<s>
Cluster	B-Algorithm
analysis	I-Algorithm
or	O
clustering	O
is	O
the	O
task	O
of	O
grouping	O
a	O
set	O
of	O
objects	O
in	O
such	O
a	O
way	O
that	O
objects	O
in	O
the	O
same	O
group	O
(	O
called	O
a	O
cluster	O
)	O
are	O
more	O
similar	O
(	O
in	O
some	O
sense	O
)	O
to	O
each	O
other	O
than	O
to	O
those	O
in	O
other	O
groups	O
(	O
clusters	O
)	O
.	O
</s>
<s>
It	O
is	O
a	O
main	O
task	O
of	O
exploratory	B-General_Concept
data	I-General_Concept
analysis	I-General_Concept
,	O
and	O
a	O
common	O
technique	O
for	O
statistical	O
data	B-General_Concept
analysis	I-General_Concept
,	O
used	O
in	O
many	O
fields	O
,	O
including	O
pattern	O
recognition	O
,	O
image	B-General_Concept
analysis	I-General_Concept
,	O
information	B-Library
retrieval	I-Library
,	O
bioinformatics	O
,	O
data	B-General_Concept
compression	I-General_Concept
,	O
computer	O
graphics	O
and	O
machine	O
learning	O
.	O
</s>
<s>
Cluster	B-Algorithm
analysis	I-Algorithm
itself	O
is	O
not	O
one	O
specific	O
algorithm	O
,	O
but	O
the	O
general	O
task	O
to	O
be	O
solved	O
.	O
</s>
<s>
Popular	O
notions	O
of	O
clusters	O
include	O
groups	O
with	O
small	O
distances	O
between	O
cluster	O
members	O
,	O
dense	O
areas	O
of	O
the	O
data	O
space	O
,	O
intervals	O
or	O
particular	O
statistical	B-General_Concept
distributions	I-General_Concept
.	O
</s>
<s>
The	O
appropriate	O
clustering	B-Algorithm
algorithm	I-Algorithm
and	O
parameter	O
settings	O
(	O
including	O
parameters	O
such	O
as	O
the	O
distance	O
function	O
to	O
use	O
,	O
a	O
density	O
threshold	O
or	O
the	O
number	O
of	O
expected	O
clusters	O
)	O
depend	O
on	O
the	O
individual	O
data	B-General_Concept
set	I-General_Concept
and	O
intended	O
use	O
of	O
the	O
results	O
.	O
</s>
<s>
Cluster	B-Algorithm
analysis	I-Algorithm
as	O
such	O
is	O
not	O
an	O
automatic	O
task	O
,	O
but	O
an	O
iterative	O
process	O
of	O
knowledge	O
discovery	O
or	O
interactive	O
multi-objective	O
optimization	O
that	O
involves	O
trial	O
and	O
failure	O
.	O
</s>
<s>
It	O
is	O
often	O
necessary	O
to	O
modify	O
data	B-General_Concept
preprocessing	I-General_Concept
and	O
model	O
parameters	O
until	O
the	O
result	O
achieves	O
the	O
desired	O
properties	O
.	O
</s>
<s>
Besides	O
the	O
term	O
clustering	O
,	O
there	O
is	O
a	O
number	O
of	O
terms	O
with	O
similar	O
meanings	O
,	O
including	O
automatic	O
classification	B-General_Concept
,	O
numerical	O
taxonomy	O
,	O
botryology	O
(	O
from	O
Greek	O
βότρυς	O
"	O
grape	O
"	O
)	O
,	O
typological	O
analysis	O
,	O
and	O
community	O
detection	O
.	O
</s>
<s>
The	O
subtle	O
differences	O
are	O
often	O
in	O
the	O
use	O
of	O
the	O
results	O
:	O
while	O
in	O
data	O
mining	O
,	O
the	O
resulting	O
groups	O
are	O
the	O
matter	O
of	O
interest	O
,	O
in	O
automatic	O
classification	B-General_Concept
the	O
resulting	O
discriminative	O
power	O
is	O
of	O
interest	O
.	O
</s>
<s>
Cluster	B-Algorithm
analysis	I-Algorithm
was	O
originated	O
in	O
anthropology	O
by	O
Driver	O
and	O
Kroeber	O
in	O
1932	O
and	O
