Crossmapping

Crossmapping is a method used in nonlinear time series analysis to infer causal relationships between variables in a dynamical system. It is most commonly associated with convergent cross mapping (CCM), which tests whether information about one variable is contained in the state space reconstructed from another variable’s time series. The approach relies on Takens’ embedding theorem, which justifies reconstructing a system’s attractor from time-delayed observations of a single variable.

The method begins by forming delay-embedded state spaces (shadow manifolds) for each observed variable. If X causes Y, the manifold reconstructed from Y should contain information about X, allowing one to estimate past states of X from the current state of Y. Practically, cross-mapping measures how well a predictor built from the target variable's manifold can predict the driver variable. A key feature is convergence: as the amount of data (library size) grows, the cross-map skill should improve if a causal link exists, whereas in the absence of causality it should not show systematic improvement.

Applications of crossmapping span ecology, climatology, physiology, and other fields where causal relationships must be inferred from observational data without explicit mechanistic models. The method is model-free and robust to nonlinear relationships but is sensitive to data quality, sampling, non-stationarity, and confounding factors. Careful implementation often includes selecting appropriate embedding parameters, testing time lags, and using surrogate data to assess significance.

Variants and extensions of CCM address practical concerns such as time-delayed effects, multivariate interactions, and noisy or short time series, broadening its applicability while maintaining a focus on detecting causality from observational data.