转移熵

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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. 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:

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。从一个过程 x 到另一个过程 y 的转移熵是通过知道 x 的过去值减少了 y 的未来值的不确定性量。更具体地说，如果在数学中用数学表示两个随机过程，用香农熵测量信息量，那么转移熵可以写成:


[math]\displaystyle{ \lt math\gt 《数学》 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), 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), T _ { x right tarrow 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) , }[/math]

</math>

数学


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]

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.

其中 h (x)是 x 的 Shannon 熵。上述转移熵的定义被其他类型的熵测度(如 r é nyi 熵)所扩展。


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:

Transfer entropy is conditional mutual information, with the history of the influenced variable [math]\displaystyle{ Y_{t-1:t-L} }[/math] in the condition:

转移熵是条件互信息，其历史变量为 y _ { t-1: t-L } </math > :


[math]\displaystyle{ \lt math\gt 《数学》 T_{X\rightarrow Y} = I(Y_t ; X_{t-1:t-L} \mid Y_{t-1:t-L}). T_{X\rightarrow Y} = I(Y_t ; X_{t-1:t-L} \mid Y_{t-1:t-L}). T _ { x right tarrow y } = i (y _ t; x _ { t-1: t-L } mid y _ { t-1: t-L }). }[/math]

</math>

数学


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]

Transfer entropy reduces to Granger causality for vector auto-regressive processes. Hence, it is advantageous when the model assumption of Granger causality doesn't hold, for example, analysis of non-linear signals. However, it usually requires more samples for accurate estimation.

对于向量自回归过程，传递熵降低到格兰杰因果关系。因此，当格兰杰因果关系的模型假设不成立时，例如，对非线性信号的分析是有利的。然而，为了精确估计，通常需要更多的样本。

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]

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.

熵公式中的概率可以用不同的方法估计(包装，最近邻) ，或者为了降低复杂度，使用非均匀嵌入。

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.

While it was originally defined for bivariate analysis, transfer entropy has been extended to multivariate forms, either conditioning on other potential source variables or considering transfer from a collection of sources, although these forms require more samples again.

虽然传递熵最初定义为双变量分析，但它已经扩展到多变量形式，或者对其他潜在源变量进行调节，或者考虑从一组源的传递，尽管这些形式再次需要更多的样本。


Transfer entropy has been used for estimation of functional connectivity of neurons[13][14][15] and social influence in social networks.[8]

Transfer entropy has been used for estimation of functional connectivity of neurons and social influence in social networks.

传递熵被用来估计社会网络中神经元的功能连通性和社会影响。

Transfer entropy is a finite version of the Directed Information which was defined in 1990 by James Massey [16] as

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{ 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]

[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

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

[18] and gambling with causal side information,[19] 

and gambling with causal side information,

和赌博与因果方面的信息,


See also


References

  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.


External links

Category:Causality

分类: 因果关系

Category:Nonlinear time series analysis

类别: 非线性时间序列分析

Category:Nonparametric statistics

类别: 无母数统计

Category:Entropy and information

类别: 熵和信息


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