introduced	O
to	O
psychology	O
by	O
Joseph	O
Zubin	O
in	O
1938	O
and	O
Robert	O
Tryon	O
in	O
1939	O
and	O
famously	O
used	O
by	O
Cattell	O
beginning	O
in	O
1943	O
for	O
trait	O
theory	O
classification	B-General_Concept
in	O
personality	O
psychology	O
.	O
</s>
<s>
The	O
notion	O
of	O
a	O
"	O
cluster	O
"	O
cannot	O
be	O
precisely	O
defined	O
,	O
which	O
is	O
one	O
of	O
the	O
reasons	O
why	O
there	O
are	O
so	O
many	O
clustering	B-Algorithm
algorithms	I-Algorithm
.	O
</s>
<s>
s	O
:	O
for	O
example	O
,	O
hierarchical	B-Algorithm
clustering	I-Algorithm
builds	O
models	O
based	O
on	O
distance	O
connectivity	O
.	O
</s>
<s>
s	O
:	O
for	O
example	O
,	O
the	O
k-means	B-Algorithm
algorithm	I-Algorithm
represents	O
each	O
cluster	O
by	O
a	O
single	O
mean	O
vector	O
.	O
</s>
<s>
s	O
:	O
clusters	O
are	O
modeled	O
using	O
statistical	B-General_Concept
distributions	I-General_Concept
,	O
such	O
as	O
multivariate	O
normal	O
distributions	O
used	O
by	O
the	O
expectation-maximization	B-Algorithm
algorithm	I-Algorithm
.	O
</s>
<s>
s	O
:	O
for	O
example	O
,	O
DBSCAN	B-Algorithm
and	O
OPTICS	B-Algorithm
defines	O
clusters	O
as	O
connected	O
dense	O
regions	O
in	O
the	O
data	O
space	O
.	O
</s>
<s>
s	O
:	O
in	O
biclustering	B-Algorithm
(	O
also	O
known	O
as	O
co-clustering	B-Algorithm
or	O
two-mode-clustering	O
)	O
,	O
clusters	O
are	O
modeled	O
with	O
both	O
cluster	O
members	O
and	O
relevant	O
attributes	O
.	O
</s>
<s>
Relaxations	O
of	O
the	O
complete	O
connectivity	O
requirement	O
(	O
a	O
fraction	O
of	O
the	O
edges	O
can	O
be	O
missing	O
)	O
are	O
known	O
as	O
quasi-cliques	O
,	O
as	O
in	O
the	O
HCS	B-Algorithm
clustering	I-Algorithm
algorithm	I-Algorithm
.	O
</s>
<s>
s	O
:	O
the	O
most	O
well	O
known	O
unsupervised	B-General_Concept
neural	B-Architecture
network	I-Architecture
is	O
the	O
self-organizing	B-Algorithm
map	I-Algorithm
and	O
these	O
models	O
can	O
usually	O
be	O
characterized	O
as	O
similar	O
to	O
one	O
or	O
more	O
of	O
the	O
above	O
models	O
,	O
and	O
including	O
subspace	O
models	O
when	O
neural	B-Architecture
networks	I-Architecture
implement	O
a	O
form	O
of	O
Principal	B-Application
Component	I-Application
Analysis	I-Application
or	O
Independent	B-Algorithm
Component	I-Algorithm
Analysis	I-Algorithm
.	O
</s>
<s>
A	O
"	O
clustering	O
"	O
is	O
essentially	O
a	O
set	O
of	O
such	O
clusters	O
,	O
usually	O
containing	O
all	O
objects	O
in	O
the	O
data	B-General_Concept
set	I-General_Concept
.	O
</s>
<s>
As	O
listed	O
above	O
,	O
clustering	B-Algorithm
algorithms	I-Algorithm
can	O
be	O
categorized	O
based	O
on	O
their	O
cluster	O
model	O
.	O
</s>
<s>
The	O
following	O
overview	O
will	O
only	O
list	O
the	O
most	O
prominent	O
examples	O
of	O
clustering	B-Algorithm
algorithms	I-Algorithm
,	O
as	O
there	O
are	O
possibly	O
over	O
100	O
published	O
clustering	B-Algorithm
algorithms	I-Algorithm
.	O
</s>
<s>
There	O
is	O
no	O
objectively	O
"	O
correct	O
"	O
clustering	B-Algorithm
algorithm	I-Algorithm
,	O
but	O
as	O
it	O
