In the sample entropy algorithm, the parameter *r* is used to
determine whether two data points, *x*_{i} and *x*_{j}, are
distinguishable or not. If
|x_{i} - *x*_{j}| ≤ *r*, then *x*_{i} and *x*_{j} are
indistinguishable. Otherwise, they are “seen” as two different data
points. In the MSE_{μ} algorithm, the *r* value is traditionally
chosen as a percentage of the SD of the original time series,
typically a value between 15 and 20%. An advantage of choosing *r* in
this way is the fact that two time series with different amplitudes
but equal correlation properties are guaranteed to have the same
sample entropy and the same MSE values. In fact, calculating sample
entropy with an *r* value that is a percentage of the time
series’ SD (e.g., 20%) is equivalent to calculating sample entropy
with a fixed *r* value (0.2) of a previously normalized time
series.

An important observation regarding the choice of *r* as a percentage
of a time series’ SD is the fact that two RR interval data points,
e.g., 625 and 633 ms, may be indistinguishable when analyzing a given
time series and distinguishable when analyzing another one. Consider
the time series A and B with SDs of 38 ms and 42 ms, respectively. If
*r* = 20% of the time series’ SD, then *r* = 7.6 ms and *r* = 8.4 ms,
for A and B, respectively. Therefore, in one case, the RR intervals
625 and 633 ms are “seen” as different (since
|625 - 633| = 8.0 > 7.6),
and in the other case, as indistinguishable, i.e., below the accepted
level of noise (since
|625 - 633| < 8.4). If one is interested in
quantifying entropy of two time series at the same level of
“resolution,” then one should choose a fixed (i.e., not
dependent on SD) *r* value. However, in doing so, the following
consideration should be kept in mind.

Let us consider two time series *A* and *B* taking values from the sets
{*a*, *b*} and {*a*, *b*, *c*}, respectively. Assume that both time
series are uncorrelated noise. For time series *A*, the probabilities of
*a* and *b* are both 1/2; and for time series *B*, the probabilities of *a*, *b*
and *c* are 1/3. To simplify the presentation, here we use Shannon
entropy. For time series *A*, the entropy is:

For time series

The entropy for time series

In conclusion, time series with a larger alphabet are more entropic
than those with a smaller alphabet and identical correlation
properties. Thus, when analyzing RR intervals time series with a fixed
*r* value, if one finds that a given time series is more entropic than
another, one cannot be sure what the source of the difference is.
Observed differences in entropy could be due to differences in the
degree of randomness of the time series, differences in their range of
values (larger/smaller alphabets) or a combination of the two.

Note that the two approaches discussed for choosing the *r* value are
both justifiable since they provide complementary information.

Our first application of MSE using a fixed *r* value was in a project
whose objective was to help forecast the need for lifesaving
interventions based solely on 15-min ECG signals: Cancio LC,
Batchinsky AI, Baker WL, et al. Combat casualties undergoing lifesaving interventions have decreased heart rate complexity at multiple time scales.
J Crit Care. 2013;28(6):1093-8.

In most studies employing MSE_{μ}, the *r* value is set to a
percentage of the SD of the C-G time series for the smallest scale
included in the analysis. Typically, the smallest scale is scale
one. Thus, the *r* value is a percentage of the original time series’
SD. This *r* value is then used to calculate the sample entropy for
all other C-G time series. A similar approach is also recommended for
MSE_{σ}, MSE_{σ2} and MSE_{MAD}. However, in these cases, the first scale to be
analyzed is not scale one. Typically, one would choose to start at
scale five or above since the coarse-graining with windows with fewer
than five data points may not retain important information pertaining
to the degree of local volatility.

The results presented in Figure 2 for both SD and variance C-G time
series follow this approach: *r* is 20% of the SD of the C-G time
series for scale 5.

MSE_{σ}, MSE_{σ2} and MSE_{MAD} analyses can also we performed using
an *r* value that is a percentage of the original time series’
SD. However, for the analysis of RR intervals time series, a value
around 20% is not adequate. Instead, values below 1% are likely more
suitable. We illustrate the issue using the RR interval time series
shown in Fig. 1. The SD of this time series is 0.133 s. Twenty percent
of this value is 0.027 s. Two data points are distinguishable if the
difference between them is larger than 0.027 s. Consider, the SD C-G
time series for scale 5 and select its median value, 0.0255 s. Only
14% of the data points in this SD C-G time series satisfy the
condition:
|x_{i} - 0.0255| > 0.027. In summary, the *r* value derived
from the original time series and used for the analysis of SD C-G time
series is so large that 86% of the points around the SD C-G median
value are indistinguishable.

Independent of which approach one chooses to follow (*r* as a
percentage of the CG time series for the first scale analyzed or as a
fixed value), an important consideration is whether or not the chosen
*r* value is too restrictive or not restrictive enough. The GMSE
algorithm outputs the number of matches with *m* and *m*+1
components. As a “rule of thumb,” if the number of matches is less
than 50 for the largest scale analyzed, then the *r* value should be
increased.

2019-01-30