Figure 1

A simulated time series $u\left[1\right],\dots ,u\left[N\right]$ is shown to illustrate the procedure for calculating sample entropy $\left(S_E\right)$ for the case $m=2$ and a given positive real value $r$. Dotted horizontal lines around data points $u\left[1\right],u\left[2\right]$, and $u\left[3\right]$ represent $u\left[1\right]\pm r$, $u\left[2\right]\pm r$, and $u\left[3\right]\pm r$, respectively. Two data points match each other, that is, they are indistinguishable, if the absolute difference between them is $⩽r$. Typically, $r$ varies between 10% and 20% of the time series SD. The symbol 엯 is used to represent data points that match the data point $u\left[1\right]$. Similarly, the symbols × and ▵ are used to represent data points that match the data points $u\left[2\right]$ and $u\left[3\right]$, respectively. Consider the two-component 엯-× template sequence $(u\left[1\right],u\left[2\right])$ and the three-component $엯\text{−}\times \text{−}\Delta$ template sequence $(u\left[1\right],u\left[2\right],u\left[3\right])$. For the segment shown, there are two 엯-× sequences, $(u\left[13\right],u\left[14\right])$ and $(u\left[43\right],u\left[44\right])$, that match the template sequence $(u\left[1\right],u\left[2\right])$, but only one 엯-×-▵ sequence that matches the template sequence $(u\left[1\right],u\left[2\right],u\left[3\right])$. Therefore, in this case, the number of sequences matching the two-component template sequences is two and the number of sequences matching the three-component template sequence is 1. These calculations are repeated for the next two-component and three-component template sequence, which are $(u\left[2\right],u\left[3\right])$ and $(u\left[2\right],u\left[3\right],u\left[4\right])$, respectively. The number of sequences that match each of the two- and three-component template sequences are again summed and added to the previous values. This procedure is then repeated for all other possible template sequences, $(u\left[3\right],u\left[4\right],u\left[5\right]),\dots ,(u\left[N\text{−}2\right],u\left[N\text{−}1\right],u\left[N\right])$, to determine the ratio between the total number of two-component template matches and the total number of three-component template matches. $S_E$ is the natural logarithm of this ratio and reflects the probability that sequences that match each other for the first two data points will also match for the next point.Reuse & Permissions

Figure 2

Schematic illustration of the coarse-graining procedure. Adapted from Ref. 8.Reuse & Permissions

Figure 3

MSE analysis of 30 simulated Gaussian distributed (mean zero, variance one) white and $1∕f$ noise time series, each with $3\times {10}^4$ data points. Symbols represent mean values of entropy for the 30 time series and error bars the SD, which in average is 0.05 for white noise and 0.02 for $1∕f$ noise. Lines represent numerical evaluation of analytic $S_E$ calculation. Note that the differences between the mean values of $S_E$ and the corresponding numerical values are less than 1%. SD is larger for $1∕f$ noise time series because of nonstationarity. Adapted from Ref. 7. (See Appendix .)Reuse & Permissions

Figure 4

Representative interbeat interval time series from (a) healthy individual (sinus rhythm), (b) subject with congestive heart failure, and (c) subject with atrial fibrillation, a highly erratic cardiac arrhythmia.Reuse & Permissions

Figure 5

MSE analysis of RR time series derived from long-term ECG recordings of healthy subjects in normal sinus rhythm, those with congestive heart failure (CHF) in sinus rhythm, and those with atrial fibrillation (AF). Symbols represent the mean values of entropy for each group and bars represent the standard error $(SE=\mathrm{SD}∕\sqrt{n})$, where $n$ is the number of subjects). Parameters to calculate $S_E$ are $m=2$ and $r=0.15$. Time series length is $2\times {10}^4$ beats. The $S_E$ values from healthy subjects are significantly ($t$-test, $p<0.05$) higher than from CHF and AF subjects for scales larger than scale 2 and scale 20, respectively.Reuse & Permissions

Figure 6

MSE analysis of RR time series derived from 24 h ECG recordings of 27 healthy young subjects, aged $34.5\pm 7.3$ years $(\text{mean}\pm \mathrm{SD})$, range 20 - 50 years, 45 healthy elderly subjects, aged $70\pm 3.97$ years, range 66 - 75 years, and 43 congestive heart failure (CHF) subjects, aged $55\pm 11.6$ years, range 22 - 78 years. (a) Waking period. For all scales the $S_E$ values from healthy young subjects are significantly ($\mathrm{t}$-test, $p<0.05$) higher than from CHF subjects. The $S_E$ values from healthy young subjects are significantly higher than from healthy elderly subjects for scales larger than scale 1. The $S_E$ values from healthy elderly subjects are significantly ($\mathrm{t}$-test, $p<0.05$) higher than from CHF subjects for scales between scales 5 and 13, inclusively. (b) Sleeping period. Both the $S_E$ values from healthy elderly and healthy young subjects are significantly ($\mathrm{t}$-test, $p<0.05$) higher than from CHF subjects for scales between scales 2 and 11, inclusively. The $S_E$ values from healthy young subjects are significantly higher than from healthy elderly subjects for scales shorter than scale 5. Symbols represent the mean values of entropy for each group and the bars represent the standard error. Parameters of $S_E$ calculation are $m=2$ and $r=0.15$. Time series length is $2\times {10}^4$ beats.Reuse & Permissions

