Biomedical Engineering Reference
In-Depth Information
C
(
ε
)
is:
N
1
i = 1
N
2
C
(
ε
)=
Θ
(
ε
−||
x i
x j
|| )
(2.130)
N
(
N
1
)
=
+
j
i
1
where Θ is the Heaviside step function,
0for x
0
Θ
(
x
)=
(2.131)
1for x
>
0
The sum counts the pairs
(
x i
,
x j
)
whose distance is smaller than ε. In the limit N
∝ε D 2 , so the correlation
and for small ε, we expect C to scale like a power law C
(
ε
)
dimension D 2 is defined by:
∂log C
(
ε
,
N
)
D 2
=
lim
ε
lim
N
(2.132)
∂logε
0
the embedding vectors are constructed by means
of the Takens theorem for a range of m values. Then one determines the correlation
sum C
In practice, from a signal x
(
n
)
(
)
(
,
)
is
inspected for the signatures of self-similarity, which is performed by construction of
a double logarithmic plot of C
ε
for the range of ε and for several embedding dimensions. Then C
m
ε
versus ε. If the curve does not change its character
for successive m we conjecture that the given m is a sufficient embedding dimension
and D 2 is found as a slope of a plot of log
(
ε
)
.
The Haussdorf dimension D H may be defined in the following way: If for the set
of points in M dimensions the minimal number of N spheres of diameter l needed to
cover the set increases like:
(
C
(
ε
,
N
))
versus log
(
ε
)
l D H
N
(
l
)
for l
0
,
(2.133)
D H is a Hausdorff dimension. D H
D 2 .
We have to bear in mind that the definition (2.132) holds in the limit N
D 2 , in most cases D H
∞,
so in practice the number of data points of the signal should be large. It has been
pointed out by [Kantz and Schreiber, 2000] that C
can be calculated automatically,
whereas a dimension may be assigned only as the result of a careful interpretation of
these curves. The correlation dimension is a tool to quantify self-similarity (fractal—
non-linear behavior) when it is known to be present. The correlation dimension can
be calculated for any kind of signal, also for a purely stochastic time series or colored
noise, which doesn't mean that these series have a non-linear character. The approach
which helps in distinguishing non-linear time series from the stochastic or linear ones
is the method of surrogate data (Sect. 1.6).
(
ε
)
2.5.3 Detrended fluctuation analysis
Detrended fluctuation analysis (DFA) quantifies intrinsic fractal-like correlation
properties of dynamic systems. A fundamental feature of a fractal system is scale-
invariance or self similarity in different scales. In DFA the variability of the signal
 
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