Biology Reference
In-Depth Information
E
XERCISE
6-8
Compute the tolerance t for sequence S1 with r
¼
0.2. What is the
tolerance for sequence S2 if r
¼
0.2?
Now suppose we have determined a tolerance value t. We say
the subsequences (a,b) and (a
0
,b
0
) are matches if |a
a
0
| <t
b
0
| <t(i.e., if the distance between each component differs by
less than the tolerance). In a regular sequence, if subsequences (a,b) and
(a
0
,b
0
) are matches, it is likely the sequences (a,b,c) and (a
0
,b
0
,c
0
) will also
be matches. Conversely, if the time series is irregular, having matches of
length 2 is not necessarily a precursor to matches of length 3. The
following definition of SampEn generalizes this observation.
and |b
For a fixed integer m, SampEn is defined as the negative natural
logarithm of the ratio A/B, where A
¼
number of matches of length
m
number of matches of length m (see Figure 6-11). For
those familiar with calculating conditional probabilities, the quantity
A/B is precisely the conditional probability that two sequences within a
tolerance r for m points remain within r of each other at the next point.
Because the computation of SampEn depends on the order of the data
points in the RRI time series, SampEn is a true time-dependent measure,
capturing temporal complexity in sequences of data. ApEn is defined
similarly, but it counts self-matches as well (see Pincus [1991] for
details).
þ
1, and B
¼
Notice that parameters m and r are critical in determining the value of
SampEn. It is, in fact, more appropriate to use the notation SampEn
¼
SampEn(m, r) to reflect the dependence. However, no guidelines exist for
optimizing their values, and this is a shortcoming in the current theory.
Bars are
r
(S.D.)
A
= number of matches of length
m+1
B
= number of matches of length
m
ApEn
≈−
ln (1+A)/(1+
B
)
SampEn =
−
ln A/
B
For regular, repeating data, A/
B
nears 1 and entropy
nears 0.
FIGURE 6-11.
Defining sample entropy of RRIs. (Courtesy of Dr. J. Randall Moorman, University of Virginia.)
