Causal Ordering for Structure Learning From Time Series

TL;DR

DOTS tackles temporal causal discovery by aggregating multiple valid causal orderings generated via diffusion-based score estimation. By embedding time series with lags, leveraging multi-scale diffusion steps to produce diverse orderings, and combining them with soft voting and CAM pruning, DOTS recovers the transitive closure ${\mathcal{G}}^+$ of the temporal DAG while mitigating single-ordering artifacts. Theoretical results show that ordering aggregation yields a robust representation of reachability and practical improvements are demonstrated: synthetic benchmarks see higher $F1$ scores (e.g., window graphs improving from 0.63 to 0.81) and real-world CausalTime data achieve the best-average $F1_S$ with substantially reduced runtimes. Overall, DOTS provides a scalable, accurate approach to temporal causal discovery, with potential extensions to non-stationary or confounded settings and further diffusion-based efficiency gains.

Abstract

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 grow. 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 Temporal Structure), using diffusion-based causal discovery for temporal data. By integrating multiple orderings, DOTS effectively recovers the transitive closure of the underlying directed acyclic graph, 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 ($d{=}\!3-\!6$ variables, $T{=}200\!-\!5{,}000$ samples), DOTS improves mean window-graph $F1$ from $0.63$ (best baseline) to $0.81$. On the CausalTime real-world benchmark ($d{=}20\!-\!36$), 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.

Figures (16)

• Figure 1: Temporal causal discovery estimates, from raw time series data (left), the underlying temporal causal DAG $\mathcal{G}$ (right).
• Figure 2: A DAG can be represented as an ordered list following the causal direction. A node in the ordering can cause any subsequent node. Searching over the space of permutations is more efficient than searching over the 2D space of matrices. However, topologically sorting nodes reduces the amount of information in the representation of causal relationships.
• Figure 3: Impact of multiple causal orderings on DAG recovery. A single (or few) ordering (left) may include extra or spurious edges, whereas aggregating multiple orderings (right) more accurately recovers the full transitive closure of the underlying DAG.
• Figure 4: DOTS pipeline for temporal causal discovery. We start with lag-embedded time-series data, apply diffusion-based single-order discovery, then extend to multiple orderings and aggregate them. The final temporal DAG below shows an example with three variables over three timesteps.
• Figure 5: Frequency emphasis of diffusion steps. A forward diffusion step decomposes ${\textnormal{x}}_k=\sqrt{\alpha_k}{\textnormal{x}}_0+ \sqrt{1-\alpha_k}\,\epsilon$. As can be seen in the Fourier domain, high values of $k$ emphasize learning of low-frequency while low values of $k$ force the network to focus on high-frequency components.

Theorems & Definitions (8)

• Proposition 1: Reconstruction of the Transitive Closure
• proof : Justification
• Example 1
• Definition 1: Partial Order
• Definition 2: Strict Partial Order
• Definition 3: Linear Extension
• Definition 4: Reachability Relation
• Definition 5: Transitive Closure