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Detecting coupling directions with transcript mutual information: A comparative study
1. | Centro de Investigación Operativa, Universidad Miguel Hernández, 03202 Elche, Spain |
2. | Department of Hypertension and Diabetology, Medical University of Gdańsk, 80-952 Gdańsk, Poland |
3. | Faculty of Applied Physics and Mathematics, Gdańsk University of Technology, 80-233 Gdańsk, Poland |
4. | IngSoft GmbH. Irrerstrasse 17. 90403 Nuernberg, Germany |
5. | Faculty of Applied Physics and Mathematics, Gdańsk University of Technology, 80-233 Gdańsk, Poland |
Causal relationships are important to understand the dynamics of coupled processes and, moreover, to influence or control the effects by acting on the causes. Among the different approaches to determine cause-effect relationships and, in particular, coupling directions in interacting random or deterministic processes, we focus in this paper on information-theoretic measures. So, we study in the theoretical part the difference between directionality indicators based on transfer entropy as well as on its dimensional reduction via transcripts in algebraic time series representations. In the applications we consider specifically the lowest dimensional case, i.e., 3-dimensional transfer entropy, which is currently one of the most popular causality indicators, and the (2-dimensional) mutual information of transcripts. Needless to say, the lower dimensionality of the transcript-based indicator can make a difference in practice, where datasets are usually small. To compare numerically the performance of both directionality indicators, synthetic data (obtained with random processes) and real world data (in the form of biomedical recordings) are used. As happened in previous related work, we found again that the transcript mutual information performs as good as, and in some cases even better than, the lowest dimensional binned and symbolic transfer entropy, the symbols being ordinal patterns.
References:
[1] |
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. |
[2] |
J. M. Amigó, S. Zambrano and M. A. F. Sanjuán.,
Detecting determinism with ordinal patterns: A comparative study, Int. J. Bifurcation and Chaos, 20 (2010), 2915-2924.
doi: 10.1142/S0218127410027453. |
[3] |
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. |
[4] |
J. M. Amigó, T. Aschenbrenner, W. Bunk and R. Monetti,
Dimensional reduction of conditional algebraic multi-information via transcripts, Inform. Sci., 278 (2014), 298-310.
doi: 10.1016/j.ins.2014.03.054. |
[5] |
J. M. Amigó, K. Keller and V. 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. |
[6] |
J. M. Amigó, R. Monetti, N. Tort-Colet and M. V. Sanchez-Vives,
Infragranular layers lead information flow during slow oscillations according to information directionality indicators, J. Comput. Neurosci, 39 (2015), 53-62.
|
[7] |
J. M. Amigó, R. Monetti, B. Graff and G. Graff, Computing algebraic transfer entropy and coupling directions via transcripts, Chaos, 26 (2016), 113115, 12pp.
doi: 10.1063/1.4967803. |
[8] |
C. Bandt and B. Pompe, Permutation entropy: A natural complexity measure for time series, Phys. Rev. Lett., 88 (2002), 174102. |
[9] |
C. Cafaro, W. M. Lord, J. Sun and E. M. Bollt, Causation entropy from symbolic representations of dynamical systems, Chaos, 25 (2015), 043106, 10pp.
doi: 10.1063/1.4916902. |
[10] |
T. M. Cover and J. A. Thomas, Elements of Information Theory, (second ed.), John Wiley & Sons, Hoboken, 2006. |
[11] |
European-Heart-Network, Cardiovascular Disease Statistics, 2017. |
[12] |
G. Graff, B. Graff, A. Kaczkowska, D. Makowiec, J. M. Amigó, J. Piskorski, K. Narkiewicz and P. Guzik,
Ordinal pattern statistics for the assessment of heart rate variability, Eur. Phys. J. Special Topics, 222 (2013), 525-534.
doi: 10.1140/epjst/e2013-01857-4. |
[13] |
B. Graff, G. Graff, D. Makowiec, A. Kaczkowska, D. Wejer, S. Budrejko, D. Kozƚ owski and K. Narkiewicz,
Entropy measures in the assessment of heart rate variability in patients with cardiodepressive vasovagal syncope, Entropy, 17 (2015), 1007-1022.
doi: 10.3390/e17031007. |
[14] |
C. W. J. Granger,
Investigating causal relations by econometric models and cross-spectral methods, Econometrica, 37 (1969), 424-438.
|
[15] |
Y. Hirata, J. M. Amigó, Y. Matsuzaka, R. Yokota, H. Mushiake and K. Aihara, Detecting causality by combined use of multiple methods: Climate and brain examples, PLos One, 11 (2016), e0158572.
doi: 10.1371/journal.pone.0158572. |
[16] |
R. Monetti, W. Bunk, T. Aschenbrenner and F. Jamitzky, Characterizing synchronization in time series using information measures extracted from symbolic representations, Phys. Rev. E, 79 (2009), 046207.
doi: 10.1103/PhysRevE.79.046207. |
[17] |
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. |
[18] |
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. |
[19] |
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. |
[20] |
A. Porta, A. Catai, A. Takahashi, V. Magagnin, T. Bassani, E. Tobaldini, P. van de Borne and N. Montano, Causal relationships between heart period and systolic arterial pressure during graded head-up tilt, Am. J. Physiol. Regul. Integr. Comp. Physiol, 300 (2011), R378–R386.
