Evaluation of causality measures based on non-uniform embedding schemes with application to the cardiovascular system

In the analysis of physiological time series it is important to investigate the inter-dependence structure among the observed variables of the system. For this, a number of measures of the so-called Granger causality have been developed, and among them information measures have gained much attention. However, information measures have to deal with the estimation of probability distributions of high-dimensional vector variables, typically formed through uniform delay embedding of the time series. Here, the focus is on measures derived after nonuniform embedding. Particularly, two schemes are considered, one based on conditional mutual information and the other based on conditional entropy. We evaluate the two measures on simulated nonlinear coupled dynamical systems and apply them to signals of heart rate variability, respiration, and oxygen concentration in the blood.