Next, we study how the scaling behavior of the components depends on the size
of the segments .
First, we consider components containing segments with
anti-correlations. For a fixed value of the fraction of the segments, we
study how
changes with
. At small scales, we observe a behavior
with a slope similar to the one for a stationary signal
with identical
anti-correlations [Fig. 8(a)]. At large scales, we observe a
crossover to random behavior (exponent
) with an increasing
crossover scale when
increases. At large scales,
we also find a vertical shift with increasing values for
when
decreases [Fig. 8(a)]. Moreover, we find that there is an
equidistant vertical shift in
when
decreases by a factor of
ten, suggesting a power-law relation between
and
at large scales.
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For components containing correlated segments with a fixed value of the
fraction we find that in the intermediate scale regime, the segment size
plays an important role in the scaling behavior of
[Fig. 8(b)]. We first focus on the intermediate scale regime when
both
and
are fixed [middle curve in
Fig. 8(b)]. We find that for a small fraction
of the correlated
segments,
has slope
, indicating random behavior
[Fig. 8(b)] which shrinks when
increases [see
Appendix 7.2, Fig. 10]. Thus, for components containing
correlated segments,
approximates at large and small scales the
behavior of a stationary signal with identical correlations (
),
while in the intermediate scale regime there is a plateau of random
behavior due to the random ``jumps'' at the borders between the non-zero and
zero segments [Fig. 5(c)]. Next, we consider the case when the
fraction of correlated segments
is fixed while the segment
size
changes. We find a vertical shift with increasing values for
when
increases [Fig. 8(b)], opposite to what we
observe for components with anti-correlated segments
[Fig. 8(a)]. Since the vertical shift in
is equidistant
when
increases by a factor of ten, our finding indicates a power-law
relationship between
and
.