Evaluating Temporal Orders for Local Non-Stationary Biological Signals Analysis: A Python Framework and Simulation Study

Background and ObjectiveAccurately 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. MethodsThe 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. ResultsThe 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. ConclusionsThe 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.