<s>
The	O
Hurst	B-Algorithm
exponent	I-Algorithm
is	O
used	O
as	O
a	O
measure	O
of	O
long-term	O
memory	O
of	O
time	O
series	O
.	O
</s>
<s>
Studies	O
involving	O
the	O
Hurst	B-Algorithm
exponent	I-Algorithm
were	O
originally	O
developed	O
in	O
hydrology	O
for	O
the	O
practical	O
matter	O
of	O
determining	O
optimum	O
dam	O
sizing	O
for	O
the	O
Nile	O
river	O
's	O
volatile	O
rain	O
and	O
drought	O
conditions	O
that	O
had	O
been	O
observed	O
over	O
a	O
long	O
period	O
of	O
time	O
.	O
</s>
<s>
The	O
name	O
"	O
Hurst	B-Algorithm
exponent	I-Algorithm
"	O
,	O
or	O
"	O
Hurst	B-Algorithm
coefficient	I-Algorithm
"	O
,	O
derives	O
from	O
Harold	O
Edwin	O
Hurst	O
(	O
1880	O
–	O
1978	O
)	O
,	O
who	O
was	O
the	O
lead	O
researcher	O
in	O
these	O
studies	O
;	O
the	O
use	O
of	O
the	O
standard	O
notation	O
H	O
for	O
the	O
coefficient	O
also	O
relates	O
to	O
his	O
name	O
.	O
</s>
<s>
In	O
fractal	O
geometry	O
,	O
the	O
generalized	O
Hurst	B-Algorithm
exponent	I-Algorithm
has	O
been	O
denoted	O
by	O
H	O
or	O
Hq	O
in	O
honor	O
of	O
both	O
Harold	O
Edwin	O
Hurst	O
and	O
Ludwig	O
Otto	O
Hölder	O
(	O
1859	O
–	O
1937	O
)	O
by	O
Benoît	O
Mandelbrot	O
(	O
1924	O
–	O
2010	O
)	O
.	O
</s>
<s>
The	O
Hurst	B-Algorithm
exponent	I-Algorithm
is	O
referred	O
to	O
as	O
the	O
"	O
index	O
of	O
dependence	O
"	O
or	O
"	O
index	O
of	O
long-range	O
dependence	O
"	O
.	O
</s>
<s>
A	O
value	O
H	O
in	O
the	O
range	B-General_Concept
0.5	O
–	O
1	O
indicates	O
a	O
time	O
series	O
with	O
long-term	O
positive	O
autocorrelation	O
,	O
meaning	O
both	O
that	O
a	O
high	O
value	O
in	O
the	O
series	O
will	O
probably	O
be	O
followed	O
by	O
another	O
high	O
value	O
and	O
that	O
the	O
values	O
a	O
long	O
time	O
into	O
the	O
future	O
will	O
also	O
tend	O
to	O
be	O
high	O
.	O
</s>
<s>
A	O
value	O
in	O
the	O
range	B-General_Concept
0	O
–	O
0.5	O
indicates	O
a	O
time	O
series	O
with	O
long-term	O
switching	O
between	O
high	O
and	O
low	O
values	O
in	O
adjacent	O
pairs	O
,	O
meaning	O
that	O
a	O
single	O
high	O
value	O
will	O
probably	O
be	O
followed	O
by	O
a	O
low	O
value	O
and	O
that	O
the	O
value	O
after	O
that	O
will	O
tend	O
to	O
be	O
high	O
,	O
with	O
this	O
tendency	O
to	O
switch	O
between	O
high	O
and	O
low	O
values	O
lasting	O
a	O
long	O
time	O
into	O
the	O
future	O
.	O
</s>
<s>
The	O
Hurst	B-Algorithm
exponent	I-Algorithm
,	O
H	O
,	O
is	O
defined	O
in	O
terms	O
of	O
the	O
asymptotic	O
behaviour	O
of	O
the	O
rescaled	O
range	B-General_Concept
as	O
a	O
function	O
of	O
the	O
time	O
span	O
of	O
a	O
time	O
series	O
as	O
follows	O
;	O
</s>
<s>
H	O
is	O
directly	O
related	O
to	O
fractal	O
dimension	O
,	O
D	O
,	O
where	O
1	O
<	O
D	O
<	O
2	O
,	O
such	O
that	O
D	O
=	O
2	O
-	O
H	O
.	O
The	O
values	O
of	O
the	O
Hurst	B-Algorithm
exponent	I-Algorithm
vary	O
between	O
0	O
and	O
1	O
,	O
with	O
higher	O
values	O
indicating	O
a	O
smoother	O
trend	O
,	O
less	O
volatility	O
,	O
and	O
less	O
roughness	O
.	O
</s>
<s>
For	O
more	O
general	O
time	O
series	O
or	O
multi-dimensional	O
process	O
,	O
the	O
Hurst	B-Algorithm
exponent	I-Algorithm
and	O
fractal	O
dimension	O
can	O
be	O
chosen	O
independently	O
,	O
as	O
the	O
Hurst	B-Algorithm
exponent	I-Algorithm
represents	O
structure	O
over	O
asymptotically	O
longer	O
periods	O
,	O
while	O
fractal	O
dimension	O
represents	O
structure	O
over	O
asymptotically	O
shorter	O
periods	O
.	O
</s>
<s>
A	O
number	O
of	O
estimators	O
of	O
long-range	O
dependence	O
have	O
been	O
proposed	O
in	O
the	O
literature	O
.	O
</s>
<s>
