DCD: Decomposition-based Causal Discovery from Autocorrelated and Non-Stationary Temporal Data

Muhammad Hasan Ferdous; Ian Stockwell; Md Osman Gani

cs.LGFeb 1, 2026

DCD: Decomposition-based Causal Discovery from Autocorrelated and Non-Stationary Temporal Data

Authors: Muhammad Hasan Ferdous, Ian Stockwell, Md Osman Gani

Organizations: Causal AI Lab, Department of Information Systems, University of Maryland, Baltimore County, Baltimore, MD, USA

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.

Figures & tables

Figure 1: Overview of DCD. STL splits each series into trend, seasonal, and residual components. ADF/KPSS and HSIC tests mark time-proxy edges ( $t\to X$ ), PCMCI+ on the residuals recovers mechanistic edges ( $X\to Y$ ), and the component graphs are merged into one multi-scale graph.

Figure 2: Empirical check of Lemma 1 . (a) The true seasonal component

S_{X}

drives

S_{Y}

. (b) The true residual

R_{X}

drives

R_{Y}

. (c) STL separates the estimated seasonal

\widehat{S}_{Y}

and residual

\widehat{R}_{Y}

. (d)

S_{X}

is uncorrelated with the estimated residual

\widehat{R}_{Y}

(

\rho\approx 0.0008

Figure 3: Aggregate performance on synthetic data. DCD gives the best balance of TPR, FDR, and SHD across conditions.

Method	Lag 2			Lag 3			Lag 4
	TPR $\uparrow$	FDR $\downarrow$	SHD $\downarrow$	TPR $\uparrow$	FDR $\downarrow$	SHD $\downarrow$	TPR $\uparrow$	FDR $\downarrow$	SHD $\downarrow$
DCD	0.78 $\pm$ 0.03	0.28 $\pm$ 0.13	7.83 $\pm$ 3.90	0.79 $\pm$ 0.04	0.38 $\pm$ 0.15	10.42 $\pm$ 4.89	0.80 $\pm$ 0.05	0.48 $\pm$ 0.06	13.00 $\pm$ 3.64
PCMCI+	0.21 $\pm$ 0.11	0.81 $\pm$ 0.11	10.83 $\pm$ 3.07	0.30 $\pm$ 0.11	0.72 $\pm$ 0.22	13.50 $\pm$ 6.26	0.29 $\pm$ 0.08	0.86 $\pm$ 0.04	17.75 $\pm$ 2.70
CD-NOD	0.23 $\pm$ 0.04	0.57 $\pm$ 0.05	9.50 $\pm$ 1.73	0.23 $\pm$ 0.04	0.57 $\pm$ 0.05	9.75 $\pm$ 1.54	0.23 $\pm$ 0.04	0.57 $\pm$ 0.05	9.75 $\pm$ 1.54
CFCI	0.32 $\pm$ 0.06	0.88 $\pm$ 0.02	19.25 $\pm$ 6.18	0.30 $\pm$ 0.06	0.88 $\pm$ 0.01	24.00 $\pm$ 5.51	0.29 $\pm$ 0.08	0.88 $\pm$ 0.02	26.00 $\pm$ 5.70
CPC	0.18 $\pm$ 0.11	0.91 $\pm$ 0.06	17.08 $\pm$ 4.78	0.19 $\pm$ 0.06	0.91 $\pm$ 0.03	23.08 $\pm$ 5.26	0.20 $\pm$ 0.11	0.92 $\pm$ 0.03	26.25 $\pm$ 5.56

Table 1: Performance at different ground-truth lags (mean

\pm

std over seeds). Best value per column in bold.

