Citation: |
[1] |
S. Abe, Tsallis entropy: How unique?, Contin. Mech. Thermodyn., 16 (2004), 237-244.doi: 10.1007/s00161-003-0153-1. |
[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), 37-45.doi: 10.1016/j.cmpb.2005.06.011. |
[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), 121-165.doi: 10.1007/978-3-540-36675-1_5. |
[4] |
J. Aczél and Z. Daróczy, Charakterisierung der Entropien positiver Ordnung und der Shannonschen Entropie, Acta Math. Acad. Sci. Hung., 14 (1963), 95-121.doi: 10.1007/BF01901932. |
[5] |
R. Adler, A. Konheim and M. McAndrew, Topological entropy, Trans. Amer. Mat. Soc., 114 (1965), 309-319.doi: 10.1090/S0002-9947-1965-0175106-9. |
[6] |
R. L. Adler and B. Marcus, Topological entropy and the equivalence of dynamical systems, Mem. Amer. Math. Soc., 20 (1979), iv+84 pp.doi: 10.1090/memo/0219. |
[7] |
V. Afraimovich, M. Courbage and L. Glebsky, Directional complexity and entropy for lift mappings, Discr. Contin. Dyn. Syst. B, 20 (2015), 3385-3401. |
[8] |
L. Alsedà, J. Llibre and M. Misiurewicz, Combinatorial Dynamics and Entropy in Dimension One, World Scientific, Singapore, 2000.doi: 10.1142/4205. |
[9] |
J. M. Amigó, L. Kocarev and J. Szczepanski, Order patterns and chaos, Phys. Lett. A, 355 (2006), 27-31. |
[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), 50001, 5pp.doi: 10.1209/0295-5075/79/50001. |
[11] |
J. M. Amigó and M. B. Kennel, Forbidden ordinal patterns in higher dimensional dynamics, Physica D, 237 (2008), 2893-2899.doi: 10.1016/j.physd.2008.05.003. |
[12] |
J. M. Amigó, S. Zambrano and M. A. F. Sanjuán, Combinatorial detection of determinism in noisy time series, Europhys. Lett., 82 (2008), 60005. |
[13] |
J. M. Amigó, Permutation Complexity in Dynamical Systems-Ordinal Patterns, Permutation Entropy, and All That, Springer Series in Synergetics, Springer, Berlin Heidelberg, 2010.doi: 10.1007/978-3-642-04084-9. |
[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), 2915-2924.doi: 10.1142/S0218127410027453. |
[15] |
J. M. Amigó, R. Monetti, T. Aschenbrenner and W. Bunk, Transcripts: An algebraic approach to coupled time series, Chaos, 22 (2012), 013105, 13pp.doi: 10.1063/1.3673238. |
[16] |
J. M. Amigó, The equality of Kolmogorov-Sinai entropy and metric permutation entropy generalized, Physica D, 241 (2012), 789-793.doi: 10.1016/j.physd.2012.01.004. |
[17] |
J. M. Amigó and K. Keller, Permutation entropy: One concept, two approaches, Eur. Phys. J. Special Topics, 222 (2013), 263-273. |
[18] |
J. M. Amigó, P. Kloeden and A. Giménez, Switching systems and entropy, J. Diff. Eq. Appl., 19 (2013), 1872-1888.doi: 10.1080/10236198.2013.788166. |
[19] |
J. M. Amigó, P. E. Kloeden and A. Giménez, Entropy increase in switching systems, Entropy, 15 (2013), 2363-2383.doi: 10.3390/e15062363. |
[20] |
J. M. Amigó, T. Aschenbrenner, W. Bunk and R. Monetti, Dimensional reduction of conditional algebraic multi-information via transcripts, Inform. Sciences, 278 (2014), 298-310.doi: 10.1016/j.ins.2014.03.054. |
[21] |
J. M. Amigó and A. Giménez, A simplified algorithm for the topological entropy of multimodal maps, Entropy, 16 (2014), 627-644.doi: 10.3390/e16020627. |
[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), 20140091, 18pp.doi: 10.1098/rsta.2014.0091. |
[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), 3415-3434. |
