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

The paper introduces DCD, a decomposition-based causal discovery framework that separates multivariate time series into trend, seasonal, and residual components to perform targeted causal analysis, thereby improving the accuracy and interpretability of causal structure recovery in non-stationary and autocorrelated data compared to existing methods.

Original authors: Muhammad Hasan Ferdous, Md Osman Gani

Published 2026-04-29

📖 4 min read☕ Coffee break read

CC BY 4.0

Original authors: Muhammad Hasan Ferdous, Md Osman Gani

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to figure out what causes what in a chaotic, noisy room. You have a group of people (variables) shouting, singing, and talking over each other. Some are shouting because of a long-term trend (like a slow-building storm), some are singing a repeating chorus (seasonal patterns), and some are just having random, quick conversations (short-term fluctuations).

The problem is that if you just listen to the whole room at once, you might think two people are having a deep conversation when they are actually just reacting to the same loud siren outside. This is the problem with analyzing complex data like weather or stock markets: non-stationarity (things changing over time) and autocorrelation (things repeating themselves) create "fake" connections that fool standard detective tools.

This paper introduces a new detective tool called DCD (Decomposition-based Causal Discovery). Here is how it works, broken down into simple steps:

1. The "Noise-Canceling Headphones" Strategy

Instead of listening to the messy room all at once, DCD puts on a special pair of headphones that splits the sound into three distinct channels:

The Trend Channel: This captures the slow, long-term drift (like a rising sea level or a slow economic boom).
The Seasonal Channel: This captures the repeating loops (like the seasons changing every year or electricity usage spiking every morning).
The Residual Channel: This captures the "leftover" noise—the quick, short-term interactions between people that aren't caused by the long trends or the repeating loops.

2. Solving the Puzzle in Three Separate Rooms

Once the noise is separated, DCD investigates each channel with a different tool, because each type of noise requires a different kind of detective work:

In the Trend Room: It checks if the slow drift is just random or if it's being driven by an outside force (like time itself). It treats time as a "proxy" driver.
In the Seasonal Room: It looks for repeating cycles. If two variables dance to the same beat, it marks them as connected to the "beat" (time), not necessarily to each other directly.
In the Residual Room: This is the most important part. Now that the long trends and repeating loops are gone, the remaining data is "clean." Here, DCD uses standard causal discovery methods to find the real short-term cause-and-effect links (e.g., "When the temperature drops right now, the humidity changes right now").

3. Putting the Pieces Back Together

Finally, DCD takes the maps from all three rooms and combines them into one big, clear picture.

It draws arrows from Time to variables that are just drifting or cycling (exogenous drivers).
It draws arrows between Variables for the real, short-term mechanical causes found in the clean residual data.

Why is this better than the old way?

The paper tested this against other top-tier detective tools (like PCMCI+ and DYNOTEARS) using fake data and real-world data (Arctic sea ice and electricity transformers).

The Old Tools: When faced with the noisy room, they got confused. They thought the "siren" (seasonality) was a conversation between two people, creating a web of fake connections. They often missed the real, quick interactions because the noise drowned them out.
DCD: By cleaning the data first, it found the true connections much more accurately. It didn't get tricked by the repeating loops or the slow drifts.

The "Leakage" Safety Net

The authors admit that sometimes the "headphones" aren't perfect, and a little bit of the "seasonal song" might leak into the "residual noise" room. They proved mathematically that even if this happens, the tool doesn't crash; it just gets slightly less accurate, but still much better than the competition. It's like having a safety net that catches you even if you trip a little.

Real-World Proof

Arctic Sea Ice: DCD correctly figured out how sea ice, temperature, and radiation interact, separating the yearly winter/summer cycle from the actual cause-and-effect relationships. Other tools either missed the connections or drew confusing, messy lines.
Electricity Transformers: DCD figured out how different parts of a power station load affect each other without getting confused by the daily usage cycles.

In short: DCD is a method that says, "Don't try to solve the whole messy puzzle at once. Clean up the noise first, solve the small pieces separately, and then put them back together." This allows it to see the true causal structure in data that is usually too chaotic to understand.

1. Problem Statement

Multivariate time series in domains like finance, climate science, and healthcare often exhibit long-term trends, seasonal patterns, and short-term fluctuations. These characteristics introduce non-stationarity and autocorrelation, which severely complicate causal inference.

The Core Challenge: Existing causal discovery (CD) methods typically operate on raw observations. They struggle to disentangle long-term causal effects, seasonal dependencies, and transient interactions.
Consequences: This leads to spurious edges (false causal links) and misattributed temporal dependencies. For instance, a shared seasonal cycle between two variables might be misidentified as a direct causal link, or a long-term trend might mask short-term mechanistic relationships.
Gap: While deep learning models have improved forecasting by capturing long-term dependencies, they remain correlational. There is a lack of methods that explicitly integrate time series decomposition with causal inference to separate these temporal scales.

2. Methodology: The DCD Framework

The authors propose Decomposition-based Causal Discovery (DCD), a structured framework that separates each time series into three components—Trend ( $T$ ), Seasonal ( $S$ ), and Residual ( $R$ )—and performs targeted causal analysis on each before integrating the results.

A. Decomposition Stage

Technique: Uses STL (Seasonal-Trend decomposition using LOESS) to decompose multivariate time series $X_i(t) = T_i(t) + S_i(t) + R_i(t)$ .
Rationale: STL is chosen over rigid Fourier bases or moving averages because LOESS smoothing adapts to local non-linearities and minimizes spectral leakage (energy from one component bleeding into another), which is critical for the theoretical guarantees.
Period Selection: The seasonal period $P$ is selected via variance maximization. If no significant seasonality is found, the seasonal component is set to zero.

