第1段
Identifying causality (1) in complex systems can be difficult. Contradictions arise in many scientific contexts where variables are positively coupled at some times but at other times appear unrelated or even negatively coupled depending on system state (movie S1). Baltic Sea fisheries, for example, exhibit radically different dynamic control regimes (top-down versus bottom-up) depending on the threshold abundance of planktivores, causing the correlations between fish and zooplankton to change sign (2). Such state-dependent behavior is a defining hallmark of complex nonlinear systems (3, 4), and nonlinearity is ubiquitous in nature (3–11). 识别复杂系统中的因果关系（1）可能很困难。矛盾出现在许多科学环境中，变量有时是正耦合的，但有时是不相关的，甚至是负耦合的，这取决于系统状态（movie S1）。例如，波罗的海渔业表现出完全不同的动态控制机制（自上而下与自下而上），这取决于浮游生物的阈值丰度，导致鱼类和浮游动物之间的相关性发生变化（2）。这种依赖于状态的行为是复杂非线性系统的一个决定性标志（3，4），非线性在自然界中无处不在（3-11）。
第2段
Ephemeral or “mirage” correlations are common in even the simplest nonlinear systems (7, 11–13), such as shown in Fig. 1 for two coupled difference equations that exhibit chaotic behavior (14): 即使在最简单的非线性系统（7，11–13）中，短暂或“幻影”关联也很常见，如Fig. 1所示，两个耦合差分方程表现出混沌行为（14）：
第3段
When this happens, variables that may be positively coupled for long periods can spontaneously become anticorrelated or decoupled; this can create problems when fitting models to observational data (15). 当这种情况发生时，可能长时间正耦合的变量会自发地变成反相关或解耦；这可能会在将模型拟合到观测数据时产生问题（15）。
第4段
Although correlation is neither necessary nor sufficient to establish causation, it remains deeply ingrained in our heuristic thinking (8, 13, 16, 17). One might conclude, for example, that the variables in Fig. 1 have no causal relation because they are uncorrelated. Obviously, lack of correlation does not imply lack of causation. Because of this and for reasons just given, the use of correlation to infer causation is risky, especially as we come to recognize that nonlinear dynamics are ubiquitous. 虽然相关性既不必要也不足以确定因果关系，但它仍然深深扎根于我们的启发式思维中（8、13、16、17）。例如，我们可以得出结论，图1中的变量没有因果关系，因为它们是不相关的。显然，缺乏相关性并不意味着缺乏因果关系。因此，基于上述原因，使用相关性来推断因果关系是有风险的，特别是当我们认识到非线性动力学无处不在时。
第5段
An alternative approach, Granger causality (GC) (18), provides a framework that uses predictability as opposed to correlation to identify causation between time-series variables. GC is recognized as the primary advance on the causation problem since Berkeley (1). 另一种方法，格兰杰因果关系（GC）（18），提供了一个使用可预测性而不是相关性来识别时间序列变量之间因果关系的框架。自Berkeley（1）以来，GC被认为是因果关系问题的主要进展。
第6段
Variable X is said to “Granger cause” Y if the predictability of Y (in some idealized model) declines when X is removed from the universe of all possible causative variables, U (18). The key requirement of GC is separability, namely that information about a causative factor is independently unique to that variable (e.g., information about predator effects is not contained in time series for the prey) and can be removed by eliminating that variable from the model. Separability is characteristic of purely stochastic and linear systems, and GC can be useful for detecting interactions between strongly coupled (synchronized) variables in nonlinear systems. Separability reflects the view that systems can be understood a piece at a time rather than as a whole. 当X从所有可能的原因变量U（18）中移除时，如果Y（在某些理想化模型中）的可预测性下降，则变量X被称为“格兰杰原因”。GC的关键要求是可分性，即关于原因的信息独立于该变量（例如，关于捕食者影响的信息不包含在猎物的时间序列中），并且可以通过从模型中消除该变量来删除。可分离性是纯随机和线性系统的特征，GC可用于检测非线性系统中强耦合（同步）变量之间的相互作用。可分性反映了这样一种观点，即系统可以一次理解为一个片段，而不是作为一个整体。
第7段
