilla-García, A non-parametric test for independence based on symbolic dynamics,, J. Econ. Dyn. Control, 31 (2007), 3889. doi: 10.1016/j.jedc.2007.01.018. Google Scholar

[109]

M. Matilla-García and M. Ruiz Marín, A non-parametric independence test using permutation entropy,, J. Econ., 144 (2008), 139. doi: 10.1016/j.jeconom.2007.12.005. Google Scholar

[110]

A. M. Mesón and F. Vericat, On the Kolmogorov-like generalization of Tsallis entropy, correlation entropies and multifractal analysis,, J. Math. Phys., 43 (2002), 904. doi: 10.1063/1.1429323. Google Scholar

[111]

R. Miles and T. Ward, Directional uniformities, periodic points, and entropy,, Discr. Contin. Dyn. Syst. B, 20 (2015), 3525. Google Scholar

[112]

J. Milnor, On the entropy geometry of cellular automata,, Complex Systems, 2 (1988), 357. Google Scholar

[113]

J. Milnor and W. Thurston, On iterated maps of the interval,, in Dynamical Systems (ed. J. C. Alexander), (1342), 465. doi: 10.1007/BFb0082847. Google Scholar

[114]

M. Misiurewicz and W. Szlenk, Entropy of piecewise monotone mappings,, Studia Math., 67 (1980), 45. Google Scholar

[115]

M. Misiurewicz, Permutations and topological entropy for interval maps,, Nonlinearity, 16 (2003), 971. doi: 10.1088/0951-7715/16/3/310. Google Scholar

[116]

R. Monetti, J. M. Amigó, T. Aschenbrenner and W. Bunk, Permutation complexity of interacting dynamical systems,, Eur. Phys. J. Special Topics, 222 (2013), 421. doi: 10.1140/epjst/e2013-01850-y. Google Scholar

[117]

R. Monetti, W. Bunk, T. Aschenbrenner, S. Springer and J. M. Amigó, Information directionality in coupled time series using transcripts,, Phys. Rev. E, 88 (2013). doi: 10.1103/PhysRevE.88.022911. Google Scholar

[118]

F. C. Morabito, D. Labate, F. La Foresta, A. Bramanti, G. Morabito and I. Palamara, Multivariate multi-scale permutation entropy for complexity analysis of Alzheimer's disease EEG,, Entropy, 14 (2012), 1186. doi: 10.3390/e14071186. Google Scholar

[119]

N. Nicolaou and J. Georgiou, The use of permutation entropy to characterize sleep electroencephalograms,, Clin. EEG Neurosci., 42 (2011), 24. doi: 10.1177/155005941104200107. Google Scholar

[120]

D. Ornstein, Two Bernoulli shifts with the same entropy are isomorphic,, Adv. Math., 4 (1970), 337. doi: 10.1016/0001-8708(70)90029-0. Google Scholar

[121]

D. Ornstein, Two Bernoulli shifts with infinite entropy are isomorphic,, Adv. Math., 5 (1970), 339. doi: 10.1016/0001-8708(70)90008-3. Google Scholar

[122]

D. Ornstein and B. Weiss, Entropy and isomorphism theorem for actions of amenable groups,, J. Analyse Math., 48 (1987), 1. doi: 10.1007/BF02790325. Google Scholar

[123]

D. Ornstein, Newton's law and coin tossing,, Notices Amer. Math. Soc., 60 (2013), 450. doi: 10.1090/noti974. Google Scholar

[124]

D. Ornstein and B. Weiss, Entropy is the only finitely observable invariant,, J. Mod. Dyn., 1 (2007), 93. Google Scholar

[125]

G. Ouyang, C. Dang, D. A. Richards and X. Li, Ordinal pattern based similarity analysis for EEG recordings,, Clin. Neurophysiol, 121 (2010), 694. doi: 10.1016/j.clinph.2009.12.030. Google Scholar

[126]

S. Y. Park and A. K. Bera, Maximum entropy autoregressive conditional heretoskedasticity model,, J. Econometrics, 150 (2009), 219. doi: 10.1016/j.jeconom.2008.12.014. Google Scholar

[127]

U. Parlitz, S. Berg, S. Luther, A. Schirdewan, J. Kurths and N. Wessel, Classifying cardiac biosignals using ordinal pattern statistics and symbolic dynamics,, Comput. Biol. Med., 42 (2012), 319. doi: 10.1016/j.compbiomed.2011.03.017. Google Scholar