B. Component-Specific Causal Analysis

DCD applies different inference strategies tailored to the statistical properties of each component:

Trend Analysis (Long-term Dependence):
- Goal: Identify exogenous temporal drivers.
- Method: Applies stationarity tests (ADF and KPSS). If a trend is non-stationary, a directed edge from the time index ( $t \to X_i$ ) is added to the trend graph ( $G_T$ ). This treats time as a latent confounder for long-term dynamics.
Seasonal Analysis (Cyclic Dependence):
- Goal: Identify deterministic cyclic dependencies.
- Method: Uses the Hilbert-Schmidt Independence Criterion (HSIC) with RBF kernels to test independence between the seasonal component $S_i(t)$ and the time index $t$ . Significant dependence adds a seasonal edge ( $t \to X_i$ ) to the seasonal graph ( $G_S$ ).
Residual Analysis (Short-term Causality):
- Goal: Identify intrinsic mechanistic interactions.
- Method: After removing trends and seasonality, the residual $R_i(t)$ approximates a stationary, weakly dependent process. A constraint-based causal discovery algorithm (e.g., PCMCI+) is applied to find lagged ( $X_{t-\tau} \to Y_t$ ) and contemporaneous ( $X_t \to Y_t$ ) dependencies, forming the residual graph ( $G_R$ ).

C. Graph Integration

The final causal graph $G_{combined}$ is the union of the three component graphs: $G_{combined} = G_T \cup G_S \cup G_R$ .
Structure: The result is a multi-relational graph distinguishing:
- Time-proxy edges: Exogenous drivers ( $t \to X$ ) representing trends and seasonality.
- Mechanistic edges: Variable-to-variable dependencies ( $X \to Y$ ) representing short-term causal mechanisms.

3. Theoretical Guarantees and Identifiability

The paper establishes formal conditions under which DCD recovers the true causal structure $G^*$ :

Assumptions:
1. Causal Invariance: The residual mechanism is time-invariant and satisfies $\beta$ -mixing (weak dependence).
2. Spectral Separability: Trend, seasonal, and residual components occupy disjoint frequency bands.
3. Bounded Leakage: The correlation (leakage $\epsilon$ ) between decomposed components is small.
4. Causal Modularity: Cross-scale causal influences are negligible.
Identifiability Theorem: Under linear Gaussian dynamics, the expected Structural Hamming Distance (SHD) between the estimated graph and the ground truth is bounded by:
$E[SHD(\hat{G}, G^*)] \leq C_1 \cdot \epsilon + C_2 \cdot \alpha_n$
Where $\epsilon$ is the leakage parameter and $\alpha_n$ is the Type-I error of the downstream causal test. This proves that DCD converges to the true structure as decomposition improves ( $\epsilon \to 0$ ) and sample size increases.

4. Key Contributions

Novel Framework: Introduction of DCD, the first framework to explicitly decompose time series into trend, seasonal, and residual components for targeted causal inference at multiple temporal scales.
Theoretical Analysis: Formalization of decomposition as spectral filtering, providing identifiability guarantees and an error bound dependent on spectral leakage and test significance.
Empirical Superiority: Demonstrated performance gains over state-of-the-art baselines (PCMCI+, CD-NOD, DYNOTEARS, etc.) on both synthetic and real-world data.
Robustness: Proven that the framework degrades gracefully when spectral separability is partially violated (e.g., amplitude modulation), rather than failing catastrophically.

5. Experimental Results

The authors evaluated DCD on synthetic benchmarks and two real-world datasets: Arctic Sea Ice (climate) and ETTh1 (electricity transformer loads).

Synthetic Benchmarks:
- Metrics: DCD achieved the best balance of True Positive Rate (TPR ~0.79), False Discovery Rate (FDR ~0.38), and Structural Hamming Distance (SHD).
- Comparison: Baselines like PCMCI+ suffered from high FDR due to autocorrelation; DYNOTEARS showed high SHD and overfitting; CD-NOD had low SHD only by discarding all lagged edges.
- Scalability: DCD maintained stability across varying sample sizes, lag orders, and dimensionality ( $d=4$ to $10$), whereas baselines showed performance collapse or unstable FDR growth.
Real-World Case Studies:
- Arctic Sea Ice: DCD recovered physically coherent multi-scale dependencies (e.g., SST $\to$ Sea Ice feedback, seasonal radiation cycles) that were obscured or misoriented in raw-data baselines.
- ETTh1: DCD identified sparse, interpretable autoregressive and cross-load interactions, avoiding the dense, spurious graphs produced by other methods.
Ablation Studies:
- Preprocessing vs. Integration: Applying STL preprocessing alone to baselines improved them, but not to the level of full DCD. This confirms that the multi-scale graph integration is a distinct source of performance gain.
- Sensitivity: Performance declined linearly (gracefully) as spectral leakage increased, validating the theoretical bounds.

6. Significance and Impact

Bridging the Gap: DCD bridges the gap between time series decomposition (traditionally used for forecasting) and causal discovery.
Handling Non-Stationarity: It offers a robust solution for non-stationary data by treating trends and seasonality as exogenous drivers rather than forcing them into the causal graph as direct variable-to-variable links.
Interpretability: By separating time-proxy edges (exogenous drivers) from mechanistic edges (intrinsic dependencies), the resulting graphs are more interpretable for domain experts (e.g., climatologists or economists).
Future Directions: The authors suggest relaxing the spectral separability assumption to handle cross-frequency coupling (e.g., trends modulating seasonal amplitude) and integrating DCD with time-varying causal discovery methods.

In summary, DCD provides a mathematically grounded, empirically validated approach to uncovering causal structures in complex, non-stationary time series by explicitly modeling the distinct temporal scales of trends, seasonality, and residuals.