Abstract
<jats:p>Background and Objective: Accurately parameterizing dynamic, time-varying interactions in physiological systems is a methodological challenge, as global causal discovery methods may obscure transient, local fluctuations. This study introduces tempord, an open-source Python library designed to estimate local temporal orders and evaluate the short-term stability, directionality, and strength of causal links in non-stationary biological signals. Methods: The algorithm estimates temporal relationships by keeping one signal stationary while iteratively shifting another one within a sliding window. To parameterize optimal inter-signal shifts (causal vector, CV), the framework utilizes linear modeling or time series distance metrics. The methodology was validated through a simulation study on synthetic bivariate signals with mathematically imposed dynamic phase delays, under both deterministic and noisy conditions. Furthermore, in-vivo capabilities were demonstrated by evaluating cardiorespiratory coupling dynamics across spontaneous and music-induced relaxation breathing states. Results: The simulation study demonstrated that the extracted CV trajectories precisely aligned with ground-truth temporal delays, assessed using mean absolute error and root mean square error for both noise-free and noisy synthetic data. In-vivo application demonstrated dynamic temporal stability and the detection of minor step changes during autonomic nervous system state transitions. Conclusions: The tempord Python package bridges the gap between global causal discovery and local beat-by-beat statistical parameterization. It provides a robust "bottom-up" analytical instrument for investigating the transient mechanisms governing complex biological networks.</jats:p>