Physiologic data:RR interval data of twenty healthy "Young" (21-34 years old) and twenty healthy "Elderly" (68-85 years old) subjects were obtained from the Fantasia module of the PhysioNet database (Goldberger et al.,). Each data corresponds to a 120 min recording of the subject's electrocardiogram (ECG) when in continuous supine resting, sampled at a frequency of 250 Hz. Each group of subjects has an equal number of men and women. Each RR interval is computed by an automated algorithm from annotated heartbeats of subjects (Iyengar et al.,). After extraction of RR series of all subjects from the database, each signal segment was selected from the beginning by varying length from 50 to 1,000 beats (total 8 different lengths:50, 100, 200, 300, 400, 500, 750, and 1,000 beats corresponding to average time durations of 0.78, 1.60, 3.23, 4.88, 6.52, 8.17, 12.30, and 16.44 min, respectively) for each subject.
RR interval time-series of "Arrhythmia" and "Healthy" subjects were obtained from the MIT-BIH module of the PhysioNet database (Goldberger et al.,). The Arrhythmia Database contains 48 ECG recordings obtained from 47 subjects (Moody and Mark,). The subjects included 25 men aged 32 to 89 years and 22 women aged 23 to 89 years. The recordings were digitized at 360 samples per second per channel with 11-bit resolution over a 10 mV range. Each beat of every record was then annotated independently using a slope sensitive QRS detector (Moody and Mark,). From this, the RR interval was then computed for each subject. The Normal sinus rhythm database contains 18 long-term ECG recordings of subjects who were found to have no significant arrhythmia; they include 5 men, aged 26 to 45, and 13 women, aged 20 to 50. After extraction of RR series of all subjects from the database, each signal segment was selected from the beginning by varying length from 50 to 1,000 beats (total 8 different lengths as mentioned in previous paragraph, which corresponds to average time durations of 0.69, 1.34, 2.67, 3.98, 5.23, 6.16, 9.89, and 13.24 min, respectively) for each subject.
In this study, we compared the characteristics and performance of DistEn as a entropy measure with ApEn and SampEn. The reason that ApEn and SampEn were used for comparison was because DistEn was initially proposed to address the dependence of the existing ApEn and SampEn methods on tolerance r.
Form (N - m + 1) vectors of length m each, given by  
Take each  vector of step 1 as a template vector and find its distance from every vector of , where the distance is given by 
Then we define  where,  is the probability of a vector  to lie within a distance r of the vector 
The above steps are repeated for m+1, resulting in Phim+1(r) from which ApEn is defined as 
ApEn is an approximation of the conditional probability (Pincus,; Pincus and Goldberger,) of two segments matching at a length of m+1 if they match at m. The embedding dimension m is the length of compared segments of the input time series and r is the threshold of distance, which is fixed to match segments when they are compared with each other. Let a time series of length N be defined as {x(n) : 1 <= n <= N}. For a given value of the embedding dimension m and tolerance r, ApEn is calculated using following steps: 
In this study, we used m = 2, 3, 4, 5 and r = 0.1*SD to 1*SD with a step size of 0.1 * SD to calculate ApEn for all signals, where SD denotes standard deviation of the signal. For ease of calculation and visualization, each RR time-series was normalized to unitary variance before calculating ApEn.
SampEn is a modified version of ApEn to find the irregularity of a given signal (Richman and Moorman,). Here, self matches between vectors are avoided from the calculation and the same number of template vectors are used in m and m+1 dimensions. For a given time series data of length N, sample entropy is calculated as
where
 being the probability of a vector  to lie within a distance r of the vector ,1 <= j <= (N - m), j   i.
Similar to ApEn, we used m = 2, 3, 4, 5 and r = 0.1 * SD to 1 * SD with a step size of 0.1 * SD to calculate SampEn for all signals, where SD denotes standard deviation of the signal. For ease of calculation and visualization, each RR time-series was normalized to unitary variance before calculating SampEn.
DistEn is initially developed from SampEn with an aim of improving the inconsistency and minimizing the dependence on input parameters. The novelty behind DistEn is the assumption that the inconsistency and parameter-dependence of SampEn-based measures come from the incomplete assessment of the distribution of inter-vector distances, and that they can be eliminated by taking full advantages of the distribution property (Li et al.,). By quantifying the Shannon entropy of the probability density of inter-vector distances:an assessment that completely and globally quantifies the distribution property, the so-developed DistEn displayed improved performance as we expected (Li et al.,).
For a given time series data of length N, embedding dimension m and bin number M the distribution entropy is calculated as follows.
Form (N - m) vectors of length m each, given by  
Take each  vector of step 1 as a template vector and find its distance from vector , where the distance is given by 
A distance matrix D of size (N - m) * (N - m - 1) is formed by repeating this calculation for all ith template vectors, where 1 <= i <= (N - m).
The elements of distance matrix D are now divided into M number of equally spaced bins and the corresponding histogram is obtained.
Now, at each bin t of the histogram, its probability is estimated as ; 1 <= t <= M.
By the definition of Shannon entropy, the normalized DistEn of a given time series x(i), is defined by the expression  where pt is the probability of each bin in the histogram.
In this study, we used m = 2, 3, 4, 5 and M = 50, 100, 200, 300, 400, 500, 750, 1000, 1500, 2000 to calculate DistEn for all signals.
In our study, we used area under the ROC curve (AUC) to test the efficiency of DistEn as a feature to distinguish signals of different levels of complexity (synthetic) and RR time series belonging to different classes (physiologic). The AUC is the probability that a classifier ranks a randomly chosen instance X higher than a randomly chosen instance Y, X and Y being samples taken from two independent populations. An AUC value of 0.5 indicates that the distributions of the features are similar in the two groups with no discriminatory power. Conversely, an AUC value of 1.0 means that the distribution of the features of the two groups do not overlap at all. The AUC value was approximated numerically using the trapezoidal rules (Hanley and McNeil,) where the larger the AUC value, the better the discriminatory performance. MATLAB R2014b Statistics toolbox was used to perform all statistical operations.