[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), 061907.doi: 10.1103/PhysRevE.64.061907. |
[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), 1793-1809. |
[26] |
R. B. Ash, Information Theory, Dover Publications, New York, 1990. |
[27] |
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. |
[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, Systems and Computers, Pacific Grove, 2008, IEEE Explore, 2008, 1698-1704.doi: 10.1109/ACSSC.2008.5074715. |
[29] |
C. H. Bennet, Notes on Landauer's principle, reversible computation and Maxwell's demon, Stud. Hist. Philos. M. P., 34 (2003), 501-510.doi: 10.1016/S1355-2198(03)00039-X. |
[30] |
G. D. Birkhoff, Proof of a recurrence theorem for strongly transitive systems, PNAS, 17 (1931), 650-655.doi: 10.1073/pnas.17.12.650. |
[31] |
S. A. Borovkova, Estimation and Prediction for Nonlinear Time Series, Ph.D thesis, University of Groningen, 1998. |
[32] |
M. Boyle and D. Lind, Expansive subdynamics, Trans. Amer. Math. Soc., 349 (1997), 55-102.doi: 10.1090/S0002-9947-97-01634-6. |
[33] |
R. Bowen, Entropy for group endomorphisms and homogeneous spaces, Trans. Amer. Math. Soc., 153 (1971), 401-414.doi: 10.1090/S0002-9947-1971-0274707-X. |
[34] |
L. Bowen, A measure-conjugacy invariant for free group actions, Ann. of Math., 171 (2010), 1387-1400.doi: 10.4007/annals.2010.171.1387. |
[35] |
L. Bowen, Measure-conjugacy invariants for actions of countable sofic groups, J. Amer. Math. Soc., 23 (2010), 217-245.doi: 10.1090/S0894-0347-09-00637-7. |
[36] |
H. W. Broer and F. Takens, Dynamical Systems and Chaos, Applied Mathematical Sciences, 172, Springer, New York, 2011.doi: 10.1007/978-1-4419-6870-8. |
[37] |
M. Brin and A. Katok, On local entropy, in Geometric dynamics (ed. J. Palis), Lecture Notes in Mathematics, 1007, Springer, Berlin Heidelberg, 1983, 30-38.doi: 10.1007/BFb0061408. |
[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), 3-9.doi: 10.1007/s10072-008-0851-3. |
[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), 21-24.doi: 10.1177/155005940503600106. |
[40] |
C. Cafaro, W. M. Lord, J. Sun and E. M. Bollt, Causation entropy from symbolic representations of dynamical systems, Chaos, 25 (2015), 043106. |
[41] |
J. S. Cánovas and A. Guillamón, Permutations and time series analysis, Chaos, 19 (2009), 043103, 12pp.doi: 10.1063/1.3238256. |
[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), 046217, 7pp.doi: 10.1103/PhysRevE.70.046217. |
[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), 149-155.doi: 10.1016/S0378-4371(98)00137-X. |
[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), 235-254.doi: 10.1016/S0378-4371(98)00471-3. |
[45] |
D. Carrasco-Olivera, R. Metzger and C. A. Morales, Topological entropy for set-valued maps, Discr. Contin. Dyn. Syst. B, 20 (2015), 3461-3474. |
[46] |
H. C. Choe, Computational Ergodic Theory, Algorithms and Computation in Mathematics, 13, Springer, Berlin Heidelberg, 2005. |
[47] |
R. Clausius, The Mechanical Theory of Heat, MacMillan and Co., London, 1865. |
[48] |
M. Courbage and B. Kamiński, Space-time directional Lyapunov exponents for cellular automata, J. Statis. Phys., 124 (2006), 1499-1509.doi: 10.1007/s10955-006-9172-1. |
[49] |
T. M. Cover and J. A. Thomas, Elements of Information Theory, John Wiley & Sons, New York, 2006. |
[50] |
B. Dai and B. Hu, Minimum conditional entropy clustering: A discriminative framework for clustering, in JMLR: Workshop and Conference Proceedings, 13, Tokyo, 2010, 47-62. |
