Temporal Patterns In The Dependency Structures Of The Cardiovascular Time Series

Copula density is a function that quantifies the level of dependency between two, or more, related time series, and also visualizes their (non)linear dependency structures. This paper aims to analyze and compare different methods for copula density estimation: local (naive) estimation, kernel estimation, K nearest neighbors, Markov state approach, histograms, and Voronoi decomposition. The methods are compared by mapping the copula density into a time series (dependency level time series) and applying Sample Entropy estimates over the range of parameters. Application examples include systolic blood pressure and pulse interval signals recorded from conscious laboratory rats, treated either with vasopressin selective V1a and V2 receptor antagonists (100 ng and 500 ng) or with saline (control group). The signals are analyzed using composite multiscale entropy. It is shown that each estimation method suffers from bias, but, for each case, a stable working region can be found. It was also shown that the analysis of the dependency level time series could reveal the information that could not be extracted from the classical beat-to-beat time series, and that the copula density, transformed to real signals domain, visualizes the regions where the dependency of cardiovascular signals is exhibited the most, reflecting their mutual relationship and providing the possibility for further research.