转移熵

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Moonscar（讨论 | 贡献）2020年10月25日 (日) 17:43的版本 （Moved page from wikipedia:en:Transfer entropy (history)）
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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 [math]\displaystyle{ X_t }[/math] and [math]\displaystyle{ Y_t }[/math] for [math]\displaystyle{ t\in \mathbb{N} }[/math] denote two random processes and the amount of information is measured using Shannon's entropy, the transfer entropy can be written as:

Transfer entropy is a non-parametric statistic measuring the amount of directed (time-asymmetric) transfer of information between two random processes. and social influence in social networks.

转移熵是衡量两个随机过程之间有向(时间不对称)信息转移量的非参数统计量。以及社交网络的社会影响力。


Transfer entropy is a finite version of the Directed Information which was defined in 1990 by James Massey as

转移熵是有向信息的有限形式，1990年由 James Massey 定义为

[math]\displaystyle{ \lt math\gt I(X^n\to Y^n) =\sum_{i=1}^n I(X^i;Y_i|Y^{i-1}) }[/math], where [math]\displaystyle{ X^n }[/math] denotes the vector [math]\displaystyle{ X_1,X_2,...,X_n }[/math] and [math]\displaystyle{ Y^n }[/math] denotes [math]\displaystyle{ Y_1,Y_2,...,Y_n }[/math]. The directed information places an important role in characterizing the fundamental limits (channel capacity) of communication channels with or without feedback

I (x ^ n to y ^ n) = sum { i = 1} ^ n i (x ^ i; y _ i | y ^ { i-1}) </math > ，其中 < math > x ^ n </math > 表示向量 < math > x1，x2，... ，xn </math > 和 < math > y ^ n </math > 表示 < math > y _ 1，y _ 2，... ，y _ n </math > 。定向信息在描述有无反馈信道的基本限制(信道容量)方面起着重要作用

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), 

and gambling with causal side information,

和赌博与因果方面的信息,

</math>


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 [math]\displaystyle{ Y_{t-1:t-L} }[/math] in the condition:


[math]\displaystyle{ T_{X\rightarrow Y} = I(Y_t ; X_{t-1:t-L} \mid Y_{t-1:t-L}). }[/math]


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

[math]\displaystyle{ I(X^n\to Y^n) =\sum_{i=1}^n I(X^i;Y_i|Y^{i-1}) }[/math], where [math]\displaystyle{ X^n }[/math] denotes the vector [math]\displaystyle{ X_1,X_2,...,X_n }[/math] and [math]\displaystyle{ Y^n }[/math] denotes [math]\displaystyle{ Y_1,Y_2,...,Y_n }[/math]. 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]

Category:Causality

分类: 因果关系


Category:Nonlinear time series analysis

类别: 非线性时间序列分析

See also

Category:Nonparametric statistics

类别: 无母数统计

Category:Entropy and information

类别: 熵和信息


This page was moved from wikipedia:en:Transfer entropy. Its edit history can be viewed at 转移熵/edithistory

  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.
  2. ↑ Seth, Anil (2007). "Granger causality". Scholarpedia. Vol. 2. p. 1667. Bibcode:2007SchpJ...2.1667S. doi:10.4249/scholarpedia.1667.
  3. ↑ 3.0 3.1 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 10.1.1.183.1617. 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 (in English). 391 (10): 2971–2989. arXiv:1106.5913. Bibcode:2012PhyA..391.2971J. doi:10.1016/j.physa.2011.12.064. ISSN 0378-4371.
  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.
  8. ↑ 8.0 8.1 Ver Steeg, Greg; Galstyan, Aram (2012). Information transfer in social media. ACM. pp. 509–518. arXiv:1110.2724. Bibcode:2011arXiv1110.2724V. {{cite conference}}: Unknown parameter |booktitle= ignored (help)
  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 10.1.1.67.3585. 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.
  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.
  13. ↑ 13.0 13.1 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.
  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 10.1.1.36.5688. {{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.
  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.
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