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第1行： − − '''Transfer entropy''' is a [[non-parametric statistics|non-parametric statistic]] measuring the amount of directed (time-asymmetric) transfer of [[information]] between two [[random process]]es.<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> 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> X_t </math> and <math> Y_t </math> for <math> 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> X_t </math> and <math> Y_t </math> for <math> 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: 第26行：
第22行： − − − − where ''H''(''X'') is Shannon entropy of ''X''. The above definition of transfer entropy has been extended by other types of [[entropy (information theory)|entropy]] measures such as [[Rényi entropy]].<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> − − 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. − + − − Transfer entropy is [[conditional mutual information]],<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> with the history of the influenced variable <math>Y_{t-1:t-L}</math> in the condition: − − Transfer entropy is conditional mutual information,<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> with the history of the influenced variable <math>Y_{t-1:t-L}</math> in the condition: − − 转移熵是<font color="#ff8000">条件互信息 conditional mutual information</font>，其历史变量为 Yt−1:t−L: − − 第63行：
第45行： − − − Transfer entropy reduces to [[Granger causality]] for [[Autoregressive model|vector auto-regressive processes]].<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> Hence, it is advantageous when the model assumption of Granger causality doesn't hold, for example, analysis of [[non-linear regression|non-linear signals]].<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> However, it usually requires more samples for accurate estimation.<ref>{{cite journal|last=Pereda|first=E|author2=Quiroga, RQ |author3=Bhattacharya, J |title=Nonlinear multivariate analysis of neurophysiological signals.|journal=Progress in Neurobiology|date=Sep–Oct 2005|volume=77|issue=1–2|pages=1–37|pmid=16289760|doi=10.1016/j.pneurobio.2005.10.003|arxiv=nlin/0510077|bibcode=2005nlin.....10077P}}</ref> − − 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. 第76行：
第53行： − − While it was originally defined for [[bivariate analysis]], transfer entropy has been extended to [[Multivariate analysis|multivariate]] forms, either conditioning on other potential source variables<ref>{{cite journal|last=Lizier|first=Joseph|author2=Prokopenko, Mikhail |author3=Zomaya, Albert |title=Local information transfer as a spatiotemporal filter for complex systems|journal=Physical Review E|year=2008|volume=77|issue=2|pages=026110|doi=10.1103/PhysRevE.77.026110|pmid=18352093|arxiv=0809.3275|bibcode=2008PhRvE..77b6110L}}</ref> or considering transfer from a collection of sources,<ref name = Lizier2011>{{cite journal|last=Lizier|first=Joseph|author2=Heinzle, Jakob |author3=Horstmann, Annette |author4=Haynes, John-Dylan |author5= Prokopenko, Mikhail |title=Multivariate information-theoretic measures reveal directed information structure and task relevant changes in fMRI connectivity|journal=Journal of Computational Neuroscience|year=2011|volume=30|issue=1|pages=85–107|doi=10.1007/s10827-010-0271-2|pmid=20799057}}</ref> 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. − − − − Transfer entropy has been used for estimation of [[functional connectivity]] of [[neurons]]<ref name=Lizier2011 /><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> and [[social influence]] in [[social networks]].<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> − − Transfer entropy has been used for estimation of functional connectivity of neurons and social influence in social networks. 第107行：
第74行： − + − 参见 − * [[Conditional mutual information]] − 条件互信息 − * [[Causality]] − 因果关系 − * [[Causality (physics)]] − 因果关系（物理） − * [[Structural equation modeling]] − 结构方程模型 − * [[Rubin causal model]] − 虚拟事实模型 − * [[Mutual information]] − 互信息 + − == References ==+ − 参考 − {{Reflist|2}} + + − == External links ==+ − 外部链接 − − − * {{cite web|title=Java Information Dynamics Toolkit (JIDT)|url=https://github.com/jlizier/jidt|publisher=[[GitHub]]|date=2019-01-16}}, a toolbox, developed in [[Java (programming language)|Java]] and usable in [[MATLAB]], [[GNU Octave]] and [[Python (programming language)|Python]], for computation of transfer entropy and related information-theoretic measures in both discrete and continuous-valued data. − + + + − [[Category:Causality]]+ − Category:Causality+ − 分类: 因果关系+ − [[Category:Nonlinear time series analysis]]+ − − Category:Nonlinear time series analysis − − 类别: 非线性时间序列分析 − − − − Category:Nonparametric statistics − − 类别: 非参数统计 − − − − Category:Entropy and information − − 类别: 熵和信息 − − <noinclude> − <small>This page was moved from [[wikipedia:en:Transfer entropy]]. Its edit history can be viewed at [[转移熵/edithistory]]</small></noinclude>+ − + +
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此词条暂由Henry翻译。
此词条暂由Henry翻译。
已由Vicky审校
已由Vicky审校