[128]

W. Parry, Intrinsic Markov chains,, Trans. Amer. Math. Soc., 112 (1964), 55. doi: 10.1090/S0002-9947-1964-0161372-1. Google Scholar

[129]

Y. Pesin, Characteristic Lyapunov exponents and smooth ergodic theory,, Russian Math. Surveys, 32 (1977), 55. Google Scholar

[130]

S. M. Pincus, Approximate entropy as a measure of system complexity,, PNAS, 88 (1991), 2297. doi: 10.1073/pnas.88.6.2297. Google Scholar

[131]

S. M. Pincus and R. R. Viscarello, Approximate entropy: A regularity measure for fetal heart rate analysis,, Obstet. Gynecol., 79 (1992), 249. Google Scholar

[132]

S. M. Pincus, Approximate entropy as a measure of irregularity for psychiatric serial metrics,, Bipolar Disord., 8 (2006), 430. doi: 10.1111/j.1399-5618.2006.00375.x. Google Scholar

[133]

B. Pompe and J. Runge, Momentary information transfer as a coupling measure of time series,, Phys. Rev. E, 83 (2011). doi: 10.1103/PhysRevE.83.051122. Google Scholar

[134]

J. Poza, R. Hornero, J. Escudero, A. Fernández and C. A. Sánchez, Regional analysis of spontaneous MEG rhythms in patients with Alzheimer's disease using spectral entropies,, Ann. Biomed. Eng., 36 (2008), 141. doi: 10.1007/s10439-007-9402-y. Google Scholar

[135]

J. C. Principe, Information Theoretic Learning. Renyi's Entropy and Kernel Perspectives,, Springer, (2010). doi: 10.1007/978-1-4419-1570-2. Google Scholar

[136]

A. Rényi, On measures of entropy and information,, in Proceedings of the 4th Berkeley Symposium on Mathematics, (1961), 547. Google Scholar

[137]

J. S. Richman and J. R. Moorman, Physiological time-series analysis using approximate entropy and sample entropy,, Am. J. Physiol.-Heart C., 278 (2000), 2039. Google Scholar

[138]

S. J. Roberts, R. Everson and I. Rezek, Minimum entropy data partitioning,, in Proceedings of International Conference on Artificial Neural Networks, (1999), 844. Google Scholar

[139]

E. A. Robinson and A. Şahin, Rank-one $\mathbbZ^d$ actions and directional entropy,, Ergod. Theor. Dynam. Syst., 31 (2011), 285. doi: 10.1017/S0143385709000911. Google Scholar

[140]

O. A. Rosso, M. T. Martin and A. Plastino, Brain electrical activity analysis using wavelet-based informational tools,, Physica A, 313 (2002), 587. doi: 10.1016/S0378-4371(02)00958-5. Google Scholar

[141]

D. Rudolph and B. Weiss, Entropy and mixing for amenable group actions,, Ann. of Math., 151 (2000), 1119. doi: 10.2307/121130. Google Scholar

[142]

D. Ruelle, Statistical mechanics on a compact set with $Z^{\nu}$ action satisfying expansiveness and specification,, Trans. Amer. Math. Soc., 187 (1973), 237. Google Scholar

[143]

J. Runge, J. Heitzig, N. Marwan and J. Kurths, Quantifying causal coupling strength: A lag-specific measure for multivariate time series related to transfer entropy,, Phys. Rev. E, 86 (2012). Google Scholar

[144]

X. San Liang, Unraveling the cause-effect relation between time series,, Phys. Rev. E, 90 (2014). Google Scholar

[145]

T. Schreiber, Measuring information transfer,, Phys. Rev. Lett., 85 (2000), 461. doi: 10.1103/PhysRevLett.85.461. Google Scholar

[146]

C. E. Shannon, A mathematical theory of communication,, Bell Syst. Tech. J., 27 (1948), 379. doi: 10.1002/j.1538-7305.1948.tb01338.x. Google Scholar

[147]

Y. G. Sinai, On the notion of entropy of dynamical systems,, Dokl. Acad. Sci. USSR, 125 (1959), 768. Google Scholar

[148]

Y. G. Sinai, Flows with finite entropy,, Dokl. Acad. Sci. USSR, 125 (1959), 1200. Google Scholar

[149]

Y. G. Sinai, Gibbs measures in ergodic theory,, Uspehi Mat. Nauk, 27 (1972), 21. Google Scholar

[150]

