Discovering Causal Relationships Between Time Series With Spatial Structure

Causal discovery is the subfield of causal inference concerned with estimating the structure of cause-and-effect relationships in a system of interrelated variables, as opposed to quantifying the strength or describing the form of causal effects. As interest in causal discovery builds in fields such as ecology, public health, and environmental sciences where data are regularly collected with spatial and temporal structures, approaches must evolve to manage autocorrelation and complex confounding. As it stands, the few proposed causal discovery algorithms for spatiotemporal data require summarizing across locations, ignore spatial autocorrelation, and/or scale poorly to high dimensions. Here, we introduce our developing framework that extends time-series causal discovery to systems with spatial structure, building upon work on causal discovery across contexts and methods for handling spatial confounding in causal effect estimation. We close by outlining remaining gaps in the literature and directions for future research.

💡 Research Summary

The paper addresses the growing need for causal discovery methods that can handle data with both temporal and spatial structure. Traditional causal discovery algorithms, such as the PC algorithm, rely on conditional independence tests to prune edges from a fully connected graph and then orient the remaining edges using collider and acyclicity rules. While these methods work well for independent and identically distributed (i.i.d.) observations, they struggle when observations are autocorrelated in time or space, or when latent confounders are present.

To cope with temporal dependence, the authors review time‑series extensions of constraint‑based methods, focusing on PCMCI+. PCMCI+ separates the discovery task into a lagged stage—where only edges from past to present are considered, guaranteeing known directionality—and an instantaneous stage, where edges among variables at the same time point are examined. By exploiting the fact that past nodes cannot be colliders, PCMCI+ dramatically reduces the search space for separating sets, making the algorithm computationally feasible even when the number of lagged variables grows with the chosen maximum lag τ. However, PCMCI+ assumes that, after accounting for lagged effects, the remaining noise is i.i.d. across time. Empirical evidence shows that spatial autocorrelation violates this assumption and leads to inflated false‑positive rates.

The authors distinguish two goals for spatiotemporal causal discovery. “Latent‑pattern” discovery seeks to map how a single variable propagates across space (e.g., volcanic ash dispersion). “Latent‑mechanism” discovery aims to uncover the underlying causal network among multiple variables that may vary across locations and time—a task of particular relevance for ecology, economics, and public health where policy decisions depend on mechanistic understanding.

To address latent‑mechanism discovery, the paper draws on the joint causal inference framework, which treats spatial locations as exogenous “contexts.” By encoding each location (or dataset) as a one‑hot context variable, Joint PCMCI+ (J‑PCMCI+) can incorporate spatial heterogeneity while preserving the asymptotic consistency guarantees of PCMCI+. The authors note, however, that this approach inflates dimensionality, adds many dummy variables, and does not exploit the known spatial correlation structure, thereby increasing multiple‑testing burden without gaining statistical power.

Consequently, the authors propose augmenting the conditional independence testing stage with explicit spatial modeling. They suggest using spatial weighting schemes, Bayesian spatial filters, or graph‑Laplacian regularization to remove spatial autocorrelation from residuals before applying the Generalized Covariance Measure (GCM) test. By feeding spatially decorrelated residuals into GCM, the power of independence tests improves and the violation of PCMCI+’s i.i.d. noise assumption is mitigated.

The paper also outlines several open challenges. First, efficient algorithms for separating set search in high‑dimensional, densely sampled spatial grids are lacking. Second, simultaneous handling of temporal non‑stationarity (seasonality, trends) and spatial non‑stationarity remains an unsolved statistical problem. Third, there is no standardized benchmark suite for evaluating spatiotemporal causal discovery methods on realistic ecological, epidemiological, or economic datasets.

Future research directions suggested include developing multi‑scale space‑time graph neural networks that can learn conditional independencies implicitly, Bayesian structure learning that incorporates spatial priors, and simulation‑based validation frameworks that generate realistic spatiotemporal data with known ground‑truth causal graphs. By integrating these advances, the authors envision a robust, scalable toolbox for uncovering latent mechanisms in complex spatiotemporal systems, thereby supporting more informed decision‑making in science and policy.