<font color="#ff8000"> 转移熵 Transfer entropy</font>（也可译为<font color="#ff8000">传递熵</font>）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<font color="#ff8000"> 香农熵 Shannon entropy</font>测量，则转移熵可以写为：
<font color="#ff8000"> 转移熵 Transfer entropy</font>（也可译为<font color="#ff8000">传递熵</font>）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<font color="#ff8000"> 香农熵 Shannon entropy</font>测量，则转移熵可以写为：
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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<font color="#ff8000"> Rényi熵 Rényi entropy</font>)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>
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<font color="#ff8000"> Rényi熵 Rényi entropy</font>)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>
转移熵是<font color="#ff8000">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</font>，其历史变量为 Yt−1:t−L:
:<math>
:<math>
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对于<font color="#ff8000">向量自回归过程 vector auto-regressive processes</font>，转移熵简化为<font color="#ff8000"> 格兰杰因果关系 Granger causality</font>。<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 。
对于<font color="#ff8000">向量自回归过程 vector auto-regressive processes</font>，转移熵简化为<font color="#ff8000"> 格兰杰因果关系 Granger causality</font>。<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 。
熵公式中的概率可以用不同的方法估计，如<font color="#ff8000">分箱 binning</font>、<font color="#ff8000">最近邻 nearest neighbors</font>，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>
熵公式中的概率可以用不同的方法估计，如<font color="#ff8000">分箱 binning</font>、<font color="#ff8000">最近邻 nearest neighbors</font>，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>
虽然转移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，<ref>{{cite journal|last=Lizier|first=Joseph|author2=Prokopenko, Mikhail |author3=Zomaya, Albert |title=Local information transfer as a spatiotemporal filter for complex systems|journal=Physical Review E|year=2008|volume=77|issue=2|pages=026110|doi=10.1103/PhysRevE.77.026110|pmid=18352093|arxiv=0809.3275|bibcode=2008PhRvE..77b6110L}}</ref> 或者考虑从一组源的传递，<ref name = Lizier2011>{{cite journal|last=Lizier|first=Joseph|author2=Heinzle, Jakob |author3=Horstmann, Annette |author4=Haynes, John-Dylan |author5= Prokopenko, Mikhail |title=Multivariate information-theoretic measures reveal directed information structure and task relevant changes in fMRI connectivity|journal=Journal of Computational Neuroscience|year=2011|volume=30|issue=1|pages=85–107|doi=10.1007/s10827-010-0271-2|pmid=20799057}}</ref>尽管这些形式再次需要更多的样本。
虽然转移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，<ref>{{cite journal|last=Lizier|first=Joseph|author2=Prokopenko, Mikhail |author3=Zomaya, Albert |title=Local information transfer as a spatiotemporal filter for complex systems|journal=Physical Review E|year=2008|volume=77|issue=2|pages=026110|doi=10.1103/PhysRevE.77.026110|pmid=18352093|arxiv=0809.3275|bibcode=2008PhRvE..77b6110L}}</ref> 或者考虑从一组源的传递，<ref name = Lizier2011>{{cite journal|last=Lizier|first=Joseph|author2=Heinzle, Jakob |author3=Horstmann, Annette |author4=Haynes, John-Dylan |author5= Prokopenko, Mikhail |title=Multivariate information-theoretic measures reveal directed information structure and task relevant changes in fMRI connectivity|journal=Journal of Computational Neuroscience|year=2011|volume=30|issue=1|pages=85–107|doi=10.1007/s10827-010-0271-2|pmid=20799057}}</ref>尽管这些形式再次需要更多的样本。
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>
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>
== See also ==
== 参见 ==
* [[条件互信息]]
* [[因果关系]]
* [[因果关系（物理）]]
* [[结构方程模型]]
* [[虚拟事实模型 ]]
* {{cite web|title=Transfer Entropy Toolbox|url=http://code.google.com/p/transfer-entropy-toolbox/|publisher=[[Google Code]]}}, a toolbox, developed in [[C++]] and [[MATLAB]], for computation of transfer entropy between spike trains.
* {{cite web|title=Multivariate Transfer Entropy (MuTE) toolbox|url=https://github.com/montaltoalessandro/MuTE|publisher=[[GitHub]]|date=2019-01-09}}, a toolbox, developed in [[MATLAB]], for computation of transfer entropy with different estimators.
* [[互信息]]
== 参考 ==
{{Reflist|2}}
== 外部链接 ==
* {{cite web|title=Transfer Entropy Toolbox|url=http://code.google.com/p/transfer-entropy-toolbox/|publisher=[[Google Code]]}}, a toolbox, developed in [[C++]] and [[MATLAB]], for computation of transfer entropy between spike trains.
* {{cite web|title=Java Information Dynamics Toolkit (JIDT)|url=https://github.com/jlizier/jidt|publisher=[[GitHub]]|date=2019-01-16}}, a toolbox, developed in [[Java (programming language)|Java]] and usable in [[MATLAB]], [[GNU Octave]] and [[Python (programming language)|Python]], for computation of transfer entropy and related information-theoretic measures in both discrete and continuous-valued data.
* {{cite web|title=Multivariate Transfer Entropy (MuTE) toolbox|url=https://github.com/montaltoalessandro/MuTE|publisher=[[GitHub]]|date=2019-01-09}}, a toolbox, developed in [[MATLAB]], for computation of transfer entropy with different estimators.
[[Category:Nonparametric statistics]]
[[Category:Entropy and information]]
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[[分类: 因果关系]] [[分类: 非线性时间序列分析]] [[分类: 非参数统计]] [[分类: 熵和信息]]