[51] |
K. Denbigh, How subjective is entropy, in Maxwell's Demon, Entropy, Information, Computing (eds. H. S. Leff and A. F. Rex), Princeton University Press, Princeton, 1990, 109-115. |
[52] |
M. Denker, Finite generators for ergodic, measure-preserving transformation, Zeit. Wahr. ver. Geb., 29 (1974), 45-55.doi: 10.1007/BF00533186. |
[53] |
M. Denker, C. Grillenberger and K. Sigmund, Ergodic Theory on Compact Spaces, Lecture Notes in Mathematics, {527}, Springer, Berlin Heidelberg, 1976. |
[54] |
M. Denker and G. Keller, Rigorous statistical procedures for data from dynamical systems, J. Stat. Phys., 44 (1986), 67-93.doi: 10.1007/BF01010905. |
[55] |
E. I. Dinaburg, The relation between topological entropy and metric entropy, Soviet Math., 11 (1970), 13-16. |
[56] |
R. J. V. dos Santos, Generalization of Shannon's theorem for Tsallis entropy, J. Math. Phys., 38 (1997), 4104-4107.doi: 10.1063/1.532107. |
[57] |
J.-P. Eckmann and D. Ruelle, Ergodic theory of chaos and strange attractors, Rev. Mod. Phys., 57 (1985), 617-656.doi: 10.1103/RevModPhys.57.617. |
[58] |
M. Einsiedler and T. Ward, Ergodic Theory with a View Towards Number Theory, Graduate Texts in Mathematics, {259}, Springer, London, 2011.doi: 10.1007/978-0-85729-021-2. |
[59] |
D. K. Faddeev, On the notion of entropy of finite probability distributions, Usp. Mat. Nauk (in Russian), 11 (1956), 227-231. |
[60] |
F. Falniowski, On the connections of the generalized entropies and Kolmogorov-Sinai entropies, Entropy, 16 (2014), 3732-3753.doi: 10.3390/e16073732. |
[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), 179-187.doi: 10.1007/s11517-005-0015-z. |
[62] |
S. Furuichi, On uniqueness Theorems for Tsallis entropy and Tsallis relative entropy, IEEE Trans. Inf. Theory, 51 (2005), 3638-3645.doi: 10.1109/TIT.2005.855606. |
[63] |
L. G. Gamero, A. Plastino and M. E. Torres, Wavelet analysis and nonlinear dynamics in a nonextensive setting, Physica A, 246 (1997), 487-509.doi: 10.1016/S0378-4371(97)00367-1. |
[64] |
P. G. Gaspard and X. J. Wang, Noise, chaos, and $(\varepsilon,\tau)$-entropy per unit time, Phys. Rep., 235 (1993), 291-343.doi: 10.1016/0370-1573(93)90012-3. |
[65] |
T. N. T. Goodman, Relating topological entropy and measure entropy, Bull. London Math. Soc., 3 (1971), 176-180.doi: 10.1112/blms/3.2.176. |
[66] |
N. Gradojevic and R. Genccay, Financial applications of nonextensive entropy, IEEE Signal Process. Mag., 28 (2011), p116. |
[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), 153-158. |
[68] |
C. W. J. Granger, Investigating causal relations by econometric models and cross-spectral methods, Econometrica, 37 (1969), 424-438. |
[69] |
P. Grassberger and I. Procaccia, Dimensions and entropies of strange attractors from a fluctuating dynamics approach, Physica D, 13 (1984), 34-54.doi: 10.1016/0167-2789(84)90269-0. |
[70] |
P. Grassberger and I. Procaccia, Measuring the strangeness of strange attractors, Physica D, 9 (1983), 189-208.doi: 10.1016/0167-2789(83)90298-1. |
[71] |
P. Grassberger and I. Procaccia, Characterization of strange attractors, Phys. Rev. Lett., 50 (1983), 346-349.doi: 10.1103/PhysRevLett.50.346. |
[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), 128701. |
[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), 20006. |
