Single channel (univariate) signal - Practical Biomedical Signal Analysis Using MATLAB - page 73

in ApEn were pointed out in [Richman and Moorman, 2000] and [Costa et al., 2002]
and another measure called sample entropy (SaEn) was introduced in [Richman and
Moorman, 2000].
In the SaEn method the vectors (blocks) ξ of differing dimensions k are cre-

ated by embedding a signal in a way similar to that given in formula (2.128). The
reconstructed vectors are the state space ( k -dimensional) representation of the dy-
namics of the system. SaEn is defined as the logarithmic difference between the
probability(density) of occurrence of the vector ξ

( i )

( )

within the chosen distance r in
k dimensions and the probability of occurrence of the vector ξ

i

( )

i within the same

chosen distance in k + 1 dimension:

log ρ k

( r )

SaEn =

) ,

(2.138)

ρ k + 1 ( r

where ρ k , ρ k + 1

1 dimensions, respectively.
(MATLAB code for calculating Sample Entropy is available at: http://www.
physionet.org/physiotools/sampen/matlab/ ) . SaEn met some criticism, e.g.,
that it does not characterize the complexity of the signal completely [Govindan et al.,
2007] and soon another modifications appeared.
In the approach called multiscale entropy (MSE) [Costa et al., 2002] the procedure
involves partitioning the signal into disjoined windows and the data are averaged
inside each time window, which in fact is equivalent to downsampling, or low-pass
filtering. In the case of a continuous system the same signal sampled at two different
sampling rates would show different behavior in MSE analysis, so the method will
quantify the system behavior in different frequency ranges.
Yet in another approach SaEn was modified by introducing the time delay δ be-
tween the successive components of blocks (or subseries) [Govindan et al., 2007]; δ
was chosen as the time point at which autocorrelation function falls below 1

( r ) ( r ) —densities of occurrence in k , k +

e .In
the density estimation in order not to account for the temporally correlated points,
for each center i ,the i

/

+

δ surrounding points were discarded.
The above described entropy measures and detrended fluctuation analysis found
applications mainly in heart rate variability analysis and their utility will be discussed
in the section concerning this signal.

2.5.7 Limitations of non-linear methods
There are several pitfalls in the application of non-linear methods. One of them
is the fact that even for infinite-dimensional, stochastic signals low-dimensional esti-
mates may be obtained, which was pointed out by [Theiler et al., 1992]. The temporal
coherence of the data may be mistaken for the trace of non-linearity. Therefore be-
fore applying non-linear methods one should first check, if indeed the signals have
traces of non-linearity, since the non-linear process needs not to produce non-linear
time series. The null hypothesis about the stochastic character of the signal may be
checked by means of surrogate data technique (Sect. 1.6).
The non-linear methods rely to a large extent on the reconstruction of a phase
space. In order to construct the phase space by embedding, rather long stationary