However, as Granger (18) realized early on, this approach may be problematic in deterministic settings, especially in dynamic systems with weak to moderate coupling. For example, GC gives ambiguous results for the system in Fig. 1 (see GC calculations S1). This is because separability is not satisfied in such systems, which, unlike the tradition in economics and single-species fisheries management, need to be considered as a whole. That is to say, in deterministic dynamic systems (even noisy ones), if X is a cause for Y, information about X will be redundantly present in Y itself and cannot formally be removed from U—a consequence of Takens’ theorem (19, 20). To see this directly, we note simply that Eq. 1 can be rewritten as a model for Y(t + 1) in terms of Y(t) and Y(t – 1) (see box S1 for a worked example). Therefore, information about X(t) that is relevant to predicting Y is redundant in this system and cannot be removed simply by eliminating X as an explicit variable. When Granger’s definition is violated, GC calculations are no longer valid, leaving the question of detecting causation in such systems unanswered. 然而，正如Granger（18）早就意识到的那样，这种方法在确定性环境中可能会有问题，特别是在具有弱到中度耦合的动态系统中。例如，GC给出了图1中系统的模糊结果（参见GC calculations S1）。这是因为在这些系统中，分离性并不令人满意，与经济学和单一物种渔业管理的传统不同，这些系统需要作为一个整体来考虑。也就是说，在确定性动态系统（即使是噪声系统）中，如果X是Y的起因，那么关于X的信息将冗余地存在于Y本身，并且不能从U中正式删除——这是Takens定理（19，20）的结果。为了直接看到这一点，我们只需注意，等式1可以根据Y（t）和Y（t–1）重写为Y（t+1）的模型（参见box S1中的示例）。因此，在这个系统中，与预测Y相关的X（t）信息是多余的，不能简单地通过消除X作为显式变量来消除。当格兰杰的定义被违反时，GC计算就不再有效，这使得在这样的系统中检测因果关系的问题没有答案。
第8段
In addition to nonseparability, ecosystems differ from the systems typically studied with Granger’s approach in other important ways. First, in ecosystem dynamics, weak to moderate coupling is the norm. McCann (21) and others have developed a strong case for the ubiquity of weak coupling in ecological food webs and have demonstrated their importance for system stability. Second, ecosystems are typically subject to forcing by external driving variables such as temperature, precipitation, and upwelling [e.g., (6, 22)]. Because many species share similar abiotic environments, this can lead to correlations and apparent synchrony among noninteracting species [e.g., the Moran effect (23)], complicating the task of sorting out the real interactions from spurious correlations. It is therefore important in ecology to have methods that (i) address nonseparable systems, (ii) identify weakly coupled variables, and (iii) distinguish interactions among species from the effects of shared driving variables. 除了不可分离性之外，生态系统在其他重要方面也不同于一般用格兰杰方法研究的系统。首先，在生态系统动力学中，弱到中度耦合是常态。McCann（21）和其他人为生态食物网中普遍存在的弱耦合提出了强有力的证据，并证明了它们对系统稳定性的重要性。其次，生态系统通常受到外部驱动变量（如温度、降水量和上升流）的强迫[6,22]。由于许多物种共享相似的非生物环境，这可能导致非交互物种之间的相关性和明显的同步性[例如，Moran效应（23）]，从而使从虚假相关性中分离真实交互的任务变得复杂。因此，在生态学中，重要的是采用以下方法：（i）处理不可分离系统，（ii）识别弱耦合变量，（iii）区分物种间的相互作用和共享驱动变量的影响。
第9段
Here, we examine an approach specifically aimed at identifying causation in ecological time series. We demonstrate the principles of our approach with simple model examples, showing that the method distinguishes species interactions from the effects of shared driving variables. Finally, we apply the method to ecological data from experimental and field studies, showing how it distinguishes top-down from bottom-up control in the classic Paramecium-Didinium experiment and clarifies the ongoing debate about the nature of interactions among sardine, anchovy, and sea surface temperature in the California Current ecosystem. 在这里，我们研究了一种方法，旨在确定生态时间序列中的因果关系。我们用简单的模型例子说明了我们方法的原理，表明该方法区分了物种相互作用和共享驱动变量的影响。最后，我们将该方法应用于来自实验和现场研究的生态数据，展示了它如何在经典草履虫实验中区分自上而下和自下而上的控制，并澄清了关于加利福尼亚海流生态系统中沙丁鱼、凤尾鱼和海表温度之间相互作用性质的持续争论。
第10段
Our approach is not in competition with the many effective methods that use GC (see supplementary text); rather, it is specifically aimed at a class of system not covered by GC. As verified in GC calculations S1 to S5 and box S1, GC does not apply to this class of system. 我们的方法并没有和许多使用GC的有效方法竞争（见补充文本）；相反，它专门针对GC未涵盖的一类系统。如GC计算S1至S5和方框S1中所验证，GC不适用于此类系统。