[74] |
R. Hanel, S. Thurner and M. Gell-Mann, Generalized entropies and logarithms and their duality relations, Proceed. Nat. Acad. Scie., 109 (2012), 19151-19154.doi: 10.1073/pnas.1216885109. |
[75] |
B. Hasselblatt and A. Katok, Principal structures, in Handbook of Dynamical Systems (eds. B. Hasselblatt and A. Katok), 1A, North Holland, Amsterdam, 2002, 1-203.doi: 10.1016/S1874-575X(02)80003-0. |
[76] |
J. Havrda and F. Charvát, Quantification method of classification processes. Concept of structural $\alpha$-entropy, Kybernetika, 3 (1967), 30-35. |
[77] |
G. A. Hedlund, Endomorphisms and automorphisms of the shift dynamical system, Math. Syst. Theory, 3 (1969), 320-375.doi: 10.1007/BF01691062. |
[78] |
H. G. E. Hentschel and I. Procaccia, The infinite number of generalized dimensions of fractals and strange attractors, Physica D, 8 (1983), 435-444.doi: 10.1016/0167-2789(83)90235-X. |
[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), 317-336.doi: 10.1098/rsta.2008.0197. |
[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), 6950-6952. |
[81] |
E. T. Jaynes, Information theory and statistical mechanics, Phys. Rev., 106 (1957), 620-630.doi: 10.1103/PhysRev.106.620. |
[82] |
P. Jizba and T. Arimitsu, The world according to Rényi: Thermodynamics of multifractal systems, Ann. Phys., 312 (2004), 17-59.doi: 10.1016/j.aop.2004.01.002. |
[83] |
C. C. Jouny and G. K. Bergey, Characterization of early partial seizure onset: Frequency, complexity and entropy, Clin. Neurophysiol., 123 (2012), 658-669.doi: 10.1016/j.clinph.2011.08.003. |
[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), 187-194.doi: 10.1016/j.cmpb.2005.06.012. |
[85] |
H. Kantz and T. Schreiber, Nonlinear Time Series Analysis, Cambridge University Press, Cambridge, 2004. |
[86] |
A. Katok, Fifty years of entropy in dynamics: 1978-2007, J. Mod. Dyn., 1 (2007), 545-596.doi: 10.3934/jmd.2007.1.545. |
[87] |
K. Keller and H. Lauffer, Symbolic analysis of high-dimensional time series, Int. J. Bif. Chaos, 13 (2003), 2657-2668.doi: 10.1142/S0218127403008168. |
[88] |
K. Keller, Permutations and the Kolmogorov-Sinai entropy, Discr. Contin. Dyn. Syst. A, 32 (2012), 891-900.doi: 10.3934/dcds.2012.32.891. |
[89] |
K. Keller, A. M. Unakafov and V. A. Unakafova, Ordinal Patterns, Entropy, and EEG, Entropy, 16 (2014), 6212-6239.doi: 10.3390/e16126212. |
[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), 3507-3524. |
[91] |
A. I. Khinchin, Mathematical Foundations of Information Theory, Dover, New York, 1957. |
[92] |
A. N. Kolmogorov, A new metric invariant of transitive dynamical systems and Lebesgue space endomorphisms, Dokl. Acad. Sci. USSR, 119 (1958), 861-864. |
[93] |
S. Kolyada and L. Snoha, Topological entropy of nonautonomous dynamical systems, Random Comput. Dynam., 4 (1996), 205-233. |
[94] |
Z. S. Kowalski, Finite generators of ergodic endomorphisms, Colloq. Math., 49 (1984), 87-89. |
[95] |
Z. S. Kowalski, Minimal generators for aperiodic endomorphisms, Commentat. Math. Univ. Carol., 36 (1995), 721-725. |
[96] |
W. Krieger, On entropy and generators of measure-preserving transformations, Trans. Amer. Math. Soc., 149 (1970), 453-464.doi: 10.1090/S0002-9947-1970-0259068-3. |
[97] |
J. Kurths, A. Voss, P. Saparin, A. Witt, H. J. Kleiner and N. Wessel, Quantitative analysis of heart rate variability, Chaos, 5 (1995), 88-94.doi: 10.1063/1.166090. |
