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Signature Kernel Conditional Independence Tests in Causal Discovery for Stochastic Processes
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Develops Markov-property-based conditional independence constraints on SDE path segments and constraint-based causal discovery algorithms (full DAG with loops, CPDAG, partially observed ADMG), plus a consistent signature-kernel CI test on path space.
Problem
Causal discovery for stochastic dynamical systems modeled by path-dependent SDEs from irregularly sampled, possibly partially observed paths, where existing methods rely on discrete-time assumptions or fixed-lag models and lack practical tests for path-valued variables.
Approach
Derive CI constraints on coordinate-process path segments over past/future intervals that satisfy a global Markov property w.r.t. a time-lifted acyclic graph (with loops), enabling sound and complete PC-style discovery that uses time direction, and test these constraints with a signature-kernel permutation CI test (SDCIT/KCIPT style) with a consistency proof.
Key results
- SigKer achieves SHD of 14±4 (n=200) and 7±3 (n=400) in the bivariate linear setting, versus 74±14 for the best SCOTCH setting shown at n=200.
- In high-dimensional linear SDE causal discovery (d=3 to 50), the CPDAG-plus-post-processing variant has SHD 13±4 at d=3 and 38±7 at d=5, versus PCMCI 29±16 and 243±37.
- In pairs trading, SigKer attains Sharpe 1.500 and return 0.076 versus 0.090 Sharpe for ADF and -0.230 for Granger.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use SigKer instead of the strongest listed SCOTCH configuration | Bivariate SDE causal discovery · Linear, path-dependent, and nonlinear bivariate SDEs | SHD (× 10^2) | Helped |
| Use SigKer instead of the strongest listed SCOTCH configuration | Bivariate SDE causal discovery · Diffusion-dependent bivariate SDE | SHD (× 10^2): 72 ± 6; 63 ± 5 vs 25 ± 13; 9 ± 8 | Mixed |
| Add symmetric CI post-processing to symmetric CI discovery (ablation) | SigKer causal discovery · Causal discovery, d=3 | SHD (× 10^2): 57 ± 84 -> 13 ± 4 | Helped |
| Add symmetric CI post-processing to symmetric CI discovery (ablation) | SigKer causal discovery · Causal discovery, d=5 | SHD (× 10^2): 147 ± 11 -> 38 ± 7 | Helped |
| Add symmetric CI post-processing to symmetric CI discovery (ablation) | SigKer causal discovery · Causal discovery, d=10 | SHD (× 10^2): 436 ± 19 -> 157 ± 15 | Helped |
| Add symmetric CI post-processing to symmetric CI discovery (ablation) | SigKer causal discovery · Causal discovery, d=20 | SHD (× 10^2): 1294 ± 48 -> 725 ± 439 | Helped |
| Add symmetric CI post-processing to symmetric CI discovery (ablation) | SigKer causal discovery · Causal discovery, d=50 | SHD (× 10^2): 6005 ± 98 -> 4593 ± 93 | Helped |
| Use symmetric CI + post-processing instead of future-extended local CI (ablation) | SigKer causal discovery · Causal discovery, d=3, d=5, and d=10 | SHD (× 10^2): 26 ± 5 -> 13 ± 4; 80± 8 -> 38 ± 7; 284 ± 19 -> 157 ± 15 | Helped |
| Use symmetric CI + post-processing instead of future-extended local CI (ablation) | SigKer causal discovery · Causal discovery, d=20 and d=50 | SHD (× 10^2): 1026 ± 40 -> 725 ± 439; 4946 ± 113 -> 4593 ± 93 | Mixed |
| Use SigKer instead of SCOTCH | Causal discovery · Causal discovery, d=20 and d=50 | SHD (× 10^2): 725 ± 439 vs 370 ± 174; 4593 ± 93 vs 538 ± 70 | Mixed |
| Use SigKer instead of ADF & Granger | Pairs trading strategy · Pairs trading, 2010/01/01–2012/12/31 | maxDD: 0.022 vs 0.027 | Hurt |
| Use SigKer instead of the listed trading baselines | Pairs trading strategy · Pairs trading, 2010/01/01–2012/12/31 | return | Helped |
Surprising findings
- SCOTCH outperforms the proposed method in the diffusion-dependence setting for some hyperparameter settings.
- Counterexample shows that with cycles, a CI oracle over path segments cannot rule out a directed edge, so full-graph discovery is impossible in the cyclic case.
Limitations
- Restricted to acyclic graphs except self-loops and stationary (time-invariant) SDE coefficients.
- Causal discovery relies on a CI oracle and faithfulness assumptions; test errors affect results.
- Permutation-based CI tests are computationally dominated by permutation search.
Experimental setup
Datasets: synthetic linear SDEs synthetic nonlinear SDEs fractional Brownian motion VBR Small-Cap ETF stock prices (2010-2012)
Benchmarks: synthetic SDE bivariate settings synthetic linear SDEs with d=3-50 pairs trading on VBR Small-Cap ETF
Metrics: SHD test power return APR Sharpe ratio maxDD maxDDD
Baselines: CCM PCMCI Granger causality Lau SCOTCH
Builds on: PC algorithm FCI algorithm KCIPT SDCIT SCOTCH
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