Information Technology Reference
In-Depth Information
2
2
H
r
1
=
(
−
2
)/
2
,
3
2
H
2
r
1
+
r
2
=
(
−
2
)/
2
,
.
.
.
2
H
nr
1
+
(
n
−
1
)
r
2
+···+
2
r
n
−
1
+
r
n
=[
(
n
+
1
)
−
(
n
+
1
)
]
/
2
.
(3.211)
If we select the two equations from (
3.211
) with
n
=
m
and
n
=
m
−
2 and subtract
from their sum twice the expression with
n
=
m
−
1 we obtain
(
2
H
1
2
2
H
2
m
2
H
r
m
=
m
+
1
)
−
+
(
m
−
1
)
,
(3.212)
which is valid for all
m
≥
1
.
Consider the case of large
m
so that we can Taylor expand
the correlation coefficient
1
2
H
2
H
1
m
2
H
2
1
m
1
m
r
m
=
+
−
2
+
−
and, by combining terms, obtain to lowest order in inverse powers of the lag time [
4
]
m
2
H
−
2
r
m
≈
H
(
2
H
−
1
)
(3.213)
.
This correlation coefficient decays very slowly with lag time, much more slowly
than does an exponential correlation, for example. In Figure
3.6
this correlation coef-
ficient is graphed as a function of lag time for various values of the scaling exponent.
We can see from Figure
3.6
that scaling time series with
H
less than 0.7 have val-
ues of the correlation coefficient for
m
≤
≤
for 0
H
1
2 that are so low, less than 0.2, that it is
easy to understand why fractal signals would historically be confused with uncorrelated
noise, even though uncorrelated noise is nearly two orders of magnitude smaller in
amplitude.
>
1
0.5
0.1
0.05
0.01
0.005
1.5
2
3
5
7
10
Lag Time
n
Figure 3.6.
Here the correlation coefficient is graphed versus lag time
n
for
H
=
0
.
509
,
0
.
55
,
0
.
60 and 0.70
from the bottom up.