[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), 789-797.doi: 10.1152/ajpregu.00069.2002. |
[99] |
A. M. Law and W. D. Kelton, Simulation, Modeling, and Analysis, McGraw-Hill, Boston, 2000. |
[100] |
F. Ledrappier and L.-S. Young, The metric entropy of diffeomorphisms. II. Relations betweenentropy, exponents and dimension, Ann. Math., 122 (1985), 540-574.doi: 10.2307/1971329. |
[101] |
X. Li, G. Ouyang and D. A. Richards, Predictability analysis of absence seizures with permutation entropy, Epilepsy Res., 77 (2007), 70-74.doi: 10.1016/j.eplepsyres.2007.08.002. |
[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), 3049-3061.doi: 10.3390/e16063049. |
[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, 142-151. |
[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), p16.doi: 10.3389/fncom.2015.00016. |
[105] |
D. Lind and B. Marcus, An Introduction to Symbolic Dynamics and Coding, Cambridge University Press, Cambridge, 1995.doi: 10.1017/CBO9780511626302. |
[106] |
J. Llibre, Brief survey on the topological entropy, Discr. Contin. Dyn. Syst. B, 20 (2015), 3363-3374. |
[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), 533-542.doi: 10.1109/JSEN.2011.2115236. |
[108] |
M. Matilla-García, A non-parametric test for independence based on symbolic dynamics, J. Econ. Dyn. Control, 31 (2007), 3889-3903.doi: 10.1016/j.jedc.2007.01.018. |
[109] |
M. Matilla-García and M. Ruiz Marín, A non-parametric independence test using permutation entropy, J. Econ., 144 (2008), 139-155.doi: 10.1016/j.jeconom.2007.12.005. |
[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), 904-917.doi: 10.1063/1.1429323. |
[111] |
R. Miles and T. Ward, Directional uniformities, periodic points, and entropy, Discr. Contin. Dyn. Syst. B, 20 (2015), 3525-3545. |
[112] |
J. Milnor, On the entropy geometry of cellular automata, Complex Systems, 2 (1988), 357-385. |
[113] |
J. Milnor and W. Thurston, On iterated maps of the interval, in Dynamical Systems (ed. J. C. Alexander), Lectures Notes in Mathematics, 1342, Springer, 1988, 465-563.doi: 10.1007/BFb0082847. |
[114] |
M. Misiurewicz and W. Szlenk, Entropy of piecewise monotone mappings, Studia Math., 67 (1980), 45-63. |
[115] |
M. Misiurewicz, Permutations and topological entropy for interval maps, Nonlinearity, 16 (2003), 971-976.doi: 10.1088/0951-7715/16/3/310. |
[116] |
R. Monetti, J. M. Amigó, T. Aschenbrenner and W. Bunk, Permutation complexity of interacting dynamical systems, Eur. Phys. J. Special Topics, 222 (2013), 421-436.doi: 10.1140/epjst/e2013-01850-y. |
[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), 022911.doi: 10.1103/PhysRevE.88.022911. |
[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), 1186-1202.doi: 10.3390/e14071186. |
[119] |
N. Nicolaou and J. Georgiou, The use of permutation entropy to characterize sleep electroencephalograms, Clin. EEG Neurosci., 42 (2011), 24-28.doi: 10.1177/155005941104200107. |
[120] |
D. Ornstein, Two Bernoulli shifts with the same entropy are isomorphic, Adv. Math., 4 (1970), 337-352.doi: 10.1016/0001-8708(70)90029-0. |
[121] |
D. Ornstein, Two Bernoulli shifts with infinite entropy are isomorphic, Adv. Math., 5 (1970), 339-348.doi: 10.1016/0001-8708(70)90008-3. |
[122] |
D. Ornstein and B. Weiss, Entropy and isomorphism theorem for actions of amenable groups, J. Analyse Math., 48 (1987), 1-141.doi: 10.1007/BF02790325. |