was	O
noted	O
,	O
"	O
clustering	O
is	O
in	O
the	O
eye	O
of	O
the	O
beholder.	O
"	O
</s>
<s>
The	O
most	O
appropriate	O
clustering	B-Algorithm
algorithm	I-Algorithm
for	O
a	O
particular	O
problem	O
often	O
needs	O
to	O
be	O
chosen	O
experimentally	O
,	O
unless	O
there	O
is	O
a	O
mathematical	O
reason	O
to	O
prefer	O
one	O
cluster	O
model	O
over	O
another	O
.	O
</s>
<s>
An	O
algorithm	O
that	O
is	O
designed	O
for	O
one	O
kind	O
of	O
model	O
will	O
generally	O
fail	O
on	O
a	O
data	B-General_Concept
set	I-General_Concept
that	O
contains	O
a	O
radically	O
different	O
kind	O
of	O
model	O
.	O
</s>
<s>
For	O
example	O
,	O
k-means	B-Algorithm
cannot	O
find	O
non-convex	O
clusters	O
.	O
</s>
<s>
Connectivity-based	O
clustering	O
,	O
also	O
known	O
as	O
hierarchical	B-Algorithm
clustering	I-Algorithm
,	O
is	O
based	O
on	O
the	O
core	O
idea	O
of	O
objects	O
being	O
more	O
related	O
to	O
nearby	O
objects	O
than	O
to	O
objects	O
farther	O
away	O
.	O
</s>
<s>
At	O
different	O
distances	O
,	O
different	O
clusters	O
will	O
form	O
,	O
which	O
can	O
be	O
represented	O
using	O
a	O
dendrogram	B-Application
,	O
which	O
explains	O
where	O
the	O
common	O
name	O
"	O
hierarchical	B-Algorithm
clustering	I-Algorithm
"	O
comes	O
from	O
:	O
these	O
algorithms	O
do	O
not	O
provide	O
a	O
single	O
partitioning	O
of	O
the	O
data	B-General_Concept
set	I-General_Concept
,	O
but	O
instead	O
provide	O
an	O
extensive	O
hierarchy	O
of	O
clusters	O
that	O
merge	O
with	O
each	O
other	O
at	O
certain	O
distances	O
.	O
</s>
<s>
In	O
a	O
dendrogram	B-Application
,	O
the	O
y-axis	O
marks	O
the	O
distance	O
at	O
which	O
the	O
clusters	O
merge	O
,	O
while	O
the	O
objects	O
are	O
placed	O
along	O
the	O
x-axis	O
such	O
that	O
the	O
clusters	O
do	O
n't	O
mix	O
.	O
</s>
<s>
Popular	O
choices	O
are	O
known	O
as	O
single-linkage	B-Algorithm
clustering	I-Algorithm
(	O
the	O
minimum	O
of	O
object	O
distances	O
)	O
,	O
complete	B-Algorithm
linkage	I-Algorithm
clustering	I-Algorithm
(	O
the	O
maximum	O
of	O
object	O
distances	O
)	O
,	O
and	O
UPGMA	B-Algorithm
or	O
WPGMA	B-Algorithm
(	O
"	O
Unweighted	O
or	O
Weighted	O
Pair	O
Group	O
Method	O
with	O
Arithmetic	O
Mean	O
"	O
,	O
also	O
known	O
as	O
average	O
linkage	B-Algorithm
clustering	I-Algorithm
)	O
.	O
</s>
<s>
Furthermore	O
,	O
hierarchical	B-Algorithm
clustering	I-Algorithm
can	O
be	O
agglomerative	O
(	O
starting	O
with	O
single	O
elements	O
and	O
aggregating	O
them	O
into	O
clusters	O
)	O
or	O
divisive	O
(	O
starting	O
with	O
the	O
complete	O
data	B-General_Concept
set	I-General_Concept
and	O
dividing	O
it	O
into	O
partitions	O
)	O
.	O
</s>
<s>
These	O
methods	O
will	O
not	O
produce	O
a	O
unique	O
partitioning	O
of	O
the	O
data	B-General_Concept
set	I-General_Concept
,	O
but	O
a	O
hierarchy	O
from	O
which	O
the	O
user	O
still	O
needs	O
to	O
choose	O
appropriate	O
clusters	O
.	O
</s>
<s>
They	O
are	O
not	O
very	O
robust	O
towards	O
outliers	B-Algorithm
,	O
which	O
will	O
either	O
show	O
up	O
as	O
additional	O
