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DFA-1 on Quadratic trend
Let us suppose now a series of the type
. The integrated time series
is
 |
(38) |
As before, let us call
and
the sizes of the series
and box, respectively. The rms fluctuation function
measuring the rms fluctuation is now defined as
 |
(39) |
where
and
are the parameters of a least-squares fit of
the
-th box of size
. As before,
and
can be determined
analytically, thus giving:
 |
(40) |
 |
(41) |
Once
and
are known,
can be evaluated, giving:
 |
(42) |
As
, the dominant term
inside the square root is given by
, and then one has approximately
 |
(43) |
leading directly to an exponent 2 in the DFA analysis. An interesting
consequence derived from Eq. (43) is that,
depends on the length of signal
, and the DFA line (
vs
) for
quadratic series
of different
does not overlap (as is the case for linear trends).
Next: Bibliography
Up: Appendix
Previous: DFA-1 on linear trend
Zhi Chen
2002-08-28