VCDF: A Validated Consensus-Driven Framework for Time Series Causal Discovery

Gene Yu; Ce Guo; Wayne Luk

cs.LGFeb 22, 2026

VCDF: A Validated Consensus-Driven Framework for Time Series Causal Discovery

Authors: Gene Yu, Ce Guo, Wayne Luk

Organizations: Department of Computing, Imperial College London, London, United Kingdom

Abstract

Time series causal discovery is essential for understanding dynamic systems, yet many existing methods remain sensitive to noise, non-stationarity, and sampling variability. We propose the Validated Consensus-Driven Framework (VCDF), a simple and method-agnostic layer that improves robustness by evaluating the stability of causal relations across blocked temporal subsets. VCDF requires no modification to base algorithms and can be applied to methods such as VAR-LiNGAM and PCMCI. Experiments on synthetic datasets show that VCDF improves VAR-LiNGAM by approximately 0.08-0.12 in both window and summary F1 scores across diverse data characteristics, with gains most pronounced for moderate-to-long sequences. The framework also benefits from longer sequences, yielding up to 0.18 absolute improvement on time series of length 1000 and above. Evaluations on simulated fMRI data and IT-monitoring scenarios further demonstrate enhanced stability and structural accuracy under realistic noise conditions. VCDF provides an effective reliability layer for time series causal discovery without altering underlying modeling assumptions.

Figures & tables

Figure 1: Illustration of fold-to-fold variability and validated edges in VCDF. Consistent relations across partitions are retained; unstable ones are removed.

Figure 2: Workflow of VCDF. Each fold produces a causal estimate; stability validation aggregates them into a final graph.

Figure 3: Comparison of VAR-LiNGAM, VCDF-VARLiNGAM, PCMCI, and VCDF-PCMCI across representative synthetic settings.

Setting	Method	F1 score
Setting	Method	Window	Summary
Linear	PCMCI	0.37 $\pm$ 0.02	0.49 $\pm$ 0.03
	VCDF-PCMCI	0.43 $\pm$ 0.02	0.51 $\pm$ 0.02
	VARLiNGAM	0.64 $\pm$ 0.03	0.69 $\pm$ 0.03
	VCDF-VARLiNGAM	0.75 $\pm$ 0.03	0.80 $\pm$ 0.03
	TCDF	0.38 $\pm$ 0.03	0.43 $\pm$ 0.03

Table 1: Performance across four representative synthetic settings.

Length $T$	Method	F1 score
Length $T$	Method	Window	Summary
250	VARLiNGAM	0.65 $\pm$ 0.05	0.71 $\pm$ 0.03
	VCDF-VARLiNGAM	0.65 $\pm$ 0.04	0.67 $\pm$ 0.03
	PCMCI	0.39 $\pm$ 0.03	0.52 $\pm$ 0.03
1000	VARLiNGAM	0.64 $\pm$ 0.02	0.71 $\pm$ 0.03
	VCDF-VARLiNGAM	0.79 $\pm$ 0.04	0.82 $\pm$ 0.04

Table 2: Performance across different time-series lengths (synthetic,

n=15

Dataset	Method	F1 score
Dataset	Method	Window	Summary
fMRI	VARLiNGAM	0.48 $\pm$ 0.07	0.61 $\pm$ 0.12
	VCDF-VARLiNGAM	0.51 $\pm$ 0.14	0.57 $\pm$ 0.17
	VCDF-PCMCI	0.39 $\pm$ 0.12	0.64 $\pm$ 0.11
IT Antivirus	VARLiNGAM	—	0.35
	VCDF-VARLiNGAM	—	0.49

Table 3: Performance on domain-specific datasets (summary F1 shown where applicable).

Length $T$	VAR-LiNGAM	VCDF-VARLiNGAM
250	0.50	2.83
500	0.65	3.62
1000	0.96	5.15
2000	1.75	8.66

Table 4: Runtime comparison (seconds) on synthetic datasets with 15 variables.

Time-Series Causal Discovery

Time-series causal discovery seeks to infer which variables influence others over time and when those effects occur.

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Time-Series Causal Discovery Causal Discovery

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Causal discovery with instantaneous effects in multivariate time series is challenging, as the instantaneous structure must be acyclic. Prior methods enforce this by either separating instantaneous and lagged estimation into multi-stage pipelines or imposing algebraic acyclicity constraints via complex augmented Lagrangian optimization, both of which incur high computational cost. In this work, we propose a different approach: we learn a differentiable permutation of variables using the Gumbel--Sinkhorn operator and triangularize the instantaneous coefficient matrix of a Structural Vector Autoregressive (SVAR) model in the learned order. This converts acyclicity from a hard constraint into a parameterization and keeps it valid throughout optimization. In doing so, our method enables unified, continuous optimization with gradient-based learning, leading to improved efficiency in time--series causal discovery. Across three real-world benchmarks, our method achieves the best overall performance compared with 12 baselines in both discovery accuracy and efficiency. On the large-scale benchmark, it further demonstrates strong scalability, achieving more than a 6x speedup over competing methods.

Abstract

Time-Series Causal Discovery

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