收敛交叉映射(CCM)是最近开发的工具来回答你问的问题。它是基于工具开发的非线性时间序列分析和动力系统理论。它允许您:< br / > 1)判断两个变量之间的因果关系存在< br / > 2)建立因果关系的方向< br / > 3)即使在噪音的存在。一个有趣的应用程序,检查论文因果反馈在气候变化* (van Nes et al ., 2015),在CCM应用于基于Vostok二氧化碳和温度数据集。编辑:下面我添加了一个更详细的解释CCM给楼主,这种技术确实回答他们的问题,以及显示它具有严格的数学基础。收敛交叉映射的总体想法是基于相空间重建(F。塔肯斯,1981;h . Abarbanel, 1996]。数字1到5解释背后的想法相空间重建,需要理解CCM。数字6到8简单解释CCM。 References are listed at the bottom for more depth. 1) A physical system that is described by a set of equations (e.g. conservation of mass, momentum, etc) has a phase space. The solution to the system of equations is a trajectory through (or subset of) the phase space. 2) An attractor is a subset of the phase space that the trajectories/solutions evolve toward. 3) If you know the governing attractor, then you have all solutions of the system for all time. 4) Taken’s theorem says that one can reconstruct the attractor of the system based on a single observable. For example, if temperature, pressure, and velocity are the three variables of the system, then you only need measurements from one of these variables to reconstruct the attractor of the system. [Video 1] 5) The reconstructed attractor is not exactly the “true” attractor, but it has a direct 1:1 mapping to the true attractor. [Video 2] 6) If two observables belong to the same system, then they each have a reconstructed attractor with a direct mapping to the true attractor. The reconstructed attractors also have a direct mapping to one another. [Video 3] 7) It is then possible to make predictions on one observable, based on the reconstructed attractor of the other observable, if they are in fact from the same attractor (causally related). [Video 4] 8) Last, a series of tests/predictions with the data help to establish the direction of, strength of, and linearity of the interactions between the two variables. This is detailed in the papers [Sugihara et al., 2012] and [van Nes et al., 2015]. To answer your question "given two observed variables, how do you tell if an third variable is simply forcing the two observed variables, making them appear correlated? First, the process of phase space reconstruction would yield an estimate of the "embedding dimension", which is an estimate of the dimension of the phase space (how many variables there). In the CCM framework, a one-way forcing relationship between the two known variables (v1 and v2) would be attractor for v1 can make skillful predictions of v2, but attractor v2 can not make skillful predictions for v1. Contingent upon the situation where you have an idea of what the third variable is (v3), *I think* what you could do is the following, take the reconstructed attractor of v3 and make predictions on both v1 and v2, and show that v3 has more predictive power on v1 (compared to v2), and that v3 has more predictive power on v2 (compared to v1). I'm not sure about this though. Also, if the forcing (v3) is thought to be linear, you could simply remove/detrend v3 from v1 and v2, as is done when you remove seasonality from temperature data. Note: There is code available in MatLab to mess around with this technique called CauseMap [Maher and Hernandez, 2015]. I believe you can find similar codes in R as well. References
 video 1: https://www.youtube.com/watch?v=7ucgQE3SO0o
 video 2: https://www.youtube.com/watch?v=rs3gYeZeJcw
 video 3: https://www.youtube.com/watch?v=NrFdIz-D2yM
 video 4: https://www.youtube.com/watch?v=8DikuwwPWsY
 Sugihara et al., 2012, *Detecting causality in complex ecosystems*.
 van Nes et al., 2015, *Causal feedbacks in climate change*.
 Abarbanel, Analysis of Observed Chaotic Data,1996, Springer publishing. 
 Takens, 1981, *Detecting strange attractors in turbulence*
 Maher, and Hernandez, 2015, *CauseMap: fast inference of causality from complex time series*