D. Smirnov, Spurious causalities with transfer entropy,, Phys. Rev. E, 87 (2013). Google Scholar

[151]

M. Smorodinsky, Ergodic Theory, Entropy,, Lectures Notes in Mathematics, (1971). Google Scholar

[152]

R. Sneddon, The Tsallis entropy of natural information,, Physica A, 386 (2007), 101. doi: 10.1016/j.physa.2007.05.065. Google Scholar

[153]

V. Srinivasan, C. Eswaran and N. Sriraam, Approximate entropy-based epileptic EEG detection using artificial neural networks,, IEEE T. Inf. Technol. B., 11 (2007), 288. doi: 10.1109/TITB.2006.884369. Google Scholar

[154]

H. Suyari, Generalization of Shannon-Khinchin axioms to nonextensive systems and the uniqueness theorem for the nonextensive entropy,, IEEE T. Inform. Theory, 50 (2004), 1783. doi: 10.1109/TIT.2004.831749. Google Scholar

[155]

F. Takens, Detecting strange attractors in turbulence,, in Dynamical Systems and Turbulence (eds. D. A. Rand and L. S. Young), (1981), 366. Google Scholar

[156]

F. Takens and E. Verbitskiy, Rényi entropies of aperiodic dynamical systems,, Israel J. Math., 127 (2002), 279. doi: 10.1007/BF02784535. Google Scholar

[157]

F. Takens and E. Verbitskiy, Generalized entropies: Rényi and correlation integral approach,, Nonlinearity, 11 (1998), 771. doi: 10.1088/0951-7715/11/4/001. Google Scholar

[158]

S. Tong, A. Bezerianos, A. Malhotra, Y. Zhu and N. Thakor, Parameterized entropy analysis of EEG following hypoxicischemic brain injury,, Phys. Lett. A, 314 (2003), 354. doi: 10.1016/S0375-9601(03)00949-6. Google Scholar

[159]

M. E. Torres and L. G. Gamero, Relative complexity changes in time series using information measures,, Physica A, 286 (2000), 457. doi: 10.1016/S0378-4371(00)00309-5. Google Scholar

[160]

B. Tóthmérész, Comparison of different methods for diversity ordering,, J. Veg. Sci., 6 (1995), 283. Google Scholar

[161]

C. Tsallis, Possible generalization of Boltzmann-Gibbs statistics,, J. Stat. Phys., 52 (1988), 479. doi: 10.1007/BF01016429. Google Scholar

[162]

C. Tsallis, The nonadditive entropy Sq and its applications in physics and elsewhere: Some remarks,, Entropy, 13 (2011), 1765. doi: 10.3390/e13101765. Google Scholar

[163]

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

[164]

V. A. Unakafova, Investigating Measures of Complexity for Dynamical Systems and for Time Series,, Ph.D thesis, (2015). Google Scholar

[165]

E. A. Verbitskiy, Generalized Entropies in Dynamical Systems,, Ph.D thesis, (2000). Google Scholar

[166]

A. Voss, S. Schulz, R. Schroeder, M. Baumert and P. Caminal, Methods derived from nonlinear dynamics for analysing heart rate variability,, Phil. Trans. R. Soc. A, 367 (2009), 277. doi: 10.1098/rsta.2008.0232. Google Scholar

[167]

P. Walters, An Introduction to Ergodic Theory,, Springer Verlag, (1982). Google Scholar

[168]

B. Weiss, Subshifts of finite type and sofic systems,, Monats. Math., 77 (1973), 462. doi: 10.1007/BF01295322. Google Scholar

[169]

B. Weiss, Entropy and actions of sofic groups,, Discr. Contin. Dyn. Syst. B, 20 (2015), 3375. Google Scholar

[170]

N. Wiener, The theory of prediction,, in Modern Mathematics for the Engineer (ed. E. F. Beckenbach), (1956). Google Scholar

[171]

A. D. Wissner-Gross and C. E. Freer, Causal entropic forces,, Phys. Rev. Lett., 110 (2013). doi: 10.1103/PhysRevLett.110.168702. Google Scholar

[172]

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. doi: 10.3390/e14081553. Google Scholar

[173]

D. Zhang, X. Jia, H. Ding, D. Ye and N. V. Thakor, Application of Tsallis entropy to EEG: Quantifying the presence of burst suppression after asphyxial cardiac arrest in rats,, IEEE Trans. Bio.-Med. Eng., 57 (2010), 867. Google Scholar

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