*Science*. It is called "Detecting Causality in Complex Ecosystems" [2], and the authors include George Sugihara and Robert May [3]. By "causality", the authors mean differentiating truly causal events from intermittent correlative events (termed "mirage" correlations in the paper, see Figure 1). And by "complex ecosystem", the authors mean comparisons between two or more time-series traces [4], although they use real-world examples [5] in the paper to test their model. The idea of uncovering causality in two or more time series is typically done using an approach called Granger causality, in which the events of one time series predict the events in another time series given some degree of lag [6].

**Figure 1.**Three instances of mirage correlations between two time series (which can occur in a range of systems, from ecological to financial).

**COURTESY:**Figure 1 in [2].

However, in cases where coupling between the two systems under analysis is weak [6], a new approach called Convergent Cross Mapping (CCM) can be used to detect causality. The CCM approach measures the extent to which the historical record (e.g. a prior time-series) can predict a current time-series [see 7]. This approach relies on the principle of cross-prediction: the current time-series must causally influence the prior time series via feedback or transitive couplings (see Figure 2). The success of this approach also depends heavily on convergence within complex systems [8] and the ability to reconstruct the state space for both time-series (Figure 3) using historical and current information [9].

**Figure 2.**Cases and examples of coupling between dynamical systems and/or variables.

**COURTESY:**Figure 4 in [2].

**Figure 3.**

**LEFT:**Example of the CCM method for three time-series sharing the same attractor basin manifold.

**RIGHT:**an example of the Simplex projection method (see notes [3] and [9]).

In my opinion, this is a very interesting and perhaps even landmark paper. You should read it and save it to your Mendeley (or similar application) library immediately.

**NOTES:**

[2] Formal citation: Sugihara, G., May, R., Ye, H., Hsieh, C-H., Deyle, E., Fogarty, M., and Munch, S. (2012). Detecting Causality in Complex Ecosystems. Science, 338, 496.

[3] Both are legends in the field of ecology. Figure 3 shows an example of their previously introduced (and theory-based) method called Simplex analysis.

[4] one example: an n-dimensional phase space trajectory such as a Lorentz attractor.

[6] MATLAB code for Granger causality can be found here (basic analysis) and here (toolbox for inferring network connectivity).

[7] What type of complex systems are most amenable to the CCM method? Nonseperable, weakly connected dynamical systems. Which are, according to the authors, something beyond the scope of (linear) Granger causality analysis.

[8] convergence is to be contrasted with Lyapunov divergence (characterized by an exponent), where two systems begin at the same initial condition and diverge over time.

[9] YouTube animation of this process from the Sugihara Lab. In implementing the CCM method, an algorithm based on the simplex projection is used to generate a nearest-neighbor solution for kernel density estimation is used. More details can be found in the Supplementary Materials.

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