Convergent Cross Mapping (CCM)
Also known as: CCM, Cross-Convergent Mapping, Empirical Dynamic Modelling Causality, Yakınsak Çapraz Haritalama
Convergent Cross Mapping (CCM) is a nonlinear, state-space method for detecting causality between time-series variables embedded in a shared dynamical system. Introduced by George Sugihara and colleagues in their landmark 2012 Science paper, CCM exploits Takens' embedding theorem: if variable X causally influences Y, the historical record of Y contains enough information to recover the states of X. Causality is confirmed when cross-map skill improves—converges—as the time-series library grows longer.
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When to use it
CCM is appropriate when variables are suspected to interact within a nonlinear, coupled dynamical system—common in ecology, neuroscience, climate science, and physiology. It requires continuous, reasonably long univariate time series (typically ≥100 observations) and stationarity or near-stationarity. CCM is unsuitable for linear-independent systems well covered by Granger causality, for short or heavily noisy series, or when confounding by a shared external driver is not separately ruled out. Alternative methods include transfer entropy for information-theoretic approaches and convergent cross-mapping variants such as CCM with seasonal decomposition.
Strengths & limitations
- Detects causality in nonlinear and weakly coupled systems where Granger causality fails
- Requires no parametric model assumptions about the underlying dynamics
- Handles bidirectional and asymmetric causation within a unified framework
- Grounded in Takens' embedding theorem, providing a rigorous mathematical foundation
- Requires sufficiently long time series; performance degrades markedly with fewer than ~50–100 observations
- Optimal embedding dimension E and lag τ must be determined separately (e.g., via simplex projection or autocorrelation), adding preprocessing complexity
- Cannot distinguish direct causation from indirect causation mediated by a common driver without additional controls
- Sensitive to observational noise and missing data, which can inflate or suppress cross-map skill estimates
Frequently asked
How does CCM differ from Granger causality?
Granger causality tests whether past values of X improve linear predictions of Y in a regression model, making it best suited to separable, linear systems. CCM, by contrast, infers causality from shared attractor geometry in state space and is designed for coupled nonlinear systems where the variables are non-separable. In such systems Granger causality can produce false negatives or false positives that CCM avoids.
How do I choose the embedding dimension E and time lag τ?
E is typically selected using the simplex projection method: reconstruct M_X for E = 1, 2, … and choose the E that maximises one-step-ahead forecast skill. The time lag τ is often set to the first minimum of the autocorrelation function or the first zero crossing of the mutual information function. The rEDM R package and pyEDM Python library provide automated routines for both.
What library length is needed for reliable CCM results?
As a practical guideline, at least 50–100 observations are usually required to form meaningful neighbour sets in E-dimensional space; many published ecological studies use 100–500 time points. Convergence itself requires varying L across a wide range, so the total series should be substantially longer than the minimum library size. Very short series may show spurious convergence or fail to distinguish signal from noise.
Sources
- Sugihara, G., et al. (2012). Detecting causality in complex ecosystems. Science, 338(6106), 496–500. DOI: 10.1126/science.1227079 ↗
How to cite this page
ScholarGate. (2026, June 2). Convergent Cross Mapping (CCM). ScholarGate. https://scholargate.app/en/causal-inference/convergent-cross-mapping
Which method?
Set this method beside its closest kin and read them side by side — the library lays the books on the table; the choice is yours.
- Granger CausalityEconometrics↔ compare
- Recurrence Quantification AnalysisComplex Systems↔ compare
- Transfer EntropyCausal inference↔ compare