Transfer entropy

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Transfer entropy is a non-parametric statistic measuring the amount of directed (time-asymmetric) transfer of information between two random processes.[1][2][3] Transfer entropy from a process X to another process Y is the amount of uncertainty reduced in future values of Y by knowing the past values of X given past values of Y. More specifically, if and for denote two random processes and the amount of information is measured using Shannon's entropy, the transfer entropy can be written as:

where H(X) is Shannon entropy of X. The above definition of transfer entropy has been extended by other types of entropy measures such as Rényi entropy.[3][4]

Transfer entropy is conditional mutual information,[5][6] with the history of the influenced variable in the condition:

Transfer entropy reduces to Granger causality for vector auto-regressive processes.[7] Hence, it is advantageous when the model assumption of Granger causality doesn't hold, for example, analysis of non-linear signals.[8][9] However, it usually requires more samples for accurate estimation.[10] The probabilities in the entropy formula can be estimated using different approaches (binning, nearest neighbors) or, in order to reduce complexity, using a non-uniform embedding.[11] While it was originally defined for bivariate analysis, transfer entropy has been extended to multivariate forms, either conditioning on other potential source variables[12] or considering transfer from a collection of sources,[13] although these forms require more samples again.

Transfer entropy has been used for estimation of functional connectivity of neurons[13][14][15] and social influence in social networks.[8] Transfer entropy is a finite version of the Directed Information which was defined in 1990 by James Massey[16] as , where denotes the vector and denotes . The directed information places an important role in characterizing the fundamental limits (channel capacity) of communication channels with or without feedback [17] [18] and gambling with causal side information.[19]

See also[edit]


  1. ^ Schreiber, Thomas (1 July 2000). "Measuring information transfer". Physical Review Letters. 85 (2): 461–464. arXiv:nlin/0001042. Bibcode:2000PhRvL..85..461S. doi:10.1103/PhysRevLett.85.461. PMID 10991308. S2CID 7411376.
  2. ^ Seth, Anil (2007). "Granger causality". Scholarpedia. Vol. 2. p. 1667. Bibcode:2007SchpJ...2.1667S. doi:10.4249/scholarpedia.1667.
  3. ^ a b Hlaváčková-Schindler, Katerina; Palus, M; Vejmelka, M; Bhattacharya, J (1 March 2007). "Causality detection based on information-theoretic approaches in time series analysis". Physics Reports. 441 (1): 1–46. Bibcode:2007PhR...441....1H. CiteSeerX doi:10.1016/j.physrep.2006.12.004.
  4. ^ Jizba, Petr; Kleinert, Hagen; Shefaat, Mohammad (2012-05-15). "Rényi's information transfer between financial time series". Physica A: Statistical Mechanics and Its Applications. 391 (10): 2971–2989. arXiv:1106.5913. Bibcode:2012PhyA..391.2971J. doi:10.1016/j.physa.2011.12.064. ISSN 0378-4371. S2CID 51789622.
  5. ^ Wyner, A. D. (1978). "A definition of conditional mutual information for arbitrary ensembles". Information and Control. 38 (1): 51–59. doi:10.1016/s0019-9958(78)90026-8.
  6. ^ Dobrushin, R. L. (1959). "General formulation of Shannon's main theorem in information theory". Uspekhi Mat. Nauk. 14: 3–104.
  7. ^ Barnett, Lionel (1 December 2009). "Granger Causality and Transfer Entropy Are Equivalent for Gaussian Variables". Physical Review Letters. 103 (23): 238701. arXiv:0910.4514. Bibcode:2009PhRvL.103w8701B. doi:10.1103/PhysRevLett.103.238701. PMID 20366183. S2CID 1266025.
  8. ^ a b Ver Steeg, Greg; Galstyan, Aram (2012). "Information transfer in social media". Proceedings of the 21st international conference on World Wide Web (WWW '12). ACM. pp. 509–518. arXiv:1110.2724. Bibcode:2011arXiv1110.2724V.
  9. ^ Lungarella, M.; Ishiguro, K.; Kuniyoshi, Y.; Otsu, N. (1 March 2007). "Methods for quantifying the causal structure of bivariate time series". International Journal of Bifurcation and Chaos. 17 (3): 903–921. Bibcode:2007IJBC...17..903L. CiteSeerX doi:10.1142/S0218127407017628.
  10. ^ Pereda, E; Quiroga, RQ; Bhattacharya, J (Sep–Oct 2005). "Nonlinear multivariate analysis of neurophysiological signals". Progress in Neurobiology. 77 (1–2): 1–37. arXiv:nlin/0510077. Bibcode:2005nlin.....10077P. doi:10.1016/j.pneurobio.2005.10.003. PMID 16289760. S2CID 9529656.
  11. ^ Montalto, A; Faes, L; Marinazzo, D (Oct 2014). "MuTE: A MATLAB Toolbox to Compare Established and Novel Estimators of the Multivariate Transfer Entropy". PLOS ONE. 9 (10): e109462. Bibcode:2014PLoSO...9j9462M. doi:10.1371/journal.pone.0109462. PMC 4196918. PMID 25314003.
  12. ^ Lizier, Joseph; Prokopenko, Mikhail; Zomaya, Albert (2008). "Local information transfer as a spatiotemporal filter for complex systems". Physical Review E. 77 (2): 026110. arXiv:0809.3275. Bibcode:2008PhRvE..77b6110L. doi:10.1103/PhysRevE.77.026110. PMID 18352093. S2CID 15634881.
  13. ^ a b Lizier, Joseph; Heinzle, Jakob; Horstmann, Annette; Haynes, John-Dylan; Prokopenko, Mikhail (2011). "Multivariate information-theoretic measures reveal directed information structure and task relevant changes in fMRI connectivity". Journal of Computational Neuroscience. 30 (1): 85–107. doi:10.1007/s10827-010-0271-2. PMID 20799057. S2CID 3012713.
  14. ^ Vicente, Raul; Wibral, Michael; Lindner, Michael; Pipa, Gordon (February 2011). "Transfer entropy—a model-free measure of effective connectivity for the neurosciences". Journal of Computational Neuroscience. 30 (1): 45–67. doi:10.1007/s10827-010-0262-3. PMC 3040354. PMID 20706781.
  15. ^ Shimono, Masanori; Beggs, John (October 2014). "Functional clusters, hubs, and communities in the cortical microconnectome". Cerebral Cortex. 25 (10): 3743–57. doi:10.1093/cercor/bhu252. PMC 4585513. PMID 25336598.
  16. ^ Massey, James (1990). "Causality, Feedback And Directed Information" (ISITA). CiteSeerX {{cite journal}}: Cite journal requires |journal= (help)
  17. ^ Permuter, Haim Henry; Weissman, Tsachy; Goldsmith, Andrea J. (February 2009). "Finite State Channels With Time-Invariant Deterministic Feedback". IEEE Transactions on Information Theory. 55 (2): 644–662. arXiv:cs/0608070. doi:10.1109/TIT.2008.2009849. S2CID 13178.
  18. ^ Kramer, G. (January 2003). "Capacity results for the discrete memoryless network". IEEE Transactions on Information Theory. 49 (1): 4–21. doi:10.1109/TIT.2002.806135.
  19. ^ Permuter, Haim H.; Kim, Young-Han; Weissman, Tsachy (June 2011). "Interpretations of Directed Information in Portfolio Theory, Data Compression, and Hypothesis Testing". IEEE Transactions on Information Theory. 57 (6): 3248–3259. arXiv:0912.4872. doi:10.1109/TIT.2011.2136270. S2CID 11722596.

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