<s>
A	O
multifractal	B-Algorithm
system	I-Algorithm
is	O
a	O
generalization	O
of	O
a	O
fractal	B-Application
system	O
in	O
which	O
a	O
single	O
exponent	O
(	O
the	O
fractal	B-Application
dimension	O
)	O
is	O
not	O
enough	O
to	O
describe	O
its	O
dynamics	O
;	O
instead	O
,	O
a	O
continuous	O
spectrum	O
of	O
exponents	O
(	O
the	O
so-called	O
singularity	O
spectrum	O
)	O
is	O
needed	O
.	O
</s>
<s>
Multifractal	B-Algorithm
systems	I-Algorithm
are	O
common	O
in	O
nature	O
.	O
</s>
<s>
The	O
origin	O
of	O
multifractality	B-Algorithm
in	O
sequential	O
(	O
time	O
series	O
)	O
data	O
has	O
been	O
attributed	O
to	O
mathematical	O
convergence	O
effects	O
related	O
to	O
the	O
central	O
limit	O
theorem	O
that	O
have	O
as	O
foci	O
of	O
convergence	O
the	O
family	O
of	O
statistical	O
distributions	O
known	O
as	O
the	O
Tweedie	O
exponential	O
dispersion	O
models	O
,	O
as	O
well	O
as	O
the	O
geometric	O
Tweedie	O
models	O
.	O
</s>
<s>
The	O
first	O
convergence	O
effect	O
yields	O
monofractal	O
sequences	O
,	O
and	O
the	O
second	O
convergence	O
effect	O
is	O
responsible	O
for	O
variation	O
in	O
the	O
fractal	B-Application
dimension	O
of	O
the	O
monofractal	O
sequences	O
.	O
</s>
<s>
Multifractal	B-Algorithm
analysis	I-Algorithm
is	O
used	O
to	O
investigate	O
datasets	O
,	O
often	O
in	O
conjunction	O
with	O
other	O
methods	O
of	O
fractal	B-Application
and	O
lacunarity	O
analysis	O
.	O
</s>
<s>
The	O
technique	O
entails	O
distorting	O
datasets	O
extracted	O
from	O
patterns	O
to	O
generate	O
multifractal	B-Algorithm
spectra	O
that	O
illustrate	O
how	O
scaling	O
varies	O
over	O
the	O
dataset	O
.	O
</s>
<s>
Multifractal	B-Algorithm
analysis	I-Algorithm
has	O
been	O
used	O
to	O
decipher	O
the	O
generating	O
rules	O
and	O
functionalities	O
of	O
complex	O
networks	O
.	O
</s>
<s>
Multifractal	B-Algorithm
analysis	I-Algorithm
techniques	O
have	O
been	O
applied	O
in	O
a	O
variety	O
of	O
practical	O
situations	O
,	O
such	O
as	O
predicting	O
earthquakes	O
and	O
interpreting	O
medical	O
images	O
.	O
</s>
<s>
In	O
a	O
multifractal	B-Algorithm
system	I-Algorithm
,	O
the	O
behavior	O
around	O
any	O
point	O
is	O
described	O
by	O
a	O
local	O
power	O
law	O
:	O
</s>
<s>
The	O
ensemble	O
formed	O
by	O
all	O
the	O
points	O
that	O
share	O
the	O
same	O
singularity	O
exponent	O
is	O
called	O
the	O
singularity	O
manifold	O
of	O
exponent	O
h	O
,	O
and	O
is	O
a	O
fractal	B-Application
set	O
of	O
fractal	B-Application
dimension	O
the	O
singularity	O
spectrum	O
.	O
</s>
<s>
In	O
practice	O
,	O
the	O
multifractal	B-Algorithm
behaviour	O
of	O
a	O
physical	O
system	O
is	O
not	O
directly	O
characterized	O
by	O
its	O
singularity	O
spectrum	O
.	O