[123] |
D. Ornstein, Newton's law and coin tossing, Notices Amer. Math. Soc., 60 (2013), 450-459.doi: 10.1090/noti974. |
[124] |
D. Ornstein and B. Weiss, Entropy is the only finitely observable invariant, J. Mod. Dyn., 1 (2007), 93-105. |
[125] |
G. Ouyang, C. Dang, D. A. Richards and X. Li, Ordinal pattern based similarity analysis for EEG recordings, Clin. Neurophysiol, 121 (2010), 694-703.doi: 10.1016/j.clinph.2009.12.030. |
[126] |
S. Y. Park and A. K. Bera, Maximum entropy autoregressive conditional heretoskedasticity model, J. Econometrics, 150 (2009), 219-230.doi: 10.1016/j.jeconom.2008.12.014. |
[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), 319-327.doi: 10.1016/j.compbiomed.2011.03.017. |
[128] |
W. Parry, Intrinsic Markov chains, Trans. Amer. Math. Soc., 112 (1964), 55-66.doi: 10.1090/S0002-9947-1964-0161372-1. |
[129] |
Y. Pesin, Characteristic Lyapunov exponents and smooth ergodic theory, Russian Math. Surveys, 32 (1977), 55-112. |
[130] |
S. M. Pincus, Approximate entropy as a measure of system complexity, PNAS, 88 (1991), 2297-2301.doi: 10.1073/pnas.88.6.2297. |
[131] |
S. M. Pincus and R. R. Viscarello, Approximate entropy: A regularity measure for fetal heart rate analysis, Obstet. Gynecol., 79 (1992), 249-255. |
[132] |
S. M. Pincus, Approximate entropy as a measure of irregularity for psychiatric serial metrics, Bipolar Disord., 8 (2006), 430-440.doi: 10.1111/j.1399-5618.2006.00375.x. |
[133] |
B. Pompe and J. Runge, Momentary information transfer as a coupling measure of time series, Phys. Rev. E, 83 (2011), 051122.doi: 10.1103/PhysRevE.83.051122. |
[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), 141-152.doi: 10.1007/s10439-007-9402-y. |
[135] |
J. C. Principe, Information Theoretic Learning. Renyi's Entropy and Kernel Perspectives, Springer, New York, 2010.doi: 10.1007/978-1-4419-1570-2. |
[136] |
A. Rényi, On measures of entropy and information, in Proceedings of the 4th Berkeley Symposium on Mathematics, Statistics and Probability (ed. J. Neyman), University of California Press, Berkeley, 1961, 547-561. |
[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), 2039-2049. |
[138] |
S. J. Roberts, R. Everson and I. Rezek, Minimum entropy data partitioning, in Proceedings of International Conference on Artificial Neural Networks, 1999, 844-849. |
[139] |
E. A. Robinson and A. Şahin, Rank-one $\mathbbZ^d$ actions and directional entropy, Ergod. Theor. Dynam. Syst., 31 (2011), 285-299.doi: 10.1017/S0143385709000911. |
[140] |
O. A. Rosso, M. T. Martin and A. Plastino, Brain electrical activity analysis using wavelet-based informational tools, Physica A, 313 (2002), 587-608.doi: 10.1016/S0378-4371(02)00958-5. |
[141] |
D. Rudolph and B. Weiss, Entropy and mixing for amenable group actions, Ann. of Math., 151 (2000), 1119-1150.doi: 10.2307/121130. |
[142] |
D. Ruelle, Statistical mechanics on a compact set with $Z^{\nu}$ action satisfying expansiveness and specification, Trans. Amer. Math. Soc., 187 (1973), 237-251. |
[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), 061121. |
[144] |
X. San Liang, Unraveling the cause-effect relation between time series, Phys. Rev. E, 90 (2014), 052150. |
[145] |
T. Schreiber, Measuring information transfer, Phys. Rev. Lett., 85 (2000), 461-464.doi: 10.1103/PhysRevLett.85.461. |
[146] |