clusters	O
or	O
even	O
cause	O
other	O
clusters	O
to	O
merge	O
(	O
known	O
as	O
"	O
chaining	O
phenomenon	O
"	O
,	O
in	O
particular	O
with	O
single-linkage	B-Algorithm
clustering	I-Algorithm
)	O
.	O
</s>
<s>
In	O
the	O
general	O
case	O
,	O
the	O
complexity	O
is	O
for	O
agglomerative	B-Algorithm
clustering	I-Algorithm
and	O
for	O
divisive	O
clustering	O
,	O
which	O
makes	O
them	O
too	O
slow	O
for	O
large	O
data	B-General_Concept
sets	I-General_Concept
.	O
</s>
<s>
For	O
some	O
special	O
cases	O
,	O
optimal	O
efficient	O
methods	O
(	O
of	O
complexity	O
)	O
are	O
known	O
:	O
SLINK	B-Algorithm
for	O
single-linkage	O
and	O
CLINK	O
for	O
complete-linkage	B-Algorithm
clustering	I-Algorithm
.	O
</s>
<s>
In	O
centroid-based	O
clustering	O
,	O
each	O
cluster	O
is	O
represented	O
by	O
a	O
central	O
vector	O
,	O
which	O
is	O
not	O
necessarily	O
a	O
member	O
of	O
the	O
data	B-General_Concept
set	I-General_Concept
.	O
</s>
<s>
When	O
the	O
number	B-Algorithm
of	I-Algorithm
clusters	I-Algorithm
is	O
fixed	O
to	O
k	O
,	O
k-means	B-Algorithm
clustering	I-Algorithm
gives	O
a	O
formal	O
definition	O
as	O
an	O
optimization	O
problem	O
:	O
find	O
the	O
k	O
cluster	O
centers	O
and	O
assign	O
the	O
objects	O
to	O
the	O
nearest	O
cluster	O
center	O
,	O
such	O
that	O
the	O
squared	O
distances	O
from	O
the	O
cluster	O
are	O
minimized	O
.	O
</s>
<s>
A	O
particularly	O
well	O
known	O
approximate	O
method	O
is	O
Lloyd	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
,	O
often	O
just	O
referred	O
to	O
as	O
"	O
k-means	B-Algorithm
algorithm	I-Algorithm
"	O
(	O
although	O
another	O
algorithm	O
introduced	O
this	O
name	O
)	O
.	O
</s>
<s>
Variations	O
of	O
k-means	B-Algorithm
often	O
include	O
such	O
optimizations	O
as	O
choosing	O
the	O
best	O
of	O
multiple	O
runs	O
,	O
but	O
also	O
restricting	O
the	O
centroids	O
to	O
members	O
of	O
the	O
data	B-General_Concept
set	I-General_Concept
(	O
k-medoids	B-Algorithm
)	O
,	O
choosing	O
medians	O
(	O
k-medians	B-Algorithm
clustering	I-Algorithm
)	O
,	O
choosing	O
the	O
initial	O
centers	O
less	O
randomly	O
(	O
k-means	B-Algorithm
++	I-Algorithm
)	O
or	O
allowing	O
a	O
fuzzy	O
cluster	O
assignment	O
(	O
fuzzy	B-Algorithm
c-means	I-Algorithm
)	O
.	O
</s>
<s>
Most	O
k-means-type	O
algorithms	O
require	O
the	O
number	B-Algorithm
of	I-Algorithm
clusters	I-Algorithm
–	O
k	O
–	O
to	O
be	O
specified	O
in	O
advance	O
,	O
which	O
is	O
considered	O
to	O
be	O
one	O
of	O
the	O
biggest	O
drawbacks	O
of	O
these	O
algorithms	O
.	O
</s>
<s>
K-means	B-Algorithm
has	O
a	O
number	O
of	O
interesting	O
theoretical	O
properties	O
.	O
</s>
<s>
First	O
,	O
it	O
partitions	O
the	O
data	O
space	O
into	O
a	O
structure	O
known	O
as	O
a	O
Voronoi	B-Architecture
diagram	I-Architecture
.	O
</s>
<s>
Second	O
,	O
it	O
is	O
conceptually	O
close	O
to	O
nearest	O
neighbor	O
classification	B-General_Concept
,	O
and	O
as	O
such	O
is	O
popular	O
in	O
machine	O
learning	O
.	O
</s>
<s>
Third	O
,	O
it	O
can	O
be	O
seen	O
as	O