Method	$d=4$			$d=6$			$d=8$			$d=10$
	TPR $\uparrow$	FDR $\downarrow$	SHD $\downarrow$	TPR $\uparrow$	FDR $\downarrow$	SHD $\downarrow$	TPR $\uparrow$	FDR $\downarrow$	SHD $\downarrow$	TPR $\uparrow$	FDR $\downarrow$	SHD $\downarrow$
DCD	0.84 $\pm$ 0.06	0.20 $\pm$ 0.15	4.11 $\pm$ 2.32	0.77 $\pm$ 0.00	0.42 $\pm$ 0.11	10.78 $\pm$ 3.49	0.79 $\pm$ 0.00	0.46 $\pm$ 0.06	12.44 $\pm$ 2.30	0.76 $\pm$ 0.00	0.44 $\pm$ 0.04	14.33 $\pm$ 1.58
PCMCI+	0.34 $\pm$ 0.16	0.69 $\pm$ 0.26	8.44 $\pm$ 4.50	0.19 $\pm$ 0.10	0.90 $\pm$ 0.05	14.11 $\pm$ 3.72	0.25 $\pm$ 0.00	0.80 $\pm$ 0.05	14.78 $\pm$ 2.64	0.30 $\pm$ 0.00	0.79 $\pm$ 0.06	18.78 $\pm$ 3.35
CD-NOD	0.23 $\pm$ 0.01	0.50 $\pm$ 0.00	8.67 $\pm$ 0.50	0.18 $\pm$ 0.00	0.60 $\pm$ 0.00	12.00 $\pm$ 0.00	0.29 $\pm$ 0.00	0.60 $\pm$ 0.00	8.00 $\pm$ 0.00	0.22 $\pm$ 0.00	0.60 $\pm$ 0.00	10.00 $\pm$ 0.00
CFCI	0.26 $\pm$ 0.06	0.89 $\pm$ 0.01	18.56 $\pm$ 5.57	0.35 $\pm$ 0.10	0.87 $\pm$ 0.02	18.67 $\pm$ 1.66	0.29 $\pm$ 0.06	0.88 $\pm$ 0.02	24.56 $\pm$ 4.53	0.31 $\pm$ 0.03	0.88 $\pm$ 0.02	30.56 $\pm$ 3.24
CPC	0.07 $\pm$ 0.06	0.95 $\pm$ 0.03	18.44 $\pm$ 4.98	0.23 $\pm$ 0.12	0.90 $\pm$ 0.04	17.11 $\pm$ 3.14	0.24 $\pm$ 0.04	0.88 $\pm$ 0.04	24.67 $\pm$ 5.41	0.20 $\pm$ 0.00	0.91 $\pm$ 0.02	28.33 $\pm$ 4.36

Table 2: Performance at different numbers of variables

d

(mean

\pm

std over seeds). Best value per column in bold.

Figure 4: Causal graphs inferred by (a) DCD, (b) PCMCI+, (c) CD-NOD, (d) BOSS-LiNGAM, (e) CFCI, (f) CPC, and (g) DYNOTEARS on Arctic sea ice. DCD captures seasonal (S) and time-dependent (T) influences explicitly.

Method	Input	TPR $\uparrow$	FDR $\downarrow$	SHD $\downarrow$
PCMCI+	Raw	0.45 $\pm$ 0.08	0.81 $\pm$ 0.02	28.5 $\pm$ 2.1
PCMCI+	Residuals	0.83 $\pm$ 0.11	0.59 $\pm$ 0.04	11.2 $\pm$ 1.6
DYNOTEARS	Raw	0.67 $\pm$ 0.05	0.85 $\pm$ 0.02	25.0 $\pm$ 1.8
DYNOTEARS	Residuals	0.79 $\pm$ 0.06	0.68 $\pm$ 0.03	14.8 $\pm$ 1.7
DCD	Multi-scale	1.00 $\pm$ 0.00	0.50 $\pm$ 0.09	6.0 $\pm$ 1.7

Table 3: Isolation test (

d{=}6

n{=}1000

, mean

\pm

std over 3 seeds). “Raw” keeps trend and seasonality, “Residuals” runs the method on the STL residuals

R_{i}(t)

only, and “Multi-scale” is full DCD.

Appendix figures & tables8 assets

Supplementary material from the paper’s appendix.

