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转移熵 (查看源代码)

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转移熵 Transfer entropy（也可译为传递熵）是衡量两个随机过程之间有向（时间不对称）信息传递量的非参数统计量。<ref>{{cite journal|last=Schreiber|first=Thomas|title=Measuring information transfer|journal=Physical Review Letters|date=1 July 2000|volume=85|issue=2|pages=461–464|doi=10.1103/PhysRevLett.85.461|pmid=10991308|arxiv=nlin/0001042|bibcode=2000PhRvL..85..461S}}</ref><ref name=Scholarpedia >{{cite encyclopedia |year= 2007 |title = Granger causality |volume = 2 |issue = 7 |pages = 1667 |last= Seth |first=Anil|encyclopedia=[[Scholarpedia]] |url=http://www.scholarpedia.org/article/Granger_causality|doi=10.4249/scholarpedia.1667 |bibcode=2007SchpJ...2.1667S|doi-access= free }}</ref><ref name=Schindler07>{{cite journal|last=Hlaváčková-Schindler|first=Katerina|author2=Palus, M |author3=Vejmelka, M |author4= Bhattacharya, J |title=Causality detection based on information-theoretic approaches in time series analysis|journal=Physics Reports|date=1 March 2007|volume=441|issue=1|pages=1–46|doi=10.1016/j.physrep.2006.12.004|bibcode=2007PhR...441....1H|citeseerx=10.1.1.183.1617}}</ref>过程X到过程Y的转移熵是指在给定过去值Y得到过去值X时，Y值不确定性的减少量。更具体地，如果Xt和Yt（t∈N）表示两个随机过程，且信息量用 香农熵 Shannon entropy测量，则转移熵可以写为：

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<math>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)</math>,

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~~:<math>~~

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<math>

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~~《数学》~~

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~~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),~~

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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),

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~~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) ,~~

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</math>

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~~</math>~~

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数学

其中 H (x)是 x 的香农熵。上述转移熵的定义已被其他类型的熵测度(如 Rényi熵 Rényi entropy)所扩展。<ref name =" Schindler07"/><ref>{{Cite journal|last=Jizba|first=Petr|last2=Kleinert|first2=Hagen|last3=Shefaat|first3=Mohammad|date=2012-05-15|title=Rényi's information transfer between financial time series|journal=Physica A: Statistical Mechanics and Its Applications|language=en|volume=391|issue=10|pages=2971–2989|doi=10.1016/j.physa.2011.12.064|issn=0378-4371|arxiv=1106.5913|bibcode=2012PhyA..391.2971J}}</ref>

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转移熵是条件<ref name = Wyner1978>{{cite journal|last=Wyner|first=A. D. |title=A definition of conditional mutual information for arbitrary ensembles|journal=Information and Control|year=1978|volume=38|issue=1|pages=51–59|doi=10.1016/s0019-9958(78)90026-8|doi-access=free}}</ref><ref name = Dobrushin1959>{{cite journal|last=Dobrushin|first=R. L. |title=General formulation of Shannon's main theorem in information theory|journal=Uspekhi Mat. Nauk|year=1959|volume=14|pages=3–104}}</ref>互信息 conditional mutual information，其历史变量为 Yt−1:t−L:

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:<math>

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

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~~<math>~~

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~~《数学》~~

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

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~~T_{X\rightarrow Y} = I(Y_t ; X_{t-1:t-L} \mid Y_{t-1:t-L}).~~

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~~T _ { x right tarrow y } = i (y _ t; x _ { t-1: t-L } mid y _ { t-1: t-L }).~~

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~~</math>~~

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</math>

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数学

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对于向量自回归过程 vector auto-regressive processes，转移熵简化为 格兰杰因果关系 Granger causality。<ref name=Equal>{{cite journal|last=Barnett|first=Lionel|title=Granger Causality and Transfer Entropy Are Equivalent for Gaussian Variables|journal=Physical Review Letters|date=1 December 2009|volume=103|issue=23|doi=10.1103/PhysRevLett.103.238701|bibcode=2009PhRvL.103w8701B|pmid=20366183|page=238701|arxiv=0910.4514}}</ref>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时，转移熵就更具优势。<ref name=Greg/><ref>{{cite journal|last=Lungarella|first=M.|author2=Ishiguro, K. |author3=Kuniyoshi, Y. |author4= Otsu, N. |title=Methods for quantifying the causal structure of bivariate time series|journal=International Journal of Bifurcation and Chaos|date=1 March 2007|volume=17|issue=3|pages=903–921|doi=10.1142/S0218127407017628|bibcode=2007IJBC...17..903L|citeseerx=10.1.1.67.3585}}</ref>然而，它通常需要更多的样本才能进行准确估计。

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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.<ref>{{cite journal|last=Montalto|first=A|author2=Faes, L |author3=Marinazzo, D |title=MuTE: A MATLAB Toolbox to Compare Established and Novel Estimators of the Multivariate Transfer Entropy.|journal=PLOS ONE|date=Oct 2014|pmid=25314003|doi=10.1371/journal.pone.0109462|volume=9|issue=10|pmc=4196918|page=e109462|bibcode=2014PLoSO...9j9462M}}</ref>