a	O
variation	O
of	O
model	O
based	O
clustering	O
,	O
and	O
Lloyd	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
as	O
a	O
variation	O
of	O
the	O
Expectation-maximization	B-Algorithm
algorithm	I-Algorithm
for	O
this	O
model	O
discussed	O
below	O
.	O
</s>
<s>
Centroid-based	O
clustering	O
problems	O
such	O
as	O
k-means	B-Algorithm
and	O
k-medoids	B-Algorithm
are	O
special	O
cases	O
of	O
the	O
uncapacitated	O
,	O
metric	O
facility	O
location	O
problem	O
,	O
a	O
canonical	O
problem	O
in	O
the	O
operations	O
research	O
and	O
computational	O
geometry	O
communities	O
.	O
</s>
<s>
A	O
convenient	O
property	O
of	O
this	O
approach	O
is	O
that	O
this	O
closely	O
resembles	O
the	O
way	O
artificial	O
data	B-General_Concept
sets	I-General_Concept
are	O
generated	O
:	O
by	O
sampling	O
random	O
objects	O
from	O
a	O
distribution	O
.	O
</s>
<s>
While	O
the	O
theoretical	O
foundation	O
of	O
these	O
methods	O
is	O
excellent	O
,	O
they	O
suffer	O
from	O
one	O
key	O
problem	O
known	O
as	O
overfitting	B-Error_Name
,	O
unless	O
constraints	O
are	O
put	O
on	O
the	O
model	O
complexity	O
.	O
</s>
<s>
One	O
prominent	O
method	O
is	O
known	O
as	O
Gaussian	O
mixture	O
models	O
(	O
using	O
the	O
expectation-maximization	B-Algorithm
algorithm	I-Algorithm
)	O
.	O
</s>
<s>
Here	O
,	O
the	O
data	B-General_Concept
set	I-General_Concept
is	O
usually	O
modeled	O
with	O
a	O
fixed	O
(	O
to	O
avoid	O
overfitting	B-Error_Name
)	O
number	O
of	O
Gaussian	O
distributions	O
that	O
are	O
initialized	O
randomly	O
and	O
whose	O
parameters	O
are	O
iteratively	O
optimized	O
to	O
better	O
fit	O
the	O
data	B-General_Concept
set	I-General_Concept
.	O
</s>
<s>
However	O
,	O
these	O
algorithms	O
put	O
an	O
extra	O
burden	O
on	O
the	O
user	O
:	O
for	O
many	O
real	O
data	B-General_Concept
sets	I-General_Concept
,	O
there	O
may	O
be	O
no	O
concisely	O
defined	O
mathematical	O
model	O
(	O
e.g.	O
</s>
<s>
In	O
density-based	O
clustering	O
,	O
clusters	O
are	O
defined	O
as	O
areas	O
of	O
higher	O
density	O
than	O
the	O
remainder	O
of	O
the	O
data	B-General_Concept
set	I-General_Concept
.	O
</s>
<s>
The	O
most	O
popular	O
density	O
based	O
clustering	O
method	O
is	O
DBSCAN	B-Algorithm
.	O
</s>
<s>
Another	O
interesting	O
property	O
of	O
DBSCAN	B-Algorithm
is	O
that	O
its	O
complexity	O
is	O
fairly	O
low	O
–	O
it	O
requires	O
a	O
linear	O
number	O
of	O
range	O
queries	B-Library
on	O
the	O
database	O
–	O
and	O
that	O
it	O
will	O
discover	O
essentially	O
the	O
same	O
results	O
(	O
it	O
is	O
deterministic	B-General_Concept
for	O
core	O
and	O
noise	O
points	O
,	O
but	O
not	O
for	O
border	O
points	O
)	O
in	O
each	O
run	O
,	O
therefore	O
there	O
is	O
no	O
need	O
to	O
run	O
it	O
multiple	O
times	O
.	O
</s>
<s>
OPTICS	B-Algorithm
is	O
a	O
generalization	O
of	O
DBSCAN	B-Algorithm
that	O
removes	O
the	O
need	O
to	O
choose	O
an	O
appropriate	O
value	O
for	O
the	O
range	O
parameter	O
,	O
and	O
produces	O
a	O
hierarchical	O
result	O
related	O
to	O
that	O
of	O
