Independence and symbolic independence of nonstationary heartbeat series during atrial fibrillation

Heartbeat intervals during atrial fibrillation are commonly believed to form a series of almost independent variables. The series extracted from 24 h Holter recordings show a nonstationary behavior. Because of nonstationarity it is difficult to give a quantitative measure of independence. In this paper, we use and compare two methods for this. The first is a classical method which models a nonstationary series using a linear Gaussian state space model. In this framework, the independence is tested on the stationary sequence of the residuals. The second method codes data into permutations and tests the uniformity of their distribution. This test assumes as null hypothesis a weaker form of independence which we call symbolic independence. We discuss some advantages of symbolic independence in the context of heartbeat series. We analyze the time series of heartbeat intervals from 24 h Holter recordings of nine subjects with chronic atrial fibrillation and find that the detrended series is a zero or one memory process for 83% of regular segments and is symbolically independent for 93% of segments.