A Kalman Filter-Based Framework for Granger Causality Assessment: Application in Tracking Maternal-Fetal Heart Rate...

Fetal and maternal heart rates (FHR and MHR) are routinely monitored during labor and delivery as critical biomarkers of fetal-maternal well-being. Assessing their directed coupling can reveal interactions that are not apparent from either signal alone. This coupling is often nonstationary, changing rapidly with uterine contractions and the stage of labor. Therefore, classical tools for detecting coupling often fail in this context. We introduce a Kalman filter (KF)-based, time-varying autoregressive (TVAR) framework for nonstationary Granger causality (GC) assessment and evaluate it for MHR-FHR coupling analysis. The method models autoregressive (AR) coefficients of univariate and bivariate predictors as latent states in a linear state-space model and tracks them recursively using an optimized KF, yielding continuous coefficient trajectories with confidence intervals. Time-varying GC is obtained by comparing one-step-ahead prediction errors (innovations) computed with and without including the other signal's history. In the stationary limit, the proposed method is shown to reduce to classical fixed-parameter AR estimation, ensuring consistency with conventional GC. The framework is generic, naturally supports online/streaming analysis, enables multivariate extensions (e.g., incorporating uterine activity or respiration to probe common-input effects), and is generalizable to nonlinear models via extended/unscented Kalman filtering or particle methods. Evaluated on a public fetal ECG dataset (ten 20-min pregnancy records and twelve 5-min labor records), KF-TVAR GC identifies brief, statistically significant MHR-FHR coupling episodes that stationary GC fails to localize. Across subjects, coupling is generally stronger from maternal-to-fetal during late pregnancy and shifts toward fetal-to-maternal during labor.