# Transfer entropy

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 $X_t$ and $Y_t$ for $t\in \mathbb{N}$ denote two random processes and the amount of information is measured using Shannon's entropy, the transfer entropy can be written as:

$T_{X\rightarrow Y} = H\left( Y_t \mid Y_{t-1:t-L}\right) - H\left( Y_t \mid Y_{t-1:t-L}, X_{t-1:t-L}\right),$

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]

Transfer entropy is conditional mutual information,[4][5] with the history of the influenced variable $Y_{t-1:t-L}$ in the condition. Transfer entropy reduces to Granger causality for vector auto-regressive processes.[6] Hence, it is advantageous when the model assumption of Granger causality doesn't hold, for example, analysis of non-linear signals.[7][8] However, it usually requires more samples for accurate estimation.[9] While it was originally defined for bivariate analysis, transfer entropy has been extended to multivariate forms, either conditioning on other potential source variables[10] or considering transfer from a collection of sources,[11] although these forms require more samples again.

Transfer entropy has been used for estimation of functional connectivity of neurons[11][12] and social influence in social networks.[7]

## References

1. ^ Schreiber, Thomas (1 July 2000). "Measuring Information Transfer". Physical Review Letters 85 (2): 461–464. doi:10.1103/PhysRevLett.85.461.
2. ^ Seth, Anil (2007). "Granger causality". Scholarpedia. 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. doi:10.1016/j.physrep.2006.12.004.
4. ^ 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.
5. ^ Dobrushin, R. L. (1959). "General formulation of Shannon's main theorem in information theory". Ushepi Mat. Nauk. 14: 3–104.
6. ^ Barnett, Lionel (1 December 2009). "Granger Causality and Transfer Entropy Are Equivalent for Gaussian Variables". Physical Review Letters 103 (23). doi:10.1103/PhysRevLett.103.238701.
7. ^ a b Ver Steeg, Greg; Galstyan, Aram (2012). http://arxiv.org/abs/1110.2724 |url= missing title (help). Proceedings of the 21st international conference on World Wide Web (WWW '12). ACM. pp. 509–518.
8. ^ 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 (03): 903–921. doi:10.1142/S0218127407017628.
9. ^ Pereda, E; Quiroga, RQ; Bhattacharya, J (Sep–Oct 2005). "Nonlinear multivariate analysis of neurophysiological signals.". Progress in neurobiology 77 (1-2): 1–37. doi:10.1016/j.pneurobio.2005.10.003. PMID 16289760.
10. ^ Lizier, Joseph; Prokopenko, Mikhail; Zomaya, Albert (2008). "Local information transfer as a spatiotemporal filter for complex systems". Physical Review E 77 (2): 026110. doi:10.1103/PhysRevE.77.026110.
11. ^ 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.
12. ^ 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.