</s>
<s>
Indeed	O
,	O
multifractal	B-Algorithm
signals	O
generally	O
obey	O
a	O
scale	O
invariance	O
property	O
that	O
yields	O
power-law	O
behaviours	O
for	O
multiresolution	O
quantities	O
,	O
depending	O
on	O
their	O
scale	O
.	O
</s>
<s>
For	O
multifractal	B-Algorithm
objects	O
,	O
one	O
usually	O
observes	O
a	O
global	O
power-law	O
scaling	O
of	O
the	O
form	O
:	O
</s>
<s>
Using	O
so-called	O
multifractal	B-Algorithm
formalism	O
,	O
it	O
can	O
be	O
shown	O
that	O
,	O
under	O
some	O
well-suited	O
assumptions	O
,	O
there	O
exists	O
a	O
correspondence	O
between	O
the	O
singularity	O
spectrum	O
and	O
the	O
multi-scaling	O
exponents	O
through	O
a	O
Legendre	O
transform	O
.	O
</s>
<s>
Multifractal	B-Algorithm
systems	I-Algorithm
are	O
often	O
modeled	O
by	O
stochastic	O
processes	O
such	O
as	O
multiplicative	B-Algorithm
cascades	I-Algorithm
.	O
</s>
<s>
This	O
evolution	O
is	O
often	O
called	O
statistical	O
intermittency	O
and	O
betrays	O
a	O
departure	O
from	O
Gaussian	B-Application
models	O
.	O
</s>
<s>
Modelling	O
as	O
a	O
multiplicative	B-Algorithm
cascade	I-Algorithm
also	O
leads	O
to	O
estimation	O
of	O
multifractal	B-Algorithm
properties	O
.	O
</s>
<s>
A	O
maximum	O
likely	O
fit	O
of	O
a	O
multiplicative	B-Algorithm
cascade	I-Algorithm
to	O
the	O
dataset	O
not	O
only	O
estimates	O
the	O
complete	O
spectrum	O
but	O
also	O
gives	O
reasonable	O
estimates	O
of	O
the	O
errors	O
.	O
</s>
<s>
Multifractal	B-Algorithm
spectra	O
can	O
be	O
determined	O
from	O
box	B-Algorithm
counting	I-Algorithm
on	O
digital	O
images	O
.	O
</s>
<s>
First	O
,	O
a	O
box	B-Algorithm
counting	I-Algorithm
scan	O
is	O
done	O
to	O
determine	O
how	O
the	O
pixels	O
are	O
distributed	O
;	O
then	O
,	O
this	O
"	O
mass	O
distribution	O
"	O
becomes	O
the	O
basis	O
for	O
a	O
series	O
of	O
calculations	O
.	O
</s>
<s>
The	O
chief	O
idea	O
is	O
that	O
for	O
multifractals	B-Algorithm
,	O
the	O
probability	O
of	O
a	O
number	O
of	O
pixels	O
,	O
appearing	O
in	O
a	O
box	O
,	O
varies	O
as	O
box	B-Algorithm
size	I-Algorithm
,	O
to	O
some	O
exponent	O
,	O
which	O
changes	O
over	O
the	O
image	O
,	O
as	O
in	O
(	O
NB	B-Protocol
:	O
For	O
monofractals	O
,	O
in	O
contrast	O
,	O
the	O
exponent	O
does	O
not	O
change	O
meaningfully	O
over	O
the	O
set	O
)	O
.	O
</s>
<s>
Multifractal	B-Algorithm
analysis	I-Algorithm
has	O
been	O
successfully	O
used	O
in	O
many	O
fields	O
,	O
including	O
physical	O
,	O
information	O
,	O
and	O
biological	O
sciences	O
.	O
</s>
<s>
Multifractal	B-Algorithm
analysis	I-Algorithm
has	O
been	O
used	O
in	O
several	O
scientific	O
fields	O
to	O
characterize	O
various	O
types	O
of	O
datasets	O
.	O
</s>
<s>
In	O
essence	O
,	O
