Causal Ordering for Structure Learning from Time Series

Predicting causal structure from time series data is crucial for understanding complex phenomena in physiology, brain connectivity, climate dynamics, and socio-economic behaviour. Causal discovery in time series is hindered by the combinatorial complexity of identifying true causal relationships, especially as the number of variables and time points grows. A common approach to simplify the task is the so-called ordering-based methods. Traditional ordering methods inherently limit the representational capacity of the resulting model. In this work, we fix this issue by leveraging multiple valid causal orderings, instead of a single one as standard practice. We propose DOTS (Diffusion Ordered TemporalStructure), using diffusion-based causal discovery for temporal data. By integrating multiple orderings, DOTS effectively recovers the transitive closure of the underlying directed acyclic graph (DAG), mitigating spurious artifacts inherent in single-ordering approaches. We formalise the problem under standard assumptions such as stationarity and the additive noise model, and leverage score matching with diffusion processes to enable efficient Hessian estimation. Extensive experiments validate the approach. Empirical evaluations on synthetic and real-world datasets demonstrate that DOTS outperforms state-of-the-art baselines, offering a scalable and robust approach to temporal causal discovery. On synthetic benchmarks spanning d=3−6 variables, T=200−5,000 samples and up to three lags, DOTS improves mean window-graph F1 from 0.63 (best baseline) to 0.81. On the CausalTime real-world benchmark (Medical, AQI, Traffic; d=20−36), while baselines remain the best on individual datasets, DOTS attains the highest average summary-graph F1 while halving runtime relative to graph-optimisation methods. These results establish DOTS as a scalable and accurate solution for temporal causal discovery. Code is available at https://github.com/CHAI-UK/DOTS.

Recommended citation: Sanchez, P., Machlanski, D., McDonagh, S., & Tsaftaris, S. A. (2025). "Causal Ordering for Structure Learning from Time Series." Transactions on Machine Learning Research.
Download Paper