doi: 10.1152/ajpregu.00553.2010. |
[21] |
T. Schreiber,
Measuring information transfer, Phys. Rev. Lett., 85 (2000), 461-464.
doi: 10.1103/PhysRevLett.85.461. |
[22] |
D. Smirnov, Spurious causalities with transfer entropy, Phys. Rev. E, 87 (2013), 042917.
doi: 10.1103/PhysRevE.87.042917. |
[23] |
M. Staniek and K. Lehnertz, Symbolic transfer entropy, Phys. Rev. Lett. 100 (2008), 158101.
doi: 10.1103/PhysRevLett.100.158101. |
[24] |
N. Wiener, Modern Mathematics for Engineers, McGraw-Hill, New York, 1956. |
show all references
References:
[1] |
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. |
[2] |
J. M. Amigó, S. Zambrano and M. A. F. Sanjuán.,
Detecting determinism with ordinal patterns: A comparative study, Int. J. Bifurcation and Chaos, 20 (2010), 2915-2924.
doi: 10.1142/S0218127410027453. |
[3] |
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. |
[4] |
J. M. Amigó, T. Aschenbrenner, W. Bunk and R. Monetti,
Dimensional reduction of conditional algebraic multi-information via transcripts, Inform. Sci., 278 (2014), 298-310.
doi: 10.1016/j.ins.2014.03.054. |
[5] |
J. M. Amigó, K. Keller and V. 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. |
[6] |
J. M. Amigó, R. Monetti, N. Tort-Colet and M. V. Sanchez-Vives,
Infragranular layers lead information flow during slow oscillations according to information directionality indicators, J. Comput. Neurosci, 39 (2015), 53-62.
|
[7] |
J. M. Amigó, R. Monetti, B. Graff and G. Graff, Computing algebraic transfer entropy and coupling directions via transcripts, Chaos, 26 (2016), 113115, 12pp.
doi: 10.1063/1.4967803. |
[8] |
C. Bandt and B. Pompe, Permutation entropy: A natural complexity measure for time series, Phys. Rev. Lett., 88 (2002), 174102. |
[9] |
C. Cafaro, W. M. Lord, J. Sun and E. M. Bollt, Causation entropy from symbolic representations of dynamical systems, Chaos, 25 (2015), 043106, 10pp.
doi: 10.1063/1.4916902. |
[10] |
T. M. Cover and J. A. Thomas, Elements of Information Theory, (second ed.), John Wiley & Sons, Hoboken, 2006. |
[11] |
European-Heart-Network, Cardiovascular Disease Statistics, 2017. |
[12] |
G. Graff, B. Graff, A. Kaczkowska, D. Makowiec, J. M. Amigó, J. Piskorski, K. Narkiewicz and P. Guzik,
Ordinal pattern statistics for the assessment of heart rate variability, Eur. Phys. J. Special Topics, 222 (2013), 525-534.
doi: 10.1140/epjst/e2013-01857-4. |
[13] |
B. Graff, G. Graff, D. Makowiec, A. Kaczkowska, D. Wejer, S. Budrejko, D. Kozƚ owski and K. Narkiewicz,
Entropy measures in the assessment of heart rate variability in patients with cardiodepressive vasovagal syncope, Entropy, 17 (2015), 1007-1022.
doi: 10.3390/e17031007. |
[14] |
C. W. J. Granger,
Investigating causal relations by econometric models and cross-spectral methods, Econometrica, 37 (1969), 424-438.
|
[15] |
Y. Hirata, J. M. Amigó, Y. Matsuzaka, R. Yokota, H. Mushiake and K. Aihara, Detecting causality by combined use of multiple methods: Climate and brain examples, PLos One, 11 (2016), e0158572.
doi: 10.1371/journal.pone.0158572. |
[16] |
R. Monetti, W. Bunk, T. Aschenbrenner and F. Jamitzky, Characterizing synchronization in time series using information measures extracted from symbolic representations, Phys. Rev. E, 79 (2009), 046207.
doi: 10.1103/PhysRevE.79.046207. |
[17] |
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. |
[18] |
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. |
[19] |
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. |
[20] |
A. Porta, A. Catai, A. Takahashi, V. Magagnin, T. Bassani, E. Tobaldini, P. van de Borne and N. Montano, Causal relationships between heart period and systolic arterial pressure during graded head-up tilt, Am. J. Physiol. Regul. Integr. Comp. Physiol, 300 (2011), R378–R386.
doi: 10.1152/ajpregu.00553.2010. |
[21] |
T. Schreiber,
Measuring information transfer, Phys. Rev. Lett., 85 (2000), 461-464.
doi: 10.1103/PhysRevLett.85.461. |
[22] |
D. Smirnov, Spurious causalities with transfer entropy, Phys. Rev. E, 87 (2013), 042917.
doi: 10.1103/PhysRevE.87.042917. |
[23] |
M. Staniek and K. Lehnertz, Symbolic transfer entropy, Phys. Rev. Lett. 100 (2008), 158101.
doi: 10.1103/PhysRevLett.100.158101. |
[24] |
N. Wiener, Modern Mathematics for Engineers, McGraw-Hill, New York, 1956. |





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