<s>
In	O
stochastic	O
processes	O
,	O
chaos	O
theory	O
and	O
time	O
series	O
analysis	O
,	O
detrended	B-Algorithm
fluctuation	I-Algorithm
analysis	I-Algorithm
(	O
DFA	O
)	O
is	O
a	O
method	O
for	O
determining	O
the	O
statistical	O
self-affinity	O
of	O
a	O
signal	O
.	O
</s>
<s>
The	O
obtained	O
exponent	O
is	O
similar	O
to	O
the	O
Hurst	B-Algorithm
exponent	I-Algorithm
,	O
except	O
that	O
DFA	O
may	O
also	O
be	O
applied	O
to	O
signals	O
whose	O
underlying	O
statistics	O
(	O
such	O
as	O
mean	O
and	O
variance	O
)	O
or	O
dynamics	O
are	O
non-stationary	B-Algorithm
(	O
changing	O
with	O
time	O
)	O
.	O
</s>
<s>
It	O
is	O
related	O
to	O
measures	O
based	O
upon	O
spectral	O
techniques	O
such	O
as	O
autocorrelation	O
and	O
Fourier	B-Algorithm
transform	I-Algorithm
.	O
</s>
<s>
introduced	O
DFA	O
in	O
1994	O
in	O
a	O
paper	O
that	O
has	O
been	O
cited	O
over	O
3,000	O
times	O
as	O
of	O
2022	O
and	O
represents	O
an	O
extension	O
of	O
the	O
(	O
ordinary	O
)	O
fluctuation	O
analysis	O
(	O
FA	O
)	O
,	O
which	O
is	O
affected	O
by	O
non-stationarities	O
.	O
</s>
<s>
Next	O
,	O
is	O
divided	O
into	O
time	O
windows	O
of	O
length	O
samples	O
each	O
,	O
and	O
a	O
local	O
least	B-Algorithm
squares	I-Algorithm
straight-line	O
fit	O
(	O
the	O
local	O
trend	O
)	O
is	O
calculated	O
by	O
minimising	O
the	O
squared	O
errors	O
within	O
each	O
time	O
window	O
.	O
</s>
<s>
The	O
scaling	O
exponent	O
is	O
calculated	O
as	O
the	O
slope	O
of	O
a	O
straight	O
line	O
fit	O
to	O
the	O
log-log	O
graph	O
of	O
against	O
using	O
least-squares	B-Algorithm
.	O
</s>
<s>
This	O
exponent	O
is	O
a	O
generalization	O
of	O
the	O
Hurst	B-Algorithm
exponent	I-Algorithm
.	O
</s>
<s>
The	O
multifractal	B-Algorithm
generalization	O
(	O
MF-DFA	O
)	O
uses	O
a	O
variable	O
moment	O
and	O
provides	O
.	O
</s>
<s>
intended	O
this	O
scaling	O
exponent	O
as	O
a	O
generalization	O
of	O
the	O
classical	O
Hurst	B-Algorithm
exponent	I-Algorithm
.	O
</s>
<s>
The	O
classical	O
Hurst	B-Algorithm
exponent	I-Algorithm
corresponds	O
to	O
the	O
second	O
moment	O
for	O
stationary	O
cases	O
and	O
to	O
the	O
second	O
moment	O
minus	O
1	O
for	O
nonstationary	O
cases	O
.	O
</s>
<s>
Multifractal	B-Algorithm
systems	I-Algorithm
scale	O
as	O
a	O
function	O
.	O
</s>
<s>
To	O
uncover	O
multifractality	B-Algorithm
,	O
Multifractal	B-Algorithm
Detrended	B-Algorithm
Fluctuation	I-Algorithm
Analysis	I-Algorithm
is	O
one	O
possible	O
method	O
.	O
</s>
<s>
For	O
fractional	O
Gaussian	O
noise	O
(	O
FGN	O
)	O
,	O
we	O
have	O
,	O
and	O
thus	O
,	O
and	O
,	O
where	O
is	O
the	O
Hurst	B-Algorithm
exponent	I-Algorithm
.	O
</s>
<s>
For	O
fractional	O
Brownian	O
motion	O
(	O
FBM	O
)	O
,	O
we	O
have	O
,	O
and	O
thus	O
,	O
and	O
,	O
where	O
is	O
the	O
Hurst	B-Algorithm
exponent	I-Algorithm
.	O
</s>
<s>
Furthermore	O
,	O
a	O
combination	O
of	O
techniques	O
including	O
MLE	O
,	O
rather	O
than	O
least-squares	B-Algorithm
has	O
been	O
shown	O
to	O
better	O
approximate	O
the	O
scaling	O
,	O
or	O
power-law	O
,	O
exponent	O
.	O
</s>
<s>
Also	O
,	O
there	O
are	O
many	O
scaling	O
exponent-like	O
quantities	O
that	O
can	O
be	O
measured	O
for	O
a	O
self-similar	O
time	O
series	O
,	O
including	O
the	O
divider	O
dimension	O
and	O
Hurst	B-Algorithm
exponent	I-Algorithm
.	O
</s>