linkage	B-Algorithm
clustering	I-Algorithm
.	O
</s>
<s>
DeLi-Clu	O
,	O
Density-Link-Clustering	O
combines	O
ideas	O
from	O
single-linkage	B-Algorithm
clustering	I-Algorithm
and	O
OPTICS	B-Algorithm
,	O
eliminating	O
the	O
parameter	O
entirely	O
and	O
offering	O
performance	O
improvements	O
over	O
OPTICS	B-Algorithm
by	O
using	O
an	O
R-tree	B-Library
index	O
.	O
</s>
<s>
The	O
key	O
drawback	O
of	O
DBSCAN	B-Algorithm
and	O
OPTICS	B-Algorithm
is	O
that	O
they	O
expect	O
some	O
kind	O
of	O
density	O
drop	O
to	O
detect	O
cluster	O
borders	O
.	O
</s>
<s>
On	O
data	B-General_Concept
sets	I-General_Concept
with	O
,	O
for	O
example	O
,	O
overlapping	O
Gaussian	O
distributions	O
–	O
a	O
common	O
use	O
case	O
in	O
artificial	O
data	O
–	O
the	O
cluster	O
borders	O
produced	O
by	O
these	O
algorithms	O
will	O
often	O
look	O
arbitrary	O
,	O
because	O
the	O
cluster	O
density	O
decreases	O
continuously	O
.	O
</s>
<s>
On	O
a	O
data	B-General_Concept
set	I-General_Concept
consisting	O
of	O
mixtures	O
of	O
Gaussians	O
,	O
these	O
algorithms	O
are	O
nearly	O
always	O
outperformed	O
by	O
methods	O
such	O
as	O
EM	B-Algorithm
clustering	I-Algorithm
that	O
are	O
able	O
to	O
precisely	O
model	O
this	O
kind	O
of	O
data	O
.	O
</s>
<s>
Mean-shift	B-Algorithm
is	O
a	O
clustering	O
approach	O
where	O
each	O
object	O
is	O
moved	O
to	O
the	O
densest	O
area	O
in	O
its	O
vicinity	O
,	O
based	O
on	O
kernel	B-General_Concept
density	I-General_Concept
estimation	I-General_Concept
.	O
</s>
<s>
Similar	O
to	O
k-means	B-Algorithm
clustering	I-Algorithm
,	O
these	O
"	O
density	O
attractors	O
"	O
can	O
serve	O
as	O
representatives	O
for	O
the	O
data	B-General_Concept
set	I-General_Concept
,	O
but	O
mean-shift	B-Algorithm
can	O
detect	O
arbitrary-shaped	O
clusters	O
similar	O
to	O
DBSCAN	B-Algorithm
.	O
</s>
<s>
Due	O
to	O
the	O
expensive	O
iterative	O
procedure	O
and	O
density	O
estimation	O
,	O
mean-shift	B-Algorithm
is	O
usually	O
slower	O
than	O
DBSCAN	B-Algorithm
or	O
k-Means	B-Algorithm
.	O
</s>
<s>
Besides	O
that	O
,	O
the	O
applicability	O
of	O
the	O
mean-shift	B-Algorithm
algorithm	I-Algorithm
to	O
multidimensional	O
data	O
is	O
hindered	O
by	O
the	O
unsmooth	O
behaviour	O
of	O
the	O
kernel	B-General_Concept
density	I-General_Concept
estimate	I-General_Concept
,	O
which	O
results	O
in	O
over-fragmentation	O
of	O
cluster	O
tails	O
.	O
</s>
<s>
The	O
grid-based	O
technique	O
is	O
used	O
for	O
a	O
multi-dimensional	O
data	B-General_Concept
set	I-General_Concept
.	O
</s>
<s>
Steps	O
involved	O
in	O
grid-based	O
clustering	B-Algorithm
algorithm	I-Algorithm
are	O
:	O
</s>
<s>
Among	O
them	O
are	O
CLARANS	O
,	O
and	O
BIRCH	B-Algorithm
.	O
</s>
<s>
With	O
the	O
recent	O
need	O
to	O
process	O
larger	O
and	O
larger	O
data	B-General_Concept
sets	I-General_Concept
(	O
also	O
known	O
as	O
big	B-Application
data	I-Application
)	O
,	O
the	O
willingness	O
to	O
trade	O
semantic	O
meaning	O
of	O
the	O
generated	O
clusters	O
for	O
performance	O
has	O
been	O
