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Previous: Superposition law for DFA
DFA-1 on linear trend
Let us suppose a linear time series
. The integrated signal
is
![\begin{displaymath}
y_{L}(i)=\sum_{j=1}^{i}A_{\rm L}j=A_{\rm L}\allowbreak \frac{i^{2}+i}{2}
\end{displaymath}](img421.png) |
(32) |
Let as call
the size of the series and
the size of the box. The rms fluctuation
as a function of
and
is
![\begin{displaymath}
F_{\rm L}(n)=A_{\rm L}\sqrt{\frac{1}{N_{max}}\sum_{k=1}^{N_...
...1)n+1}^{kn}\left(\frac{i^{2}+i}{2}-(a_{k}+b_{k}i)\right)^{2}}
\end{displaymath}](img422.png) |
(33) |
where
and
are the parameters of a least-squares fit of the
-th box of size
.
and
can be determined analytically,
thus giving:
![\begin{displaymath}
a_{k}=1-\frac{1}{12}n^{2}+\frac{1}{2}n^{2}k+\frac{1}{12}n-\frac{1}{2}%%
k^{2}n^{2}
\end{displaymath}](img425.png) |
(34) |
![\begin{displaymath}
b_{k}=1-\frac{1}{2}n+kn+\frac{1}{2}
\end{displaymath}](img426.png) |
(35) |
With these values,
can be evaluated analytically:
![\begin{displaymath}
F_{\rm L}(n)=A_{\rm L}\frac{1}{60}\sqrt{\left( 5n^{4}+25n^{3}+25n^{2}-25n-30\right) }
\end{displaymath}](img427.png) |
(36) |
The dominating term inside the square root is
and then one obtains
![\begin{displaymath}
F_{\rm L}(n)\approx \frac{\sqrt{5}}{60}A_{\rm L}n^{2}
\end{displaymath}](img429.png) |
(37) |
leading directly to an exponent of 2 in the DFA. An important consequence is that, as
does not depend on
, for linear trends
with the same slope, the DFA must give exactly the same results
for series of different sizes. This is not
true for other trends, where the exponent is 2, but the factor
multiplying
can depend on
.
Next: DFA-1 on Quadratic trend
Up: Appendix
Previous: Superposition law for DFA
Zhi Chen
2002-08-28