Assessing Sample Entropy of physiological signals by the norm component matrix algorithm: Application on muscular signals during isometric contraction

Abstract

Sample Entropy (SampEn) is a popular method for assessing the unpredictability of biological signals. Its calculation requires to preliminarily set the tolerance threshold r and the embedding dimension m. Even if most studies select m=2 and r=0.2 times the signal standard deviation, this choice is somewhat arbitrary. Effects of different r and m values on SampEn have been rarely assessed, because of the high computational burden of this task. Recently, however, a fast algorithm for estimating correlation sums (Norm Component Matrix, NCM) has been proposed that allows calculating SampEn quickly over wide ranges of r and m. The aim of our work is to describe the structure of SampEn of physiological signals with different complex dynamics as a function of m and r and in relation to the correlation sum. In particular, we investigate whether the criterion of "maximum entropy" for selecting r previously proposed for Approximate Entropy, also applies to SampEn; and whether information from correlation sums provides indications for the choice of r and m. For this aim we applied the NCM algorithm on electromyographic and mechanomyographic signals during isometric muscle contraction, estimating SampEn over wide ranges of r (0.01 ≤ r ≤ 5) and m (from 1 to 11). Results indicate that the "maximum entropy" criterion to select r in Approximate Entropy cannot be applied to SampEn. However, the analysis of correlation sums alternatively suggests to choose r that at any m maximizes the number of "escaping vectors", i.e., data points effectively contributing to the SampEn estimation.

A Strategy to Reduce Bias of Entropy Estimates in Resting-State fMRI Signals

“…These commonly used parameters were typically applied to signals with slower dynamics such as heart rate; hence, some studies have suggested that these parameters are inappropriate for signals with faster dynamics, and proposed to use r values that maximize the ApEn value ( Chen et al, 2005 ; Chon et al, 2009 ). However, this maximum entropy approach was shown to be invalid for SampEn estimates ( Castiglioni et al, 2013 ).…”

Determination of Sample Entropy and Fuzzy Measure Entropy Parameters for Distinguishing Congestive Heart Failure from Normal Sinus Rhythm Subjects

“…Some selection methods for dynamic threshold were proposed to replace the constant r. A typical example for ApEn was from the study of Lu et al, regarding the threshold r from 0.01 to 1.0 times of standard deviation of time series maximizing ApEn as the optimal r [15]. A typical example for SampEn was from the study of Castiglioni et al, estimating SampEn over wide ranges of r (0.01 ≤ r ≤ 5) for determining the optimal r [17]. However, the maximum methods for determining r for both ApEn and SampEn are also based on the Heaviside function, which judges two vectors as either "similar" or "dissimilar", without any intermediate states; this could result in the poor statistical stability for entropy estimates.…”

“…So the original suggestion between 0.1 and 0.25 times the SD of the time series for r does not always demonstrate the best results for all data sets and therefore, elaborate methods to choose r have been developed. The parameter r may be chosen based on the minimization of the maximum SampEn relative error and conditional probability [8] or to provide the maximum value of ApEn [15] or SmapEn [16,17]. However, our recent study suggested that the maximum value method may not be appropriate for analyzing the nonlinear cardiovascular signals and is only appropriate for known random time series [16,18].…”

Entropy Measures as Descriptors to Identify Apneas in Rheoencephalographic Signals

“…SampEn requires the a priori definition of the same parameters listed for ApEn - N, m and r – and those are typically coincident with the ones used for ApEn (i.e., m = 2, r = 0.2). However, even though some authors consider the same criteria can be used for both SampEn and ApEn [ 49 ], other publications suggest that they should be explored independently since algorithms proposed for the choice of r in ApEn are not applicable for SampEn [ 50 ]. Moreover, appropriate values for m and r depend of the type of signal under analysis [ 49 ].…”

“…SampEn overcomes the bias problem detected in ApEn as well as its inconsistencies, such that if SampEn of one signal (x 1 ) is higher than the value obtained with another signal (x 2 ) for a pair of m and r values, a new m-r pair would still provide higher SampEn values for the signal x 1 [ 51 ]. Nonetheless, Castiglioni et al [ 50 ] detected inconsistencies in SampEn calculations when studying mechanomyographic signals for certain m values, and Yentes et al [ 28 ] published similar findings for some r choices, suggesting that under certain conditions SampEn can also be affected by inconsistencies.…”