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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