Appendix

Method	Settings
DCD	STL with period $P_{i}^{\star}\in\{2,\dots,50\}$ chosen by seasonal variance, variance gate $\tau_{S}=0.1$ , seasonal smoother $15$ , trend smoother $\max(P_{i}^{\star}+1,35)$ rounded up to odd, no robustness iterations. ADF and KPSS with a constant term at $\alpha=0.05$ . Permutation HSIC with RBF kernels ( $\gamma=0.1$ ), $500$ permutations, $\alpha=0.05$ . Residual stage PCMCI+ with CMI-knn ( $\mathrm{knn}=0.2$ , $5$ shuffle neighbors, rank transform), $\mathrm{pc\_alpha}=0.05$ .
PCMCI+	CMI-knn ( $\mathrm{knn}=0.2$ , $5$ shuffle neighbors, rank transform), shuffle significance test, $\mathrm{pc\_alpha}=0.05$ , majority collider rule, no graph-wide FDR correction.
CD-NOD	causal-learn cdnod , KCI test, $\alpha=0.05$ , stable option, $\mathrm{uc\_rule}=0$ , $\mathrm{uc\_priority}=-1$ , sample index as surrogate.
CFCI, CPC	Tetrad GUI defaults, Fisher-Z test, $\alpha=0.01$ .
BOSS-LiNGAM	Tetrad GUI defaults.
DYNOTEARS	causalnex from_pandas_dynamic , $\lambda_{w}=\lambda_{a}=0.05$ ( $0.01$ on Arctic), weight threshold $0.05$ .

Appendix

Table 5: Method settings. On synthetic data every method uses the generative lag as

\tau_{\max}

. On real data

\tau_{\max}=3

for Arctic and

2

for ETTh1, except DYNOTEARS on Arctic (lag order

2

$d$	Period $P$	TPR $\uparrow$	FDR $\downarrow$	SHD $\downarrow$
	10 (Inc.)	0.500 $\pm$ 0.00	0.757 $\pm$ 0.04	8.33 $\pm$ 1.15
	15 (Opt.)	0.750 $\pm$ 0.00	0.565 $\pm$ 0.06	5.00 $\pm$ 1.00
4	20 (Inc.)	0.500 $\pm$ 0.00	0.747 $\pm$ 0.03	8.00 $\pm$ 1.00
	30 (Opt.)	0.750 $\pm$ 0.00	0.133 $\pm$ 0.23	1.67 $\pm$ 1.15
	35 (Inc.)	0.583 $\pm$ 0.14	0.720 $\pm$ 0.06	7.67 $\pm$ 1.15
	10 (Inc.)	0.722 $\pm$ 0.10	0.705 $\pm$ 0.03	12.00 $\pm$ 1.00

Appendix

Table 6: Effect of the STL period

P

(mean

\pm

std over

n\in\{500,1000,1500\}

\tau_{\max}=4

). Opt. marks periods that match the generator, Inc. marks mismatched periods.

$d$	$\tau_{\max}$	TPR $\uparrow$	FDR $\downarrow$	SHD $\downarrow$
	2	0.750 $\pm$ 0.00	0.217 $\pm$ 0.20	2.00 $\pm$ 1.00
4	4	0.750 $\pm$ 0.00	0.133 $\pm$ 0.23	1.67 $\pm$ 1.15
	6	0.750 $\pm$ 0.00	0.300 $\pm$ 0.09	2.33 $\pm$ 0.58
	2	1.000 $\pm$ 0.00	0.444 $\pm$ 0.10	5.00 $\pm$ 1.73
6	4	1.000 $\pm$ 0.00	0.490 $\pm$ 0.09	6.00 $\pm$ 2.00
	6	1.000 $\pm$ 0.00	0.475 $\pm$ 0.09	5.67 $\pm$ 2.08

Appendix

Table 7: Effect of the search depth

\tau_{\max}

(mean

\pm

std over

n\in\{500,1000,1500\}

P=30

Coupling $\lambda$	Leakage $\varepsilon$	TPR $\uparrow$	FDR $\downarrow$	SHD $\downarrow$
0.0	0.02	1.000 $\pm$ 0.00	0.444 $\pm$ 0.10	5.0 $\pm$ 1.7
0.2	0.06	0.944 $\pm$ 0.05	0.512 $\pm$ 0.06	6.8 $\pm$ 1.5
0.5	0.12	0.861 $\pm$ 0.07	0.589 $\pm$ 0.05	9.2 $\pm$ 1.9
1.0	0.25	0.722 $\pm$ 0.09	0.694 $\pm$ 0.04	13.5 $\pm$ 2.4