C. E. Shannon, A mathematical theory of communication, Bell Syst. Tech. J., 27 (1948), 379-423, 623-653.doi: 10.1002/j.1538-7305.1948.tb01338.x. |
[147] |
Y. G. Sinai, On the notion of entropy of dynamical systems, Dokl. Acad. Sci. USSR, 125 (1959), 768-771. |
[148] |
Y. G. Sinai, Flows with finite entropy, Dokl. Acad. Sci. USSR, 125 (1959), 1200-1202. |
[149] |
Y. G. Sinai, Gibbs measures in ergodic theory, Uspehi Mat. Nauk, 27 (1972), 21-64. |
[150] |
D. Smirnov, Spurious causalities with transfer entropy, Phys. Rev. E, 87 (2013), 042917. |
[151] |
M. Smorodinsky, Ergodic Theory, Entropy, Lectures Notes in Mathematics, 214, Springer, Berlin Heidelberg, 1971. |
[152] |
R. Sneddon, The Tsallis entropy of natural information, Physica A, 386 (2007), 101-118.doi: 10.1016/j.physa.2007.05.065. |
[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), 288-295.doi: 10.1109/TITB.2006.884369. |
[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), 1783-1787.doi: 10.1109/TIT.2004.831749. |
[155] |
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, Berlin Heidelberg, 1981, 366-381. |
[156] |
F. Takens and E. Verbitskiy, Rényi entropies of aperiodic dynamical systems, Israel J. Math., 127 (2002), 279-302.doi: 10.1007/BF02784535. |
[157] |
F. Takens and E. Verbitskiy, Generalized entropies: Rényi and correlation integral approach, Nonlinearity, 11 (1998), 771-782.doi: 10.1088/0951-7715/11/4/001. |
[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), 354-361.doi: 10.1016/S0375-9601(03)00949-6. |
[159] |
M. E. Torres and L. G. Gamero, Relative complexity changes in time series using information measures, Physica A, 286 (2000), 457-473.doi: 10.1016/S0378-4371(00)00309-5. |
[160] |
B. Tóthmérész, Comparison of different methods for diversity ordering, J. Veg. Sci., 6 (1995), 283-290. |
[161] |
C. Tsallis, Possible generalization of Boltzmann-Gibbs statistics, J. Stat. Phys., 52 (1988), 479-487.doi: 10.1007/BF01016429. |
[162] |
C. Tsallis, The nonadditive entropy Sq and its applications in physics and elsewhere: Some remarks, Entropy, 13 (2011), 1765-1804.doi: 10.3390/e13101765. |
[163] |
A. M. Unakafov and K. Keller, Conditional entropy of ordinal patterns, Physica D, 269 (2014), 94-102.doi: 10.1016/j.physd.2013.11.015. |
[164] |
V. A. Unakafova, Investigating Measures of Complexity for Dynamical Systems and for Time Series, Ph.D thesis, University of Luebeck, 2015. |
[165] |
E. A. Verbitskiy, Generalized Entropies in Dynamical Systems, Ph.D thesis, University of Groningen, 2000. |
[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), 277-296.doi: 10.1098/rsta.2008.0232. |
[167] |
P. Walters, An Introduction to Ergodic Theory, Springer Verlag, New York, 1982. |
[168] |
B. Weiss, Subshifts of finite type and sofic systems, Monats. Math., 77 (1973), 462-474.doi: 10.1007/BF01295322. |
[169] |
B. Weiss, Entropy and actions of sofic groups, Discr. Contin. Dyn. Syst. B, 20 (2015), 3375-3383. |
[170] |
N. Wiener, The theory of prediction, in Modern Mathematics for the Engineer (ed. E. F. Beckenbach), McGraw-Hill, New York, 1956. |
[171] |
A. D. Wissner-Gross and C. E. Freer, Causal entropic forces, Phys. Rev. Lett., 110 (2013), 168702.doi: 10.1103/PhysRevLett.110.168702. |
[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), 1553-1577.doi: 10.3390/e14081553. |
[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), 867-874. |