In this section, we study the effect of nonstationarity caused by removing
segments of a given length from a signal and stitching together the remaining
parts -- a ``cutting'' procedure often used in pre-processing data prior to
analysis. To address this question, we first generate a stationary correlated
signal (see Sec. 2) of length
and a scaling exponent
, using the
modified Fourier filtering method[63]. Next, we divide this signal into
non-overlapping segments of size
and randomly remove
some of these segments. Finally, we stitch together the remaining segments in
the signal
[Fig. 2(a)], thus obtaining a surrogate
nonstationary signal which is characterized by three parameters: the scaling
exponent
, the segment size
and the fraction of the signal
, which is removed.
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We find that the scaling behavior of such a nonstationary
signal strongly depends on the scaling exponent of the original
stationary correlated signal
. As illustrated in Fig. 2(b),
for a stationary anti-correlated signal with
, the
cutting procedure
causes a crossover in the scaling behavior of the resultant nonstationary
signal. This crossover appears even when only
of the
segments are cut out. At the scales larger than the crossover scale
the r.m.s. fluctuation function behaves
as
, which means an uncorrelated randomness, i.e., the
anti-correlation has been completely destroyed in this regime. For all
anti-correlated signals with exponent
, we observe a
similar crossover behavior. This result is surprising, since researchers often
take for granted that a cutting procedure before analysis does not
change the scaling properties of the original signal. Our simulation shows that
this assumption is not true, at least for anti-correlated signals.
Next, we investigate how the two parameters -- the segment size and the
fraction of points cut out from the signal -- control the effect of the
cutting procedure on the scaling behavior of anti-correlated signals. For
the fixed size of the segments (
), we find that the crossover scale
decreases with increasing the fraction of the
cutout segments [Fig. 2(c)]. Furthermore, for anti-correlated
signals with small values of the scaling exponent
, e.g.,
and
, we find that
and the fraction of
the cutout segments display an approximate power-law relationship. For a
fixed fraction of the removed segments, we find that the crossover scale
increases with increasing the segment size
[Fig. 2(d)]. To minimize the effect of the cutting procedure on the
correlation properties, it is advantageous to cut smaller number of segments
of larger size
. Moreover, if the segments which need to be removed are
too close (e.g., at a distance shorter than the size of the segments), it
may be advantageous to cut out both the segments and a part of the signal
between them. This will effectively increase the size of the segment
without substantially changing the fraction of the signal which is cut out,
leading to an increase in the crossover scale
. Such strategy
would minimize the effect of this type of nonstationarity on the scaling
properties of data. For small values of the scaling exponent
(
), we find that
and
follow power-law
relationships [Fig. 2(d)]. The reason we do not observe a power-law
relationship between
and
and between
and the
fraction of cutout segments for the values of the scaling exponent
close to
may be due to the fact that the crossover regime becomes
broader when it separates scaling regions with similar exponents, thus
leading to uncertainty in defining
. For a fixed
and a fixed
fraction of the removed segments [see Figs. 2(c) and (d)], we
observe that
increases with the increasing value of the scaling
exponent
, i.e., the effect of the cutting procedure on the scaling
behavior decreases when the anti-correlations in the signal are weaker
(
closer to
).
Finally, we consider the case of correlated signals with
. Surprisingly, we find that the scaling of correlated signals
is not affected by the cutting procedure. This observation remains true
independently of the segment size
-- from very small
up to very
large
segments -- even when up to
of the segments are
removed from a signal with
points [Fig. 2(e)].