Modified permutation-entropy analysis of heartbeat dynamics

Abstract

Heart rate variability (HRV) contains important information about the modulation of the cardiovascular system. Various methods of nonlinear dynamics (e.g., estimating Lyapunov exponents) and complexity measures (e.g., correlation dimension or entropies) have been applied to HRV analysis. Permutation entropy, which was proposed recently, has been widely used in many fields due to its conceptual and computational simplicity. It maps a time series onto a symbolic sequence of permutation ranks. The original permutation entropy assumes the time series under study has a continuous distribution, thus equal values are rare and can be ignored by ranking them according to their order of emergence, or broken by adding small random perturbations to ensure every symbol in a sequence is different. However, when the observed time series is digitized with lower resolution leading to a greater number of equal values, or the equalities represent certain characteristic sequential patterns of the system, it may not be rational to simply ignore or break them. In the present paper, a modified permutation entropy is proposed that, by mapping the equal value onto the same symbol (rank), allows for a more accurate characterization of system states. The application of the modified permutation entropy to the analysis of HRV is investigated using clinically collected data. Results show that modified permutation entropy can greatly improve the ability to distinguish the HRV signals under different physiological and pathological conditions. It can characterize the complexity of HRV more effectively than the original permutation entropy.

Entropic Measures of Complexity of Short-Term Dynamics of Nocturnal Heartbeats in an Aging Population

“…Our second method is based on the distribution of ordinal patterns and the resulting permutation entropy [1,9,10]. However, because we deal with recordings of 8-ms resolution, we address methods that enable us to consider events with repeated values [11].…”

Permutation Entropy: New Ideas and Challenges

“…One class of such modifications is based on adding some metric information of the corresponding delay vectors to the ordinal patterns considered. Entropy measures of that type are the fine-grained Permutation entropy proposed by Liu and Wang [7], the weighted Permutation entropy and the robust Permutation entropy introduced by Fadlallah et al [8] and Keller et al [9], respectively, Bian et al [10] have adapted Permutation entropy to the situation of delay vectors with equal components which are relevant when the number of possible values in a time series is small. Other variants are to consider Tsallis or Renyi entropies instead of the Shannon entropy which goes back to the work of Zunino et al [11] and Liang et al [12], or to integrate information of different scales as done by Li et al [13], Ouyang et al [14] and Azami and Ascodero [15].…”

Применение Энтропии Перестановок При Анализе Хаотических, Шумовых И Хаотических Зашумленных Рядов

“…Также энтропия перестановок может применяться при анализе сердечной активности [11]. Сердечные болезни часто связываются с отклонениями в изменчивости сердечного ритма и в последовательностях характерных интервалов между биениями сердца (Beat to beat interval, BBI).…”