<s>
Granular	B-General_Concept
computing	I-General_Concept
is	O
an	O
emerging	O
computing	O
paradigm	O
of	O
information	O
processing	O
that	O
concerns	O
the	O
processing	O
of	O
complex	O
information	O
entities	O
called	O
"	O
information	O
granules	O
"	O
,	O
which	O
arise	O
in	O
the	O
process	O
of	O
data	O
abstraction	O
and	O
derivation	O
of	O
knowledge	O
from	O
information	O
or	O
data	O
.	O
</s>
<s>
At	O
present	O
,	O
granular	B-General_Concept
computing	I-General_Concept
is	O
more	O
a	O
theoretical	O
perspective	O
than	O
a	O
coherent	O
set	O
of	O
methods	O
or	O
principles	O
.	O
</s>
<s>
As	O
mentioned	O
above	O
,	O
granular	B-General_Concept
computing	I-General_Concept
is	O
not	O
an	O
algorithm	O
or	O
process	O
;	O
there	O
is	O
no	O
particular	O
method	O
that	O
is	O
called	O
"	O
granular	B-General_Concept
computing	I-General_Concept
"	O
.	O
</s>
<s>
On	O
a	O
low-resolution	O
satellite	O
image	O
,	O
for	O
example	O
,	O
one	O
might	O
notice	O
interesting	O
cloud	O
patterns	O
representing	O
cyclones	B-Device
or	O
other	O
large-scale	O
weather	O
phenomena	O
,	O
while	O
in	O
a	O
higher-resolution	O
image	O
,	O
one	O
misses	O
these	O
large-scale	O
atmospheric	O
phenomena	O
but	O
instead	O
notices	O
smaller-scale	O
phenomena	O
,	O
such	O
as	O
the	O
interesting	O
pattern	O
that	O
is	O
the	O
streets	O
of	O
Manhattan	O
.	O
</s>
<s>
The	O
aim	O
of	O
granular	B-General_Concept
computing	I-General_Concept
is	O
to	O
try	O
to	O
take	O
advantage	O
of	O
this	O
fact	O
in	O
designing	O
more	O
effective	O
machine-learning	O
and	O
reasoning	O
systems	O
.	O
</s>
<s>
There	O
are	O
several	O
types	O
of	O
granularity	O
that	O
are	O
often	O
encountered	O
in	O
data	B-Application
mining	I-Application
and	O
machine	O
learning	O
,	O
and	O
we	O
review	O
them	O
below	O
:	O
</s>
<s>
One	O
type	O
of	O
granulation	O
is	O
the	O
quantization	B-Algorithm
of	O
variables	O
.	O
</s>
<s>
It	O
is	O
very	O
common	O
that	O
in	O
data	B-Application
mining	I-Application
or	O
machine-learning	O
applications	O
the	O
resolution	O
of	O
variables	O
needs	O
to	O
be	O
decreased	O
in	O
order	O
to	O
extract	O
meaningful	O
regularities	O
.	O
</s>
<s>
An	O
example	O
of	O
this	O
would	O
be	O
a	O
variable	O
such	O
as	O
"	O
outside	O
temperature	O
"	O
(	O
)	O
,	O
which	O
in	O
a	O
given	O
application	O
might	O
be	O
recorded	O
to	O
several	O
decimal	B-Architecture
places	I-Architecture
of	O
precision	B-Architecture
(	O
depending	O
on	O
the	O
sensing	O
apparatus	O
)	O
.	O
</s>
<s>
For	O
this	O
reason	O
,	O
any	O
"	O
regularity	O
"	O
which	O
our	O
learning	O
algorithms	O
might	O
detect	O
at	O
this	O
level	O
of	O
resolution	O
would	O
have	O
to	O
be	O
spurious	O
,	O
as	O
an	O
artifact	O
of	O
overfitting	B-Error_Name
.	O
</s>
<s>
Thus	O
,	O
in	O
this	O
case	O
,	O
reducing	O
resolution	O
is	O
a	O
method	O
of	O
controlling	O
overfitting	B-Error_Name
.	O
</s>
<s>
In	O
this	O
sense	O
,	O
increasing	O
granularity	O
provides	O
an	O
antidote	O
to	O
the	O
so-called	O
curse	B-Algorithm
of	I-Algorithm
dimensionality	I-Algorithm
,	O
which	O
