Conditional Local Independence Testing for Itô processes with Applications to Dynamic Causal Discovery

TL;DR

The paper tackles causal discovery in continuous-time dynamical systems by formalizing conditional local independence for Itô processes and proposing a martingale-based test. It introduces the Local Covariance Measure for Itô processes (LCM-Itô) and an estimation framework built on optimal filtering with sample splitting and cross-fitting to achieve a $\sqrt{N}$-consistent estimator of the drift-projection interactions. The main results establish the asymptotic distribution under the null, derive p-values via a Wiener-series, and prove power against local alternatives, with a practical OU-based filtering estimator enabling implementation. The approach is demonstrated on synthetic data and applied to resting-state fMRI, yielding a robust causal graph that supports dynamic causal discovery in neuroscience and potentially other domains with Itô dynamics.

Abstract

Inferring causal relationships from dynamical systems is the central interest of many scientific inquiries. Conditional local independence, which describes whether the evolution of one process is influenced by another process given additional processes, is important for causal learning in such systems. In this paper, we propose a hypothesis test for conditional local independence in Itô processes. Our test is grounded in the semimartingale decomposition of the Itô process, with which we introduce a stochastic integral process that is a martingale under the null hypothesis. We then apply a test for the martingale property, quantifying potential deviation from local independence. The test statistics is estimated using the optimal filtering equation. We show the consistency of the estimation, thereby establishing the level and power of our test. Numerical verification and a real-world application to causal discovery in brain resting-state fMRIs are conducted.

Figures (6)

• Figure 1: Type I error rate and recall of local independence tests with $d=10, 15$.
• Figure 2: Recall of local independence tests under different causal strengths.
• Figure 3: Type I error rate of the proposed test. Left: the sample size $N_0$ goes to infinity while the sampling interval $\delta$ kept fixed. Right: the sampling interval $\delta$ goes to zero.
• Figure 4: The recovered causal graph from resting-state fMRIs. We have omitted self-loops for clarity. The upper $44$ regions are cortical regions and the bottom $19$ regions (colored in purple) are subcortical regions. The cortical regions are further grouped into $6$ subgroups: Visual, Somatosensory-Motor, Auditory, (rest of) Temporal, (rest of) Posterior, and (rest of) Anterior cortex, according to glasser2016multi. The left and right sides represent the left and right hemispheres, respectively. The width of each edge is proportional to the strength of the causal interaction obtained from $\Tilde{\Phi}$.
• Figure 5: Type I error rate and recall of different test procedures under nonlinear data (upper row) and data with non-isometric noise (lower row).

Theorems & Definitions (111)

• Definition 2.1: Conditional local independence
• Proposition 2.2
• Definition 2.3: Addictive residual process
• Definition 2.4: Local Covariance Measure for Itô processes
• Proposition 2.5
• Proposition 2.6
• Definition 3.1
• Proposition 3.2
• Proposition 3.5
• Proposition 3.6