Convergent cross mapping (CCM) is a state space reconstruction (SSR)-based method designed for causal discovery in coupled time series, where Granger causality may not be applicable due to a separability assumption. However, CCM requires a large number of observations and is not robust to observation noise which limits its applicability. Moreover, in CCM and its variants, the SSR step is mostly implemented with delay embedding where the parameters for reconstruction usually need to be selected using grid search-based methods. In this paper, we propose a Bayesian version of CCM using deep Gaussian processes (DGPs), which are naturally connected with deep neural networks. In particular, we adopt the framework of SSR-based causal discovery and carry out the key steps using DGPs within a non-parametric Bayesian probabilistic framework in a principled manner. The proposed approach is first validated on simulated data and then tested on data used in obstetrics for monitoring the well-being of fetuses, i.e., fetal heart rate (FHR) and uterine activity (UA) signals in the last two hours before delivery. Our results indicate that UA affects the FHR, which agrees with recent clinical studies.
Keywords: convergent cross mapping; deep Gaussian processes; fetal heart rate; state space reconstruction; uterine activity.