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On the Recoverability of Causal Relations from Temporally Aggregated I.I.D. Data
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Theoretically and experimentally analyzes when instantaneous causal discovery on temporally aggregated i.i.d. data recovers true causal relations, defining functional and conditional-independence consistency.
Problem
Temporal aggregation of high-frequency time-delayed causal processes produces instantaneous dependence, and it is unclear whether non-temporal causal discovery on such aggregated data recovers the true causal relations.
Approach
Formalizes two consistency notions, functional consistency (aggregated model keeps an additive-noise functional form in the correct direction) and conditional independence consistency (aggregated data keeps the Markov properties of the summary graph), and gives conditions under which they hold; shows partial linearity suffices for chain/fork CI consistency while collider is naturally consistent.
Key results
- Direct LiNGAM accuracy on bivariate aggregated linear non-Gaussian data drops from near 100% at small k to about 50% (random guess) as k grows from 3 to 30.
- In the fully nonlinear CI experiment (fork structure, nonlinear-nonlinear), the target conditional independence (VI) is rejected 58% of the time, versus 5% when both relations are linear, indicating CI consistency fails under complete nonlinearity.
- In the nonlinear aggregated causal discovery experiment, PC, FCI and GES collapse from about 90% accuracy on disaggregated data, with GES retaining roughly 70% on aggregated data.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Aggregate the data (k=2) instead of using disaggregated data | PC, FCI, and GES on nonlinear causal models · Nonlinear causal discovery | accuracy: near 90% accuracy for disaggregated data; aggregated accuracy collapses, with GES maintaining around 70% | Hurt |
| Aggregate the data (k=2) instead of using disaggregated data | PC, FCI, and GES on linear causal models · Linear causal discovery | accuracy | No clear effect |
| Increase aggregation factor k | Direct LiNGAM on linear non-Gaussian data · Bivariate causal direction; k from 1 to 100 | accuracy: as k increases from 3 to 30, the accuracy drops rapidly to 50% | Hurt |
| Increase aggregation factor k | ANM on nonlinear data · Bivariate causal direction | accuracy | Hurt |
| Make both causal relationships nonlinear rather than retaining some linearity | Aggregated chain and fork models · Conditional independence consistency | CI rejection rate (VI): Chain: 5% -> 18%; Fork: 5% -> 58% | Hurt |
| Increase VAR(N_X,t) | GES on nonlinear causal discovery data · Aggregated data | accuracy: 0.98 ± 0.03 -> 0.13 ± 0.04 | Hurt |
| Increase VAR(N_Z,t) | GES on nonlinear causal discovery data · Aggregated data | accuracy: 0.94 ± 0.04 -> 0.28 ± 0.09 | Hurt |
| Use some linearity rather than making both relationships nonlinear | Aggregated chain and fork models · Conditional independence consistency | CI rejection rate (VI): Chain: 18% -> 5%; Fork: 58% -> 5% | Helped |
Surprising findings
- Even in the linear non-Gaussian case, temporal aggregation makes the instantaneous model unidentifiable because the summed noise becomes Gaussian by the central limit theorem.
- Aggregation has little effect on causal discovery in the linear case but causes a large accuracy collapse for nonlinear data.
Limitations
- No solution to the aggregation problem is provided beyond a trivial skeleton-prior experiment.
- Analysis of functional consistency for finite k is restricted to additive noise models and bivariate settings.
- Theoretical results assume the aligned instantaneous model; the approximation to time-delay models relies on large k.
Experimental setup
Datasets: synthetic data
Metrics: accuracy rejection rate at 5% significance
Baselines: PC FCI GES Direct LiNGAM ANM
Builds on: VAR model LiNGAM PC algorithm ANM KCI
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