Comparison of performance with different distance metrics in convergent cross-mapping method

Convergent Cross-Mapping (CCM) is a widely used nonlinear approach for detecting causality in complex systems, with applications in ecology, neuroscience, and economics. Traditionally, CCM employs Euclidean distance for nearest-neighbor identification, but this choice can be limited in high-dimensional, noisy, or scale-heterogeneous data. To address this, we systematically compare six metrics —Euclidean, Standardized Euclidean, Manhattan, Chebyshev, Cosine, and Correlation— within the CCM framework. Using coupled Logistic and Lorenz-Rössler systems, we assess their ability to capture causal dynamics under weak, strong, and asymmetric coupling. We further apply them to empirical data of neuronal multi-unit activity and behavioral rhythms in rodents, highlighting their performance in real-world conditions. Results reveal that no single distance metric is universally optimal: Chebyshev distance performs well for short sequences; Cosine distance excels in strong symmetric coupling; Standardized Euclidean is robust to scale differences and noise; Euclidean remains stable across settings; while Correlation distance underperforms due to its linearity. These findings underscore the importance of distance selection in CCM and provide methodological guidance for diverse applications.