relates	O
to	O
the	O
exponential	O
decrease	O
in	O
statistical	O
power	O
with	O
increase	O
in	O
number	O
of	O
dimensions	O
or	O
variable	O
cardinality	O
.	O
</s>
<s>
It	O
is	O
not	O
feasible	O
to	O
exhaustively	O
test	O
all	O
possible	O
discretization	B-Algorithm
resolutions	O
on	O
all	O
variables	O
in	O
order	O
to	O
see	O
which	O
combination	O
of	O
resolutions	O
yields	O
interesting	O
or	O
significant	O
results	O
.	O
</s>
<s>
Instead	O
,	O
the	O
feature	O
space	O
must	O
be	O
preprocessed	O
(	O
often	O
by	O
an	O
entropy	O
analysis	O
of	O
some	O
kind	O
)	O
so	O
that	O
some	O
guidance	O
can	O
be	O
given	O
as	O
to	O
how	O
the	O
discretization	B-Algorithm
process	O
should	O
proceed	O
.	O
</s>
<s>
A	O
sample	O
of	O
papers	O
that	O
address	O
the	O
problem	O
of	O
variable	O
discretization	B-Algorithm
in	O
general	O
,	O
and	O
multiple-variable	O
discretization	B-Algorithm
in	O
particular	O
,	O
is	O
as	O
follows	O
:	O
,	O
,	O
,	O
,	O
,	O
,	O
,	O
,	O
,	O
,	O
,	O
,	O
,	O
,	O
,	O
,	O
</s>
<s>
A	O
number	O
of	O
classical	O
methods	O
,	O
such	O
as	O
principal	B-Application
component	I-Application
analysis	I-Application
,	O
multidimensional	O
scaling	O
,	O
factor	O
analysis	O
,	O
and	O
structural	O
equation	O
modeling	O
,	O
and	O
their	O
relatives	O
,	O
fall	O
under	O
the	O
genus	O
of	O
"	O
variable	O
transformation.	O
"	O
</s>
<s>
Also	O
in	O
this	O
category	O
are	O
more	O
modern	O
areas	O
of	O
study	O
such	O
as	O
dimensionality	B-Algorithm
reduction	I-Algorithm
,	O
projection	B-General_Concept
pursuit	I-General_Concept
,	O
and	O
independent	B-Algorithm
component	I-Algorithm
analysis	I-Algorithm
.	O
</s>
<s>
These	O
dimensionality	B-Algorithm
reduction	I-Algorithm
methods	O
are	O
all	O
reviewed	O
in	O
the	O
standard	O
texts	O
,	O
such	O
as	O
,	O
,	O
and	O
.	O
</s>
<s>
A	O
different	O
class	O
of	O
variable	O
granulation	O
methods	O
derive	O
more	O
from	O
data	B-Algorithm
clustering	I-Algorithm
methodologies	O
than	O
from	O
the	O
linear	O
systems	O
theory	O
informing	O
the	O
above	O
methods	O
.	O
</s>
<s>
In	O
data	B-Algorithm
clustering	I-Algorithm
,	O
one	O
identifies	O
a	O
group	O
of	O
similar	O
entities	O
(	O
using	O
a	O
"	O
measure	O
of	O
similarity	O
"	O
suitable	O
to	O
the	O
domain	O
—	O
)	O
,	O
and	O
then	O
in	O
some	O
sense	O
replaces	O
those	O
entities	O
with	O
a	O
prototype	O
of	O
some	O
kind	O
.	O
</s>
<s>
Although	O
variable	B-General_Concept
clustering	I-General_Concept
methods	O
based	O
on	O
linear	O
correlation	O
have	O
been	O
proposed	O
(	O
;	O
)	O
,	O
more	O
powerful	O
methods	O
of	O
variable	B-General_Concept
clustering	I-General_Concept
are	O
based	O
on	O
the	O
mutual	O
information	O
between	O
variables	O
.	O
</s>
<s>
In	O
database	B-Application
systems	I-Application
,	O
aggregations	O
(	O
see	O
e.g.	O
</s>
<s>
OLAP	B-Application
aggregation	I-Application
and	O
Business	B-General_Concept
intelligence	I-General_Concept
systems	I-General_Concept
)	O
result	O
in	O
transforming	O
original	O
data	O
tables	O
(	O
often	O
called	O
information	O
systems	O
)	O
into	O
the	O
