Detecting Causality in the Frequency Domain with Cross-Mapping Coherence

TL;DR

The paper addresses the challenge of detecting directed and circular causality in nonlinear time series within the frequency domain. It introduces Cross-Mapping Coherence (CMC), an extension of Convergent Cross-Mapping (CCM) that replaces the evaluation metric with coherence to reveal frequency-specific causal links, aided by peak prominence to identify propagation delays and suppress Granger-like peaks. Through simulations of logistic maps, Lorenz systems, Kuramoto oscillators, and a Wilson–Cowan neural mass model, CMC demonstrates accurate directionality, sensitivity to weak couplings, and robustness to noise, with results aligning with spectral Granger causality in cortical models. The approach offers a new tool for spectral causal discovery with potential applications in neuroscience and nonlinear Earth-system dynamics, while acknowledging limitations such as the lack of a formal statistical significance framework and opportunities for time-frequency extensions.

Abstract

Understanding causal relationships within a system is crucial for uncovering its underlying mechanisms. Causal discovery methods, which facilitate the construction of such models from time-series data, hold the potential to significantly advance scientific and engineering fields. This study introduces the Cross-Mapping Coherence (CMC) method, designed to reveal causal connections in the frequency domain between time series. CMC builds upon nonlinear state-space reconstruction and extends the Convergent Cross-Mapping algorithm to the frequency domain by utilizing coherence metrics for evaluation. We tested the Cross-Mapping Coherence method using simulations of logistic maps, Lorenz systems, Kuramoto oscillators, and the Wilson-Cowan model of the visual cortex. CMC accurately identified the direction of causal connections in all simulated scenarios. When applied to the Wilson-Cowan model, CMC yielded consistent results similar to spectral Granger causality. Furthermore, CMC exhibits high sensitivity in detecting weak connections, demonstrates sample efficiency, and maintains robustness in the presence of noise. In conclusion, the capability to determine directed causal influences across different frequency bands allows CMC to provide valuable insights into the dynamics of complex, nonlinear systems.

Figures (9)

• Figure 1: Causal discovery from time series. Causal discovery algorithms try to identify causal structures between time series, whether it is independence, unidirectional direct or circular causation, or hidden confounding.
• Figure 2: Overview of Convergent Cross Mapping. (a) Overview of the CCM algorithm from a macroscopic perspective and (b - k) demonstrative stages for the coupled logistic map system $x \rightarrow y$. (b, f) The inputs consist of individual time series observations for the $x$ (blue) and $y$ (orange) variables. (c, g) State-space reconstruction employing time-delay embedding. The three-dimensional plots depict the reconstructed states of the subsystems $X$ (blue) and $Y$ (orange). (d, h) Predicted time series $\hat{x}$ (blue) and $\hat{y}$ (orange). (e, i) Evaluation of the the predictions using the coefficient of determination metrics ($R^2$). The models exhibited high prediction-quality for the $x$ (blue) variable, whereas low prediction-quality for $y$ (orange) variable. (j) Convergence as the function of library length. The $R^2$ values are converging in the $x \rightarrow y$ direction (blue), but do not converge in the $y \rightarrow x$ direction (orange), thereby indicating a unidirectional causal link. (k) Cross-mapping quality as the function of time shift between the time series (Cross-Mapping Function). The negative semi-axis denotes the causal side, where the cause precedes the effect. The positive semi-axis represents the anti-causal side, where the predictability of the effect from the cause can be identified. The peak on the blue curve at zero signifies unidirectional coupling from $x$ to $y$ with minimal propagation delay. The absence of peaks in the reversed direction indicates no causal link in the $y$ to $x$ direction (orange curve).
• Figure 3: Cross-Mapping Coherence detects causal relationship in the frequency domain and filters out spurious causality by peak prominence calculation. (a) A comparative overview of Cross-Mapping Coherence (CMC) and Convergent Cross Mapping (CCM). Both methodologies analyze two time series and are comprised of three principal stages: state reconstruction, prediction, and evaluation. Although the state reconstruction and prediction phases are identical in both methods, CMC diverges in the evaluation phase. Specifically, CMC utilizes the coherence metric to compare predicted and actual values, thereby offering spectral insights into the causal connection. (b) Significant peak determination with peak prominence calculation on the Cross-Mapping Coherence Function. The coherence value is computed for each temporal shift, followed by an examination of the resultant coherence function across individual frequency bands. An illustrative coherence function is presented at a specific frequency (blue curve). Peaks may manifest on the anti-causal side of the function (Granger peak; blue dot) if the state of the effect can be inferred from the state of the cause. This Granger peak may hinder the causal inference procedure, so if predictability is present, one has to handle maximum determination with special care. To reduce the impact of Granger peaks, we identify peaks on the causal side of the graph, and calculate the prominence value of each peak to quantify the causal effect.
• Figure 4: Causal discovery on the logistic map systems with diverse coupling scenarios using Cross-Mapping Coherence. The columns of the figure illustrate various coupling configurations: unidirectional coupling $\rightarrow$ (a, e, i, m), circular coupling $x \leftrightarrow y$ (b, f, j, n), hidden common driver $x \Lsh \Rsh y$ (c, g, k, o), and independence $x \perp y$ (d, h, l, p), respectively. The direction of the inferred relationship is color-coded: the $x \rightarrow y$ direction is indicated by blue, whereas the $y \rightarrow x$ direction is marked by orange colors. (a-d) The time-delayed CCM function reveals causal relations from bivariate time series. A pronounced peak indicates the presence of causal links in both the unidirectional (a) and circular (b) scenarios. However, the absence of distinct peaks in the hidden common driver (c) and independent (d) cases mark the lack of direct causal connections. (e-l) CMC functions for the coupled logistic map systems show spectral information for the inferred $x \rightarrow y$ (e-h) and $y \rightarrow x$ (i-l) directions. The vertical bands on the two-dimensional CMC functions in the case of unidirectional and circular couplings indicate a roughly frequency-independent causal link between the time series (e, f, j). The absence of such a band indicates that there were no direct causal link to be inferred (i). Here the colormaps are normalized to the $[0, 1]$ interval. (m-p) The peak prominence of CMC values show the proper causal couplings uniformly along the frequency axis.
• Figure 5: Dependence of CMC on simulation parameters for logistic maps with unidirectional coupling. (a) Dependence on time-series length ($D=2$, $C_{X \rightarrow Y}=0.15$, $\sigma_{\varepsilon}=0$). The CMC function is convergent, the causal link is hardly detectable for $L=400$, but clearly visible at $L=700$ simulation steps. (b) Dependence on coupling strength ($L=2000$, $D=2$, $\sigma_{\varepsilon}=0$). The causal link is detected by CMC even in case of such weak coupling as $C_{X \rightarrow Y}=0.05$. (c) Dependence on signal-to-noise ratio ($L=10000$, $D=2$, $C_{X \rightarrow Y}=0.15$ ). CMC can detect the causal link when the signal-to-noise ratio is at least between $5$ and $10$.