Parametric Estimation of Entropy Using High Order Markov Chains for Heart Rate Variability Analysis

The aim of this study is to investigate the parametric estimation of entropy and entropy rate of Heart Rate Variability (HRV) series, through the usage of Higher Order Markov Chain (HOMC) models. In HOMCs, the dynamic depends on an arbitrary number of previous steps, and not just the present state as in traditional Markov chains. After obtaining the transition probabilities, entropy and entropy rate were derived in terms of the stationary distribution. First, we empirically confirmed the convergence of the estimated values to the theoretical ones, by creating synthetic signals from HOMCs with known characteristics. Then, we tested the methodology on HRV series derived from long-term recordings of 44 patients affected by congestive heart failure and 54 normal controls. After quantization of RR series with three different strategies, metrics were estimated varying the HOMC order (up to 7) and the number of samples. As no gold standard was available, we measured the capability of entropy and entropy rate of discriminating among the two populations considered, using a support vector machine model (k = 5 fold validation). On synthetic series, the estimation error was marginal when and smaller when the MCs were tightly connected. The classification averagely scored an accuracy of about 80% in distinguishing normal and CHF patients, with a maximum value of 86.7% (AUC = 0.92).