Figure 7

MSE analysis of RR time series derived from 24 h ECG recordings during waking and sleeping periods. (a) Young healthy subjects. The $S_E$ values for the waking period are significantly ($t$-test) higher $(p<0.05)$ than for the sleeping period on scales larger than scale 7. (b) Elderly healthy subjects. The $S_E$ values for the sleeping period are significantly ($t$-test) higher $(p<0.05)$ than for the waking period on scales shorter than scale 16. (c) Congestive heart failure subjects. The $S_E$ values for the sleeping period are significantly ($t$-test) higher $(p<0.05)$ than for the waking period on all scales but scale 1. Symbols represent mean values of entropy for each group and the bars represent the standard error. Parameters of $S_E$ calculation are $m=2$ and $r=0.15$. Time series length is $2\times {10}^4$ beats.Reuse & Permissions

Figure 8

Scatter plot of the slope of the MSE curves between scale factors 6 and 20 vs the slope of the MSE curves between scale factors 1 and 5, for healthy and congestive heart failure (CHF) groups during the sleeping period. For both groups, symbols with error bars represent the mean of $y$-axis values, and the error bars the corresponding SD. The groups are well separated $(p<0.005)$.Reuse & Permissions

Figure 9

MSE results for binary files of a computer executable program (LINUX kernel) and a compressed data file. The original binary file has only two symbols, 0 and 1. However, the number of symbols in coarse-grained sequences increases with the scale factor, which introduces a characteristic artifact on the MSE curves. In order to avoid this artifact, instead of the original sequences, we analyze a derived sequence, which is constructed as follows: we divide the original sequence into consecutive nonoverlapping segments, each with 128 data points, and then calculate the number of 1’s (0’s) within each segment. Some structural information is lost since the procedure is not a one-to-one mapping. The derived sequences are expected to be more regular than the original ones. However, this procedure does not alter the conclusions drawn from our analysis.Reuse & Permissions

Figure 10

MSE results for four coding, nine noncoding DNA sequences from human chromosome 22 and 30 binary random time series. All coding sequences with more than identified $4\times {10}^3\mathrm{bp}$ were selected. The longest coding sequences has 6762 bp. All noncoding sequences with more than 6000 and fewer than 6050 bp were selected. The length of the random sequences is 6000 data points. The symbols and the error bars represent the $S_E$ mean values and SD, respectively. Due to a typical artifact that affects the MSE results of discrete sequences (Appendix ), only the entropy values for scales 1, 5, 9, 13, and 17 are plotted. Note the higher complexity of the noncoding vs coding sequences ($p=0.006$ for scale 9). The lowest entropy values are assigned to the random (white noise: mean zero, variance 1) time series mapped to a binary sequence: 1 if $x_i>0$ and 0 if $x_i<0$.Reuse & Permissions

Figure 11

Gaussian distribution. Shadowed areas centered at points $-2$ and 1 represent the probability that the distances between each of these points and any other point chosen randomly from the time series are less than or equal to $r$.Reuse & Permissions

Figure 12

Correspondence between the covariance and the shape of the contours of a bivariate Gaussian density function. If two random variables, $X_j$ and $X_k$, are independent $[C_{jk}=C(X_j,X_k)=0]$, the shapes of the contours are ellipses with major and minor axes parallel to $X_j$ and $X_k$ axes, respectively. If the variables have equal variance $({\sigma }_j={\sigma }_k)$, the shape of the contour is a circle. In contrast, if two variables are not independent, the shapes of the contours are ellipses with major and minor axes that are not aligned with the axes $X_j$ and $X_k$.Reuse & Permissions

Figure 13

The ellipse represents the contour of a bivariate Gaussian density function. The major and minor axes of the ellipse are not parallel to the axes $X_j$ and $X_k$, meaning that the random variables are correlated in this frame. However, there exists a rotation that transforms the original frame into one defined by the axes $Y_j$ and $Y_k$, which are aligned with the major and minor axes of the ellipse. Therefore, in this frame the original variables are uncorrelated.Reuse & Permissions

Figure 14

$S_E$ as a function of time series number of data points $N.r=0.15$ and $m=2$ for all time series. Symbols represent the mean values of $S_E$ for 30 simulated white and $1∕f$ noise time series, and the error bars represent the SD.Reuse & Permissions

Figure 15

$S_E$ as a function of the parameter $r$ (left plot) and $m$ (right plot). $N=3\times {10}^4$ and $r=0.15$ for all time series. Symbols represent the mean values of $S_E$ for 30 simulated $1∕f$ and white noise time series, and error bars represent the SD.Reuse & Permissions

Figure 16

Effects of different amounts of Gaussian white noise on MSE curves. The MSE curve labeled “original” corresponds to the MSE results for the RR intervals time series from a healthy subject.Reuse & Permissions

Figure 17

Contour plot showing how the percentage of outliers and their amplitude (relative to the mean value of the time series) affects the variance of the time series. Lines connect pairs of values that change the variance by the same amount.Reuse & Permissions

Figure 18

(a) The interbeat interval time series of a young healthy subject with 15 outliers that represent artifacts or missed beat detections. Note that the absolute value of the outliers is much larger than the mean RR interval. (b) The interbeat interval time series of an elderly healthy subject with frequent premature ventricular complexes (PVCs) (two are represented in the figure). (c) MSE results for the time series shown in plot (a): the solid line is the MSE result for the unfiltered time series; the dotted line is the MSE results for the same time series excluding outliers; and the dashed line is the MSE result for the original time series but using an $r$ value that is calculated by excluding the outliers. (d) MSE results for time series shown in plot (b): solid and dotted lines are the MSE results for unfiltered and filtered (PVCs removed) time series.Reuse & Permissions

Figure 19

Probability of distinguishing any two data points randomly chosen from the coarse-grained time series of binary discrete time series $(r=0.5)$.Reuse & Permissions