Causal Partial Separability: Learning Nonlinear DAGs from Multivariate Functional Data

In functional data analysis, learning directed acyclic graphs (DAGs) from multivariate functional data is crucial for identifying root causes. While recent causal discovery methods for multivariate functional data primarily rely on linear models, there is a lack of nonlinear acyclic models tailored for such data. The fundamental differences between scalars and functions prevent existing scalar nonlinear causal discovery methods from being directly applicable to functional data. To address these limitations, we propose Causal Partial Separability (CPS), a framework where functional variables admit a shared spectral basis and a common causal ordering, thereby mapping the problem to scalar DAG learning across the shared basis. We analyze the equivalence between functional DAGs and the union of blockwise DAGs, deriving the nonlinear identifiability of functional DAGs under the assumption of nonlinear Additive Noise Models (ANMs). Building on CPS, we introduce cps-fDAG, a functional DAG estimation method that combines shared spectral basis learning, searching for a common causal ordering, and hierarchical edge pruning. We establish the theoretical consistency for both the common ordering and edge pruning and provide a sufficient sample size condition. Experiments on synthetic datasets and biophysical fMRI simulation benchmarks demonstrate that cps-fDAG effectively learns causal DAGs for multivariate functional data.