Go to TMLR homepageDCD: Decomposition-based Causal Discovery from Autocorrelated and Non-Stationary Temporal Data
Abstract: Multivariate time series in domains such as finance, climate science, and healthcare often exhibit long-term trends, seasonal patterns, and short-term fluctuations, complicating causal inference under non-stationarity and autocorrelation. Existing causal discovery methods typically operate on raw observations, making them vulnerable to spurious edges and misattributed temporal dependencies. We introduce a decomposition-based causal discovery framework that separates each time series into trend, seasonal, and residual components and performs component-specific causal analysis. Trend components are assessed using stationarity tests, seasonal components using kernel-based dependence measures, and residual components using constraint-based causal discovery. The resulting component-level graphs are integrated into a unified multi-scale causal structure. This approach isolates long- and short-range causal effects, reduces spurious associations, and improves interpretability. Across extensive synthetic benchmarks and real-world climate data, our framework more accurately recovers ground-truth causal structure than state-of-the-art baselines, particularly under strong non-stationarity and temporal autocorrelation.
Submission Type: Regular submission (no more than 12 pages of main content)
Changes Since Last Submission: We have implemented a major revision addressing all three reviewers' feedback. The changes span theoretical formalism, empirical completeness, and presentation.
THEORETICAL CONTRIBUTIONS (Addressing Reviewer Q91o)
1. Complete Rewrite of Section 3 (Theory): Discarded heuristic spectral arguments and introduced a formal Multi-Scale Structural Causal Model (MS-SCM) with explicit latent variables and exogenous periodic drivers.
2. Non-Circular Assumptions: Replaced the invalid "Bounded Leakage" assumption with a standard "Estimator Consistency" assumption (A3). Added intuitive explanations for Spectral Separability (A2) connecting it back to trend/seasonal/residual mechanisms.
3. Rigorous Identifiability Theorem: Replaced heuristic lemmas with a Theorem of Asymptotic Structural Identifiability (Theorem 1) with a complete three-step proof showing that conditioning on decomposed components is equivalent to blocking back-door paths from latent drivers.
4. Added Remark on Non-Linear Extensions clarifying that Linear Gaussianity (A3) is used for efficiency of the estimator, not as a fundamental limit of the framework.
EMPIRICAL COMPLETENESS (Addressing Reviewer qyzU)
5. Added DYNOTEARS as a Baseline: Full integration across all synthetic tables (Tables 1, 2), aggregate performance figure (Figure 3), real-world qualitative graphs (Figure 7g), and detailed failure-mode analysis in Sections 4.4 and 4.5.
6. Sensitivity to Spectral Separability (New Appendix E.4): Added a controlled experiment inducing amplitude modulation S_i(t) = A_i[1 + λT_i(t)]sin(2πt/P) to test graceful decline. Results show monotonic degradation (TPR 1.00 → 0.72, SHD 5.0 → 13.5) rather than catastrophic failure as λ varies from 0 to 1.0.
7. Isolation Test for Multi-Scale Integration (New Appendix E.3): Ran PCMCI+ and DYNOTEARS on STL residuals alone. STL preprocessing improves both (PCMCI+: SHD 28.5 → 11.2; DYNOTEARS: SHD 25.0 → 14.8) but neither matches DCD's multi-scale integration (SHD 6.0), confirming the gains arise from integration, not just preprocessing.
8. Robustness to Decomposition Misalignment (New Appendix E.1): Perturbed STL period P around true periodicities across d ∈ {4,6,8}. Demonstrates graceful decline: at d=4, optimal P=30 gives SHD 1.67 while misaligned P=10 still gives SHD 8.33 (still better than all baselines).
9. Clarified Edge Interpretation: Explicitly defined t → X_i as "time-proxy edges" (exogenous temporal forcing / latent time-dependent drivers) as distinct from "mechanistic edges" (variable-to-variable X → Y), addressing the SCM-semantics concern.
10. Qualitative Framing: Reframed real-world results as "Qualitative Case Studies" (Sections 4.4, 4.5) rather than strong validation, acknowledging the absence of ground truth.
11. Discussion of Failure Modes: Expanded the Limitations section into five distinct subsections covering spectral separability violation, exogenous treatment of trends, stability of residual structure, decomposition quality on short series, and real-world evaluation.
PRESENTATION AND CLARITY (Addressing Reviewer KkWU)
12. Softened Tone in Related Work: Revised discussion of Granger causality and other baselines; acknowledged linearity assumptions in our own theoretical bounds.
13. Clarified Assumptions: Added intuitive explanations connecting spectral separability to the trend/seasonal/residual decomposition. Defined the period P explicitly in Section 3.1.
14. Moved Variable Definitions: All variables including Y(t) and U(t) now defined in the main text and lemma statements, not in proofs.
15. Explained Component-wise Learning: Clarified how Assumption A1 (Structural Independence) ensures that removing exogenous components does not introduce latent variable bias.
16. Expanded STL Description: Added overview of Seasonal-Trend decomposition using LOESS, including the inner/outer loop structure and robustness properties.
17. Unified CI Tests for Fair Comparison: Clarified in Section 4.2 and Appendix C that baselines (PCMCI+) were evaluated using the same CI tests (Partial Correlation and CMI-knn) as DCD, isolating performance gains to the decomposition strategy.
18. Enhanced Dataset Descriptions: Added Table 3 listing all 11 variables in the Arctic Sea Ice dataset, 7 variables in ETTh1, with units, temporal resolution (1979–2018, monthly), and full configuration details.
19. Fixed Figure Legibility: Improved resolution and layout of all real-world graph figures (Figure 7). Broken cross-references (Table ??) corrected.
20. Restructured Appendix: Consolidated all ablation studies into a single organized appendix (Appendix E) with six subsections: period misalignment, causal depth, multi-scale integration, spectral separability, sample size, and lag/dimensionality effects.
Assigned Action Editor: ~Fabio_Stella1
Submission Number: 6893
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