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~~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，如分箱 binning、最近邻 nearest neighbors，或为了降低复杂度，使用非均匀嵌入方法。<ref>{{cite journal|last=Montalto|first=A|author2=Faes, L |author3=Marinazzo, D |title=MuTE: A MATLAB Toolbox to Compare Established and Novel Estimators of the Multivariate Transfer Entropy.|journal=PLOS ONE|date=Oct 2014|pmid=25314003|doi=10.1371/journal.pone.0109462|volume=9|issue=10|pmc=4196918|page=e109462|bibcode=2014PLoSO...9j9462M}}</ref>

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转移熵被用于估计神经元的功能连接<ref>{{cite journal|last=Vicente|first=Raul|author2=Wibral, Michael |author3=Lindner, Michael |author4= Pipa, Gordon |title=Transfer entropy—a model-free measure of effective connectivity for the neurosciences |journal=Journal of Computational Neuroscience|date=February 2011|volume=30|issue=1|pages=45–67|doi=10.1007/s10827-010-0262-3|pmid=20706781|pmc=3040354}}</ref><ref name = Shimono2014>{{cite journal|last=Shimono|first=Masanori|author2=Beggs, John |title=Functional clusters, hubs, and communities in the cortical microconnectome |url=https://cercor.oxfordjournals.org/content/early/2014/10/21/cercor.bhu252.full |journal=Cerebral Cortex|date= October 2014|volume=25|issue=10|pages=3743–57|doi=10.1093/cercor/bhu252 |pmid=25336598 |pmc=4585513}}</ref>和社交网络的社交影响。<ref name=Greg>{{cite conference |arxiv=1110.2724|title= Information transfer in social media|last1= Ver Steeg |first1= Greg|last2=Galstyan|first2= Aram |year= 2012|publisher= [[Association for Computing Machinery|ACM]]|booktitle= Proceedings of the 21st international conference on World Wide Web (WWW '12) |pages= 509–518 |bibcode=2011arXiv1110.2724V}}</ref>

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Transfer entropy is a finite version of the [[Directed Information]] which was defined in 1990 by [[James Massey]] <ref>{{cite journal|last1=Massey|first1=James|title=Causality, Feedback And Directed Information|date=1990|issue=ISITA|citeseerx=10.1.1.36.5688}}</ref> as

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~~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年由詹姆斯·梅西 James Massey<ref>{{cite journal|last1=Massey|first1=James|title=Causality, Feedback And Directed Information|date=1990|issue=ISITA|citeseerx=10.1.1.36.5688}}</ref>定义为

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<math>I(X^n\to Y^n) =\sum_{i=1}^n I(X^i;Y_i|Y^{i-1})</math>, where <math>X^n</math> denotes the vector <math>X_1,X_2,...,X_n</math> and <math>Y^n</math> denotes <math>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 <ref>{{cite journal|last1=Permuter|first1=Haim Henry|last2=Weissman|first2=Tsachy|last3=Goldsmith|first3=Andrea J.|title=Finite State Channels With Time-Invariant Deterministic Feedback|journal=IEEE Transactions on Information Theory|date=February 2009|volume=55|issue=2|pages=644–662|doi=10.1109/TIT.2008.2009849|arxiv=cs/0608070}}</ref>

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<math>I(X^n\to Y^n) =\sum_{i=1}^n I(X^i;Y_i|Y^{i-1})</math>, where <math>X^n</math> denotes the vector<math>X_1,X_2,...,X_n</math>and <math>Y^n</math> denotes <math>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 and gambling with causal side information,

I(Xn→Yn)=∑ni=1I(Xi;Yi|Yi−1)，其中 Xn表示向量X1,X2,...,Xn和Yn表示 Y1,Y2,...,Yn。有向信息在描述有无反馈<ref>{{cite journal|last1=Permuter|first1=Haim Henry|last2=Weissman|first2=Tsachy|last3=Goldsmith|first3=Andrea J.|title=Finite State Channels With Time-Invariant Deterministic Feedback|journal=IEEE Transactions on Information Theory|date=February 2009|volume=55|issue=2|pages=644–662|doi=10.1109/TIT.2008.2009849|arxiv=cs/0608070}}</ref> <ref>{{cite journal|last1=Kramer|first1=G.|title=Capacity results for the discrete memoryless network|journal=IEEE Transactions on Information Theory|date=January 2003|volume=49|issue=1|pages=4–21|doi=10.1109/TIT.2002.806135}}</ref>信道的基本限制(信道容量)与基于因果信息赌博<ref>{{cite journal|last1=Permuter|first1=Haim H.|last2=Kim|first2=Young-Han|last3=Weissman|first3=Tsachy|title=Interpretations of Directed Information in Portfolio Theory, Data Compression, and Hypothesis Testing|journal=IEEE Transactions on Information Theory|date=June 2011|volume=57|issue=6|pages=3248–3259|doi=10.1109/TIT.2011.2136270|arxiv=0912.4872}}</ref>方面起着重要作用。

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<ref>{{cite journal|last1=Kramer|first1=G.|title=Capacity results for the discrete memoryless network|journal=IEEE Transactions on Information Theory|date=January 2003|volume=49|issue=1|pages=4–21|doi=10.1109/TIT.2002.806135}}</ref> and [[gambling]] with causal side information,<ref>{{cite journal|last1=Permuter|first1=Haim H.|last2=Kim|first2=Young-Han|last3=Weissman|first3=Tsachy|title=Interpretations of Directed Information in Portfolio Theory, Data Compression, and Hypothesis Testing|journal=IEEE Transactions on Information Theory|date=June 2011|volume=57|issue=6|pages=3248–3259|doi=10.1109/TIT.2011.2136270|arxiv=0912.4872}}</ref>

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== 参见 ==

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