Multiscale cumulative residual distribution entropy and its applications on heart rate time series
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ORIGINAL PAPER
Multiscale cumulative residual distribution entropy and its applications on heart rate time series Xuegeng Mao
. Pengjian Shang . Albert C. Yang . Chung-Kang Peng
Received: 29 October 2019 / Accepted: 31 August 2020 Ó Springer Nature B.V. 2020
Abstract Distribution entropy has been proved to reveal stability for short time series and to distinguish different classes of series by complexity. However, there still exists some drawbacks. For example, it does not consider the possible causality underlying the data, which may not precisely identify deterministic from stochastic processes. In addition, cumulative residual entropy can successfully solve such problems and identify randomness and complexity of time series quite clearly. We therefore combine distribution entropy with cumulative residual entropy named cumulative residual distribution entropy (CRDE), aiming at considering both distribution and values of distances in the state space. CRDE can detect the temporal and spatial structures of the series after adding multiscale analysis. Results show that the combined method can characterize series from stochastic system (white noise and 1/f noise) and deterministic system (chaotic and periodic series). Then, we apply it to physiological signals, and the result is consistent with the one that loss of complexity at larger scales is related to aging and disease. X. Mao (&) P. Shang School of Science, Beijing Jiaotong University, Beijing 100044, People’s Republic of China e-mail: [email protected] X. Mao A. C. Yang C.-K. Peng H. A. Rey Institute for Nonlinear Dynamics in Medicine, Beth Israel Deaconess Medical Center/Harvard Medical School, Boston, MA 02215, USA
Keywords Distribution entropy Cumulative residual entropy Cumulative distribution function Cumulative residual distribution entropy
1 Introduction A real-world system includes numerous components with greatly nonlinear interactions, and it is continuously being improved over time. It is essential to create quantitative methods in the macroscopic level of nonlinear and dynamical systems. Therefore, complexity is usually used to quantify the emergent properties of systems with multiple interacting components and quantifying the complexity of dynamical systems has always been a hot topic in scientific fields. Earlier researches including dimensional analysis and entropy calculation have been applied to prove the existence of the deterministic chaos from data [1, 2], but sufficient amount of data points was required for the application and there were also other problems remaining to be solved. Based upon the issues, Pincus proposed a preliminary mathematical algorithm, approximate entropy (ApEn), to identify the inherent complexity [3, 4]. ApEn is associated with Kolmogorov entropy, which is used to quantify the ratio of generating new information [5] and has been broadly used in physiological datasets [6, 7]. Then, to solve the problems
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that ApEn is sensitive to
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