Runge Jakob, Heitzig Jobst, Marwan Norbert, Kurths Jürgen
Potsdam Institute for Climate Impact Research, 14473 Potsdam, Germany.
Phys Rev E Stat Nonlin Soft Matter Phys. 2012 Dec;86(6 Pt 1):061121. doi: 10.1103/PhysRevE.86.061121. Epub 2012 Dec 17.
While it is an important problem to identify the existence of causal associations between two components of a multivariate time series, a topic addressed in Runge, Heitzig, Petoukhov, and Kurths [Phys. Rev. Lett. 108, 258701 (2012)], it is even more important to assess the strength of their association in a meaningful way. In the present article we focus on the problem of defining a meaningful coupling strength using information-theoretic measures and demonstrate the shortcomings of the well-known mutual information and transfer entropy. Instead, we propose a certain time-delayed conditional mutual information, the momentary information transfer (MIT), as a lag-specific measure of association that is general, causal, reflects a well interpretable notion of coupling strength, and is practically computable. Rooted in information theory, MIT is general in that it does not assume a certain model class underlying the process that generates the time series. As discussed in a previous paper [Runge, Heitzig, Petoukhov, and Kurths, Phys. Rev. Lett. 108, 258701 (2012)], the general framework of graphical models makes MIT causal in that it gives a nonzero value only to lagged components that are not independent conditional on the remaining process. Further, graphical models admit a low-dimensional formulation of conditions, which is important for a reliable estimation of conditional mutual information and, thus, makes MIT practically computable. MIT is based on the fundamental concept of source entropy, which we utilize to yield a notion of coupling strength that is, compared to mutual information and transfer entropy, well interpretable in that, for many cases, it solely depends on the interaction of the two components at a certain lag. In particular, MIT is, thus, in many cases able to exclude the misleading influence of autodependency within a process in an information-theoretic way. We formalize and prove this idea analytically and numerically for a general class of nonlinear stochastic processes and illustrate the potential of MIT on climatological data.
虽然识别多元时间序列的两个组成部分之间因果关联的存在是一个重要问题（Runge、Heitzig、Petoukhov和Kurths在《物理评论快报》108, 258701 (2012)中探讨了该主题），但以有意义的方式评估它们关联的强度更为重要。在本文中，我们专注于使用信息论度量来定义有意义的耦合强度的问题，并展示了著名的互信息和转移熵的缺点。相反，我们提出了一种特定的时间延迟条件互信息，即瞬时信息传递（MIT），作为一种特定滞后的关联度量，它具有一般性、因果性，反映了一个易于解释的耦合强度概念，并且在实际中是可计算的。基于信息论，MIT具有一般性，因为它不假设生成时间序列的过程有特定的模型类别。如前一篇论文[Runge、Heitzig、Petoukhov和Kurths，《物理评论快报》108, 258701 (2012)]所讨论的，图形模型的一般框架使MIT具有因果性，因为它仅对在其余过程条件下不独立的滞后分量给出非零值。此外，图形模型允许对条件进行低维表述，这对于可靠估计条件互信息很重要，因此使MIT在实际中可计算。MIT基于源熵的基本概念，我们利用它来产生一个耦合强度的概念，与互信息和转移熵相比，这个概念在许多情况下易于解释，因为它仅取决于两个分量在特定滞后时的相互作用。特别地，因此，MIT在许多情况下能够以信息论的方式排除过程中自相关性的误导性影响。我们针对一般类别的非线性随机过程进行了形式化并通过解析和数值方法证明了这一想法，并在气候学数据上展示了MIT的潜力。
