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applied to raw data and the correlation coefficient is in the interval
−
1
≤
r
n
≤
1for
every
n
.
We want to construct a simple analytic form for (
3.204
) that captures any long-time
correlation pattern in the data. One way to determine the correlations in the time series,
aside from the brute-force method of directly evaluating (
3.204
) is by aggregating the
data into groups of increasing size. In this way the scaling behavior changes as a func-
tion of the number of data points we aggregate and it is this dependence we seek to
exploit. Segment the sequence of data points into adjacent groups of
n
elements each so
that the
n
nearest neighbors of the
j
th data point aggregate to form
Q
(
n
)
j
=
Q
nj
+
Q
nj
−
1
+···+
Q
nj
−
n
−
1
.
(3.205)
The average over the aggregated sets yields
n
1
n
Q
(
n
)
j
Q
(
n
)
=
=
nQ
.
(3.206)
j
=
1
We have used the fact that the sum yields
N Q
and the brackets
n
denote the closest
integer determined by
N
n
. The ratio of the standard deviation to the mean value is
denoted as the relative dispersion and is given by [
4
]
/
Va r
Q
(
n
)
Q
(
n
)
Va r
Q
(
n
)
nQ
.
R
(
n
)
=
=
(3.207)
If the time series scales the relative dispersion satisfies the relation
R
(
n
)
=
n
H
−
1
R
(
1
)
,
(3.208)
which results from the standard deviation scaling as
n
H
and the mean increasing linearly
with lag time. The variance of the coarse-grained data set is determined in analogy to
the nearest-neighbor averaging to be
n
Q
(
n
)
j
2
1
n
Q
(
n
)
j
Va r
Q
(
n
)
=
−
j
=
1
n
−
1
=
n
Va r
Q
+
2
1
(
n
−
j
)
Cov
n
Q
.
(3.209)
j
=
Substituting (
3.208
) for the relative dispersion into (
3.209
) allows us, after a little
algebra, to write
j
=
1
(
n
−
1
n
2
H
−
1
=
n
+
2
n
−
j
)
r
n
.
(3.210)
Equation (
3.210
) provides a recursion relation for the correlation coefficients from
which we construct the hierarchy of relations