tables	O
with	O
different	O
semantics	O
of	O
rows	O
and	O
columns	O
,	O
wherein	O
the	O
rows	O
correspond	O
to	O
the	O
groups	O
(	O
granules	O
)	O
of	O
original	O
tuples	O
and	O
the	O
columns	O
express	O
aggregated	O
information	O
about	O
original	O
values	O
within	O
each	O
of	O
the	O
groups	O
.	O
</s>
<s>
The	O
origins	O
of	O
the	O
granular	B-General_Concept
computing	I-General_Concept
ideology	O
are	O
to	O
be	O
found	O
in	O
the	O
rough	O
sets	O
and	O
fuzzy	O
sets	O
literatures	O
.	O
</s>
<s>
Just	O
as	O
in	O
the	O
case	O
of	O
value	O
granulation	O
(	O
discretization/quantization	O
)	O
,	O
it	O
is	O
possible	O
that	O
relationships	O
(	O
dependencies	O
)	O
may	O
emerge	O
at	O
one	O
level	O
of	O
granularity	O
that	O
are	O
not	O
present	O
at	O
another	O
.	O
</s>
<s>
The	O
objects	O
that	O
can	O
be	O
definitively	O
categorized	O
according	O
to	O
concept	O
structure	O
based	O
on	O
are	O
those	O
in	O
the	O
set	O
and	O
since	O
there	O
are	O
six	O
of	O
these	O
,	O
the	O
dependency	O
of	O
on	O
,	O
This	O
might	O
be	O
considered	O
an	O
interesting	O
dependency	O
in	O
its	O
own	O
right	O
,	O
but	O
perhaps	O
in	O
a	O
particular	O
data	B-Application
mining	I-Application
application	O
only	O
stronger	O
dependencies	O
are	O
desired	O
.	O
</s>
<s>
Finding	O
the	O
"	O
right	O
"	O
concept	O
resolution	O
is	O
a	O
tricky	O
problem	O
for	O
which	O
many	O
methods	O
have	O
been	O
proposed	O
(	O
e.g.	O
,	O
AIC	O
,	O
BIC	B-General_Concept
,	O
MDL	O
,	O
etc	O
.	O
</s>
<s>
Granular	B-General_Concept
computing	I-General_Concept
can	O
be	O
conceived	O
as	O
a	O
framework	O
of	O
theories	O
,	O
methodologies	O
,	O
techniques	O
,	O
and	O
tools	O
that	O
make	O
use	O
of	O
information	O
granules	O
in	O
the	O
process	O
of	O
problem	O
solving	O
.	O
</s>
<s>
In	O
this	O
sense	O
,	O
granular	B-General_Concept
computing	I-General_Concept
is	O
used	O
as	O
an	O
umbrella	O
term	O
to	O
cover	O
topics	O
that	O
have	O
been	O
studied	O
in	O
various	O
fields	O
in	O
isolation	O
.	O
</s>
<s>
By	O
examining	O
all	O
of	O
these	O
existing	O
studies	O
in	O
light	O
of	O
the	O
unified	O
framework	O
of	O
granular	B-General_Concept
computing	I-General_Concept
and	O
extracting	O
their	O
commonalities	O
,	O
it	O
may	O
be	O
possible	O
to	O
develop	O
a	O
general	O
theory	O
for	O
problem	O
solving	O
.	O
</s>
<s>
In	O
a	O
more	O
philosophical	O
sense	O
,	O
granular	B-General_Concept
computing	I-General_Concept
can	O
describe	O
a	O
way	O
of	O
thinking	O
that	O
relies	O
on	O
the	O
human	O
ability	O
to	O
perceive	O
the	O
real	O
world	O
under	O
various	O
levels	O
of	O
granularity	O
(	O
i.e.	O
,	O
abstraction	O
)	O
in	O
order	O
to	O
abstract	O
and	O
consider	O
only	O
those	O
things	O
that	O
serve	O
a	O
specific	O
interest	O
and	O
to	O
switch	O
among	O
different	O
granularities	O
.	O
</s>
<s>
Granular	B-General_Concept
computing	I-General_Concept
is	O
thus	O
essential	O
in	O
human	O
problem	O
solving	O
and	O
hence	O
has	O
a	O
very	O
significant	O
impact	O
on	O
the	O
design	O
and	O
implementation	O
of	O
intelligent	O
systems	O
.	O
</s>