multifractal	B-Algorithm
analysis	I-Algorithm
applies	O
a	O
distorting	O
factor	O
to	O
datasets	O
extracted	O
from	O
patterns	O
,	O
to	O
compare	O
how	O
the	O
data	O
behave	O
at	O
each	O
distortion	O
.	O
</s>
<s>
This	O
is	O
done	O
using	O
graphs	O
known	O
as	O
multifractal	B-Algorithm
spectra	O
,	O
analogous	O
to	O
viewing	O
the	O
dataset	O
through	O
a	O
"	O
distorting	O
lens	O
"	O
,	O
as	O
shown	O
in	O
the	O
illustration	O
.	O
</s>
<s>
Several	O
types	O
of	O
multifractal	B-Algorithm
spectra	O
are	O
used	O
in	O
practise	O
.	O
</s>
<s>
One	O
practical	O
multifractal	B-Algorithm
spectrum	O
is	O
the	O
graph	O
of	O
DQ	O
vs	O
Q	O
,	O
where	O
DQ	O
is	O
the	O
generalized	O
dimension	O
for	O
a	O
dataset	O
and	O
Q	O
is	O
an	O
arbitrary	O
set	O
of	O
exponents	O
.	O
</s>
<s>
The	O
expression	O
generalized	O
dimension	O
thus	O
refers	O
to	O
a	O
set	O
of	O
dimensions	O
for	O
a	O
dataset	O
(	O
detailed	O
calculations	O
for	O
determining	O
the	O
generalized	O
dimension	O
using	O
box	B-Algorithm
counting	I-Algorithm
are	O
described	O
below	O
)	O
.	O
</s>
<s>
The	O
image	O
shows	O
D(Q )	O
spectra	O
from	O
a	O
multifractal	B-Algorithm
analysis	I-Algorithm
of	O
binary	O
images	O
of	O
non-	O
,	O
mono-	O
,	O
and	O
multi-fractal	O
sets	O
.	O
</s>
<s>
As	O
is	O
the	O
case	O
in	O
the	O
sample	O
images	O
,	O
non	O
-	O
and	O
mono-fractals	O
tend	O
to	O
have	O
flatter	O
D(Q )	O
spectra	O
than	O
multifractals	B-Algorithm
.	O
</s>
<s>
D( Q	O
=	O
0	O
)	O
is	O
equal	O
to	O
the	O
capacity	O
dimension	O
,	O
which	O
—	O
in	O
the	O
analysis	O
shown	O
in	O
the	O
figures	O
here	O
—	O
is	O
the	O
box	B-Algorithm
counting	I-Algorithm
dimension	O
.	O
</s>
<s>
This	O
relates	O
to	O
the	O
"	O
multi	O
"	O
in	O
multifractal	B-Algorithm
,	O
where	O
multifractals	B-Algorithm
have	O
multiple	O
dimensions	O
in	O
the	O
D(Q )	O
versus	O
Q	O
spectra	O
,	O
but	O
monofractals	O
stay	O
rather	O
flat	O
in	O
that	O
area	O
.	O
</s>
<s>
Another	O
useful	O
multifractal	B-Algorithm
spectrum	O
is	O
the	O
graph	O
of	O
versus	O
(	O
see	O
calculations	O
)	O
.	O
</s>
<s>
These	O
graphs	O
generally	O
rise	O
to	O
a	O
maximum	O
that	O
approximates	O
the	O
fractal	B-Application
dimension	O
at	O
Q	O
=	O
0	O
,	O
and	O
then	O
fall	O
.	O
</s>
<s>
Like	O
DQ	O
versus	O
Q	O
spectra	O
,	O
they	O
also	O
show	O
typical	O
patterns	O
useful	O
for	O
comparing	O
non-	O
,	O
mono-	O
,	O
and	O
multi-fractal	O
patterns	O
.	O
</s>
<s>
In	O
particular	O
,	O
for	O
these	O
spectra	O
,	O
non	O
-	O
and	O
mono-fractals	O
converge	O
on	O
certain	O
values	O
,	O
whereas	O
the	O
spectra	O
from	O
multifractal	B-Algorithm
patterns	O
typically	O
form	O
humps	O
over	O
a	O
broader	O
area	O
.	O
</s>
