Causal explanations of outliers in systems with lagged time-dependencies

TL;DR

This work addresses root-cause explanations in time-dependent systems with memory by extending the causal root-cause analysis (CRCA) to lagged dependencies within a structural causal model (SCM). It introduces two finite-unfolding strategies for the infinite time-dependency graph: a truncation approach that preserves all causal mechanisms up to a maximum lag $L$, and a non-truncation approach that approximates dangling dependencies. Using a memory-afflicted energy DGP with injections that generate peaks, the study shows that both variants can localize root-causes across features and time, with the truncated model excelling in time localization for longer delays ($L \ge 7$) and the non-truncated model often achieving better feature localization due to more attributable nodes. The findings provide a blueprint for applying CRCA to energy systems and highlight a trade-off between mechanism fidelity and temporal reach, with computational cost dominated by Shapley-value calculations. This work paves the way for more scalable, time-aware causal explanations of energy peaks and suggests future directions toward recurrent causal automata.

Abstract

Root-cause analysis in controlled time dependent systems poses a major challenge in applications. Especially energy systems are difficult to handle as they exhibit instantaneous as well as delayed effects and if equipped with storage, do have a memory. In this paper we adapt the causal root-cause analysis method of Budhathoki et al. [2022] to general time-dependent systems, as it can be regarded as a strictly causal definition of the term "root-cause". Particularly, we discuss two truncation approaches to handle the infinite dependency graphs present in time-dependent systems. While one leaves the causal mechanisms intact, the other approximates the mechanisms at the start nodes. The effectiveness of the different approaches is benchmarked using a challenging data generation process inspired by a problem in factory energy management: the avoidance of peaks in the power consumption. We show that given enough lags our extension is able to localize the root-causes in the feature and time domain. Further the effect of mechanism approximation is discussed.

Figures (18)

• Figure 1: Dependencies of an exemplary system with $X^1_t \coloneqq f_1(X^2_{t-2}, X^3_{t-1}\,|\, N^1)$, $X^2_t \coloneqq f_2(X^1_t\,|\,N^2)$ and $X^3_t \coloneqq f_3(X^2_t\,|\,N^3)$.
• Figure 2: Unfolding and truncation of example graph shown in Figure \ref{['fig:dep_graph_summary']} with $L=2$: The rectangular nodes represent the dangling parents $\mathop{\mathrm{Pa}}\nolimits_{[t-2,t]}(3) = \{X^2_{t-4},\, X^2_{t-3},\, X^3_{t-3}\}$ on which we condition in the truncation model whereas the noise terms $N_{[t-L,\,t]} = \{(N_{t-2}, N_{t-1}, N_t)\}$ of the circular nodes are assessed in the attribution analysis. In the non-truncated case the mechanisms and noise distributions of the dangling nodes are adapted the remaining dependencies (e.g. $X^2_{t-3} \to X^3_{t-3}$ remains while $X^2_{t-3}$ has no more parents) and then also employed for the attribution analysis.
• Figure 3: System summary graph of the DGP
• Figure 4: Exemplary DGP output: The dotted horizontal line in the $\mathop{\mathrm{Grid}}\nolimits$ graph signifies the peak limit
• Figure 5: Exemplary system propagation with (orange) and without (blue) injection. The most relevant system nodes are shown. The injection starts at the vertical dotted line and the peak limit is given by the horizontal dotted line. Peaks caused by the injection (marked with $X$) are determined by comparison with the baseline.