Here we show how the DFA results for any two signals and
[denoted as
and
] relate with the DFA result for the
sum of these two signals
[denoted as
, where
is the box
length (scale of analysis)]. In the general cases, we find
. When the two signals are not
correlated, we find that the following superposition rule is valid:
. Here we derive these relations.
First we summarize again the procedure of the DFA method[3]. It
includes the following steps: starting with an original signal of length
, we integrate and
obtain
, where
is the mean of
. Next, we divide
into non-overlapping boxes of equal length
. In each box we
fit the signal
using a polynomial function
, where
is the
coordinate corresponding to the
th signal point. We calculate
the r.m.s. fluctuation function
.
To prove the superposition rule, we first focus on one particular box along
the signal. In order to find the analytic expression of best fit in this box,
we write
From Eqs. (21) we determine
.
For the signals ,
and
after the integration, in each box we have
In the general case, we can utilize the Cauchy inequality
From Eqs. (21) for , in every box we have
. Thus we obtain
where
and
fluctuate around zero.
When
and
are not correlated, the value of the third
term in Eq. (27) is close to zero and we obtain the following
superposition rule