The	O
oldest	O
and	O
best-known	O
is	O
the	O
so-called	O
rescaled	O
range	B-General_Concept
(	O
R/S	O
)	O
analysis	O
popularized	O
by	O
Mandelbrot	O
and	O
Wallis	O
and	O
based	O
on	O
previous	O
hydrological	O
findings	O
of	O
Hurst	O
.	O
</s>
<s>
Alternatives	O
include	O
DFA	B-Algorithm
,	O
Periodogram	O
regression	O
,	O
aggregated	O
variances	O
,	O
local	O
Whittle	O
's	O
estimator	O
,	O
wavelet	O
analysis	O
,	O
both	O
in	O
the	O
time	O
domain	O
and	O
frequency	O
domain	O
.	O
</s>
<s>
To	O
estimate	O
the	O
Hurst	B-Algorithm
exponent	I-Algorithm
,	O
one	O
must	O
first	O
estimate	O
the	O
dependence	O
of	O
the	O
rescaled	O
range	B-General_Concept
on	O
the	O
time	O
span	O
n	O
of	O
observation	O
.	O
</s>
<s>
The	O
average	O
rescaled	O
range	B-General_Concept
is	O
then	O
calculated	O
for	O
each	O
value	O
of	O
n	O
.	O
</s>
<s>
For	O
a	O
(	O
partial	O
)	O
time	O
series	O
of	O
length	O
,	O
,	O
the	O
rescaled	O
range	B-General_Concept
is	O
calculated	O
as	O
follows	O
:	O
</s>
<s>
Compute	O
the	O
range	B-General_Concept
;	O
</s>
<s>
Compute	O
the	O
standard	B-General_Concept
deviation	I-General_Concept
;	O
</s>
<s>
The	O
Hurst	B-Algorithm
exponent	I-Algorithm
is	O
estimated	O
by	O
fitting	O
the	O
power	O
law	O
to	O
the	O
data	O
.	O
</s>
<s>
The	O
Anis-Lloyd	O
corrected	O
R/S	O
Hurst	B-Algorithm
exponent	I-Algorithm
is	O
calculated	O
as	O
0.5	O
plus	O
the	O
slope	O
of	O
.	O
</s>
<s>
No	O
asymptotic	O
distribution	O
theory	O
has	O
been	O
derived	O
for	O
most	O
of	O
the	O
Hurst	B-Algorithm
exponent	I-Algorithm
estimators	O
so	O
far	O
.	O
</s>
<s>
However	O
,	O
Weron	O
used	O
bootstrapping	B-Application
to	O
obtain	O
approximate	O
functional	O
forms	O
for	O
confidence	O
intervals	O
of	O
the	O
two	O
most	O
popular	O
methods	O
,	O
i.e.	O
,	O
for	O
the	O
Anis-Lloyd	O
corrected	O
R/S	O
analysis	O
:	O
</s>
<s>
and	O
for	O
DFA	B-Algorithm
:	O
</s>
<s>
In	O
both	O
cases	O
only	O
subseries	O
of	O
length	O
were	O
considered	O
for	O
estimating	O
the	O
Hurst	B-Algorithm
exponent	I-Algorithm
;	O
subseries	O
of	O
smaller	O
length	O
lead	O
to	O
a	O
high	O
variance	O
of	O
the	O
R/S	O
estimates	O
.	O
</s>
<s>
The	O
basic	O
Hurst	B-Algorithm
exponent	I-Algorithm
can	O
be	O
related	O
to	O
the	O
expected	O
size	O
of	O
changes	O
,	O
as	O
a	O
function	O
of	O
the	O
lag	O
between	O
observations	O
,	O
as	O
measured	O
by	O
E( |Xt+τ−Xt|2	O
)	O
.	O
</s>
<s>
Mathematically	O
,	O
in	O
one	O
technique	O
,	O
the	O
Hurst	B-Algorithm
exponent	I-Algorithm
can	O
be	O
estimated	O
such	O
that	O
:	O
</s>
<s>
Multifractal	B-Algorithm
detrended	B-Algorithm
fluctuation	I-Algorithm
analysis	I-Algorithm
is	O
one	O
method	O
to	O
estimate	O
from	O
non-stationary	O
time	O
series	O
.	O
</s>
<s>
When	O
is	O
a	O
non-linear	O
function	O
of	O
q	O
the	O
time	O
series	O
is	O
a	O
multifractal	B-Algorithm
system	I-Algorithm
.	O
</s>
<s>
An	O
analysis	O
of	O
economic	O
time	O
series	O
by	O
means	O
of	O
the	O
Hurst	B-Algorithm
exponent	I-Algorithm
using	O
rescaled	O
range	B-General_Concept
and	O
Detrended	B-Algorithm
fluctuation	I-Algorithm
analysis	I-Algorithm
is	O
conducted	O
by	O
econophysicist	O
A.F.	O
</s>
<s>
This	O
paper	O
studies	O
the	O
time	O
varying	O
character	O
of	O
Long-range	O
dependency	O
and	O
,	O
thus	O
of	O
informational	O
efficiency	O
.	O
</s>
<s>
Hurst	B-Algorithm
exponent	I-Algorithm
has	O
also	O
been	O
applied	O
to	O
the	O
investigation	O
of	O
long-range	O
dependency	O
in	O
DNA	O
,	O
and	O
photonic	O
band	O
gap	O
materials	O
.	O
</s>
