Entropy Analysis of Univariate Biomedical Signals: Review and Comparison of Methods

Abstract

Nonlinear techniques have found an increasing interest in the dynamical analysis of various kinds of systems. Among these techniques, entropy-based metrics have emerged as practical alternatives to classical techniques due to their wide applicability in different scenarios, specially to short and noisy processes. Issued from information theory, entropy approaches are of great interest to evaluate the degree of irregularity and complexity of physical, physiological, social, and econometric systems. Based on Shannon entropy and conditional entropy (CE), various techniques have been proposed; among them, approximate entropy, sample entropy, fuzzy entropy, distribution entropy, permutation entropy, and dispersion entropy are probably the most well-known. After a presentation of the basic information-theoretic functionals, these measures are detailed, together with recent proposals inspired by nearest neighbors and parametric approaches. Moreover, the role of dimension, data length, and parameters in using these measures

Feasibility of Ultra-Short Term Analysis of Heart Rate and Systolic Arterial Pressure Variability at Rest and During Stress via Time-domain and Entropy-based Measures

“…Using such notation, the static entropy quantifies the ‘static’ information contained in the current state of the process X , without considering its temporal dynamics, and can be defined as [45]: where E [·] is the expectation operator and p (·) the probability density, while H (·) denotes the entropy. The dynamic entropy (DE) instead represents the “joint” entropy of the present and past variables composing the process; therefore it provides the amount of information brought by the current sample of the series and by its past samples as well, thus giving ‘dynamic’ information on the entire process, and it can be defined as [46]: being H (·,·) the joint entropy of two random variables. Then, the conditional entropy (CE) quantifies the average uncertainty that remains about the present state of the process when its past states are known (i.e., the new information contained in the current sample that cannot be inferred from the past history), and is defined as [45]: where H (· | ·) denotes conditional entropy operator.…”

“…AR model identification has been performed via ordinary least squares method [47] to obtain estimations of regression parameters and prediction error variance, thus estimating the variance and the covariance matrices of the process. Then, denoting as 𝜎 ̂𝑋 2 the variance of the process, as 𝛴 ̂𝑋𝑛 𝑿 𝑛 𝑚 the covariance matrix of the present and past states of X, and as 𝜎 ̂𝑈 2 the prediction error variance, the above defined entropy measures can be computed as [46]:…”

“…The second estimation method (hereinafter referred as knn) is a model-free approach based on nearest neighbor metrics, which exploits the intuitive notion that the local probability density around a given data point is inversely related to the distance between the point and its neighbors. Using this approach, estimates of SE, DE and CE of the process X can be respectively computed through the following expressions [45,46]:…”

Analysis of Cardiac Pulse Arrival Time Series at Rest and during Physiological Stress

“…Information-theoretic analysis was performed to assess the static and dynamic information content of the analyzed series. Specifically, the entropy of a process , representing the quantity of information held in the current state of , is defined under the hypothesis of stationarity as [21]:…”

“…1 (c). The information storage (S) is the quantity of information held in the current state of the system attributable to its past states, measuring regularity and predictability of the time series; it is defined as follows [21]: analyzed process, and = . .…”