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DFA-1 on sinusoidal trend
Given a sinusoidal trend
(
), where
is the amplitude of the
signal and
is the period, we find that the rms fluctuation
function
does not depend on the length of the
signal
, and has the same shape for different amplitudes
and different periods [Fig. 5]. We find a
crossover at scale corresponding to the period of the sinusoidal
trend
 |
(11) |
and does not depend on the amplitude
. We
call this crossover
for convenience, as we will see later. For
, the rms fluctuation
exhibits an
apparent scaling with the same exponent as
for the
linear trend [see Eq. (7)]:
 |
(12) |
where
is a constant independent of the length
, of
the period
and the amplitude
of the sinusoidal
signal, and of the box size
. As for the linear trend
[Eq.(7)], we obtain
because
at small scales (box size
) the sinusoidal function is
dominated by a linear term. For
, due to the
periodic property of the sinusoidal trend,
is a
constant independent of the scale
:
 |
(13) |
The period
and the amplitude
also affects the
vertical shift of
in both regions. We note that in
Eqs.(12) and (13),
is proportional to the
amplitude
, a behavior which is also observed for the
linear trend [Eq. (7)].
Figure 5:
Root mean square fluctuation function
for
sinusoidal functions of length
with different
amplitude
and period
. All curves exhibit a
crossover at
, with a slope
for
, and a flat region for
. There are some spurious singularities at
(
is a positive integer) shown by the spikes.
 |
Next: DFA-1 on noise with
Up: Noise with sinusoidal trend
Previous: Noise with sinusoidal trend
Zhi Chen
2002-08-28