When experiments are impossible, scientists extract causality from observations. For systems distributed in space and time, this is especially hard: the number of grid cells often exceeds the number of time slices by orders of magnitude. Classical methods either reduce dimensionality, losing local interactions, or work with only one variable. M-CaStLe solves this problem by treating each grid point as a repetition of the same local mechanism.
M-CaStLe builds a locally encoded neighborhood structure (LENS): for each interior grid cell, it collects nine neighboring cells (Moore neighborhood) and all variables within them. These neighborhoods are arranged into a common matrix where each row is one spatial repetition. Then the parent identification phase (PIP) applies a standard causal discovery algorithm (e.g., PCMCI or DYNOTEARS) to this matrix, but searches for parents only for variables in the central cell. This trick increases the effective sample size from T to roughly T·(N−2)², where N is the grid size.
On synthetic tests with vector autoregression, M-CaStLe achieved an F1 score of up to 0.96 with one variable and maintained accuracy above 0.9 with six variables, outperforming direct spatial analysis and separate estimation. In an advection-diffusion-reaction task, the median error in estimating the transport angle was 4.76°, and the reaction graph was recovered perfectly in more than 90% of cases. For the Pinatubo eruption data with only seven time slices, M-CaStLe recovered the chain SO2 → H2SO4 → SO4 with an F1 of 0.95, whereas PCMCI on spatially averaged series found no links at all. On El Niño data, the method revealed the expected asymmetry: ocean temperature influences convection, but not the other way around.
M-CaStLe shows that local causality in multidimensional grid data can be extracted without losing spatial detail. This opens the way to analyzing fast and local phenomena, such as volcanic plumes, weather fronts, or pollutant dispersion, where global methods fail. Jointly estimating all variables in the neighborhood helps avoid errors that arise when analyzing each variable sequentially.
The method can be extended to larger-radius neighborhoods or adaptive grids, which is useful when spatial and temporal resolutions do not match. For nonstationary systems, a procedure to find stationary regions where M-CaStLe's assumptions hold is possible. Application to irregular grids, for example in hydrology or ecology, is also promising.
The results will impact Earth sciences (climate, atmospheric chemistry, oceanography), epidemiology, and hydrology, where data are often on regular grids and understanding local causal mechanisms is needed.
Immediate next steps include adapting M-CaStLe to non-square and irregular grids, and developing data-driven methods for choosing the optimal neighborhood size.
In causal inference for spatiotemporal systems, the problem of separating direct and mediated causal links under strong spatial correlation remains unsolved. M-CaStLe mitigates it through local constraints, but the question of correctly handling distant dependencies (teleconnections) remains open.
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What the work does
This paper introduces M-CaStLe, a multidimensional generalization of the CaStLe algorithm for discovering causal structures in data on regular grids. The method combines information about the spatial neighborhood of each cell and multiple variables within it, enabling joint estimation of transport and cross-variable interactions. On synthetic tests and real-world data (atmospheric chemistry after the Pinatubo eruption, El Niño phases), M-CaStLe recovers causal links more accurately than standard approaches.
Where its strength is
The strength of this work lies in exploiting spatial repetitions of local structures to increase the effective sample size, thereby overcoming the limitation of few time slices. Jointly estimating all variables in the neighborhood allows detection of both transport and cross-variable interactions without sacrificing spatial detail.
Where it could go next
Worth trying
Nearby works in our archive
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For the Pinatubo data with 7 time slices, M-CaStLe obtained 5488 effective samples per variable, which is 769 times more than the original number of observations over time.