Appendix

Table 8: DCD under increasing spectral leakage (mean

\pm

std over 3 seeds,

d=6

n=1000

P=30

\tau_{\max}=2

Method	$n=500$			$n=1000$			$n=1500$
	TPR $\uparrow$	FDR $\downarrow$	SHD $\downarrow$	TPR $\uparrow$	FDR $\downarrow$	SHD $\downarrow$	TPR $\uparrow$	FDR $\downarrow$	SHD $\downarrow$
DCD	0.79 $\pm$ 0.03	0.36 $\pm$ 0.16	10.08 $\pm$ 4.74	0.79 $\pm$ 0.05	0.38 $\pm$ 0.14	10.33 $\pm$ 4.72	0.79 $\pm$ 0.04	0.40 $\pm$ 0.13	10.83 $\pm$ 4.69
PCMCI+	0.27 $\pm$ 0.08	0.79 $\pm$ 0.16	14.17 $\pm$ 5.04	0.28 $\pm$ 0.12	0.77 $\pm$ 0.18	13.92 $\pm$ 5.47	0.25 $\pm$ 0.13	0.82 $\pm$ 0.13	14.00 $\pm$ 5.20
CD-NOD	0.23 $\pm$ 0.04	0.57 $\pm$ 0.05	9.67 $\pm$ 1.61	0.23 $\pm$ 0.04	0.57 $\pm$ 0.05	9.67 $\pm$ 1.61	0.23 $\pm$ 0.04	0.57 $\pm$ 0.05	9.67 $\pm$ 1.61
CFCI	0.32 $\pm$ 0.07	0.88 $\pm$ 0.02	24.58 $\pm$ 7.27	0.28 $\pm$ 0.06	0.88 $\pm$ 0.01	22.58 $\pm$ 5.73	0.30 $\pm$ 0.08	0.88 $\pm$ 0.01	22.08 $\pm$ 6.16
CPC	0.19 $\pm$ 0.12	0.91 $\pm$ 0.04	22.75 $\pm$ 6.96	0.19 $\pm$ 0.09	0.91 $\pm$ 0.05	22.25 $\pm$ 6.15	0.18 $\pm$ 0.08	0.91 $\pm$ 0.04	21.42 $\pm$ 6.44

Appendix

Table 9: Performance at different sample sizes (mean

\pm

std). Best value per column in bold.

Figure 5: Effect of sample size on TPR, FDR, and SHD. Shaded bands show $\pm$ one standard deviation.

Figure 6: Effect of the generative lag on TPR, FDR, and SHD. Shaded bands show $\pm$ one standard deviation.

Figure 7: Effect of the number of variables on TPR, FDR, and SHD. Shaded bands show $\pm$ one standard deviation. The FDR panel starts at $0.4$ , so DCD’s value at $d=4$ ( $0.20$ , Table 2 ) lies below the plotted range.

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Time-Series Causal Discovery Causal Structure Learning Non-Stationary Time Series Time Series Transformers

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The widespread availability of complex time series data in various domains such as environmental science, epidemiology, and economics demands robust causal discovery methods that can identify intricate contemporaneous and lagged relationships in non-stationary, nonlinear, and noisy settings. Existing constraint-based methods often rely heavily on conditional independence tests that degrade for limited data samples and complex distributions, while score-based methods impose strong statistical assumptions. Recent methods address special cases such as change point detection or distribution shifts, but struggle to provide a unified solution. We propose the Transformer Integrated Temporal Causal Discovery (TTCD) Framework, a novel end-to-end approach that learns contemporaneous and lagged causal relations from non-stationary time series. TTCD introduces a Non-Stationary Feature Learner integrating temporal and frequency-domain attention with dynamic non-stationarity profiling, and a custom Causal Structure Learner. A key innovation is reconstruction-guided causal signal distillation, to distill essential causal signals through the reconstruction process of the transformer decoder, which mitigates noise and spurious correlations while preserving meaningful dependencies. The Causal Structure Learner operates on distilled reconstructed signals to infer the underlying causal graph without restrictive assumptions on noise distributions or data generation processes. Experiments on synthetic, benchmark, and real world datasets show that TTCD consistently outperforms state-of-the-art baselines in both accuracy and consistency with domain knowledge, demonstrating the approach's effectiveness for causal discovery in challenging real world contexts.

Abstract

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