increasing	O
.	O
</s>
<s>
This	O
led	O
to	O
the	O
development	O
of	O
pre-clustering	O
methods	O
such	O
as	O
canopy	B-Algorithm
clustering	I-Algorithm
,	O
which	O
can	O
process	O
huge	O
data	B-General_Concept
sets	I-General_Concept
efficiently	O
,	O
but	O
the	O
resulting	O
"	O
clusters	O
"	O
are	O
merely	O
a	O
rough	O
pre-partitioning	O
of	O
the	O
data	B-General_Concept
set	I-General_Concept
to	O
then	O
analyze	O
the	O
partitions	O
with	O
existing	O
slower	O
methods	O
such	O
as	O
k-means	B-Algorithm
clustering	I-Algorithm
.	O
</s>
<s>
For	O
high-dimensional	O
data	O
,	O
many	O
of	O
the	O
existing	O
methods	O
fail	O
due	O
to	O
the	O
curse	B-Algorithm
of	I-Algorithm
dimensionality	I-Algorithm
,	O
which	O
renders	O
particular	O
distance	O
functions	O
problematic	O
in	O
high-dimensional	O
spaces	O
.	O
</s>
<s>
This	O
led	O
to	O
new	O
clustering	B-Algorithm
algorithms	I-Algorithm
for	I-Algorithm
high-dimensional	I-Algorithm
data	I-Algorithm
that	O
focus	O
on	O
subspace	O
clustering	O
(	O
where	O
only	O
some	O
attributes	O
are	O
used	O
,	O
and	O
cluster	O
models	O
include	O
the	O
relevant	O
attributes	O
for	O
the	O
cluster	O
)	O
and	O
correlation	B-Algorithm
clustering	I-Algorithm
that	O
also	O
looks	O
for	O
arbitrary	O
rotated	O
(	O
"	O
correlated	O
"	O
)	O
subspace	O
clusters	O
that	O
can	O
be	O
modeled	O
by	O
giving	O
a	O
correlation	O
of	O
their	O
attributes	O
.	O
</s>
<s>
Examples	O
for	O
such	O
clustering	B-Algorithm
algorithms	I-Algorithm
are	O
CLIQUE	O
and	O
SUBCLU	B-Algorithm
.	O
</s>
<s>
Ideas	O
from	O
density-based	O
clustering	O
methods	O
(	O
in	O
particular	O
the	O
DBSCAN/OPTICS	O
family	O
of	O
algorithms	O
)	O
have	O
been	O
adapted	O
to	O
subspace	O
clustering	O
(	O
HiSC	O
,	O
hierarchical	O
subspace	O
clustering	O
and	O
DiSH	O
)	O
and	O
correlation	B-Algorithm
clustering	I-Algorithm
(	O
HiCO	O
,	O
hierarchical	O
correlation	B-Algorithm
clustering	I-Algorithm
,	O
4C	O
using	O
"	O
correlation	O
connectivity	O
"	O
and	O
ERiC	O
exploring	O
hierarchical	O
density-based	O
correlation	O
clusters	O
)	O
.	O
</s>
<s>
One	O
is	O
Marina	O
Meilă	O
's	O
variation	O
of	O
information	O
metric	O
;	O
another	O
provides	O
hierarchical	B-Algorithm
clustering	I-Algorithm
.	O
</s>
<s>
Also	O
belief	O
propagation	O
,	O
a	O
recent	O
development	O
in	O
computer	B-General_Concept
science	I-General_Concept
and	O
statistical	O
physics	O
,	O
has	O
led	O
to	O
the	O
creation	O
of	O
new	O
types	O
of	O
clustering	B-Algorithm
algorithms	I-Algorithm
.	O
</s>
<s>
Popular	O
approaches	O
involve	O
"	O
internal	O
"	O
evaluation	O
,	O
where	O
the	O
clustering	O
is	O
summarized	O
to	O
a	O
single	O
quality	O
score	O
,	O
"	O
external	O
"	O
evaluation	O
,	O
where	O
the	O
clustering	O
is	O
compared	O
to	O
an	O
existing	O
"	O
ground	O
truth	O
"	O
classification	B-General_Concept
,	O
"	O
manual	O
"	O
evaluation	O
by	O
a	O
human	O
expert	O
,	O
and	O
"	O
indirect	O
"	O
evaluation	O
by	O
evaluating	O
the	O
utility	O
of	O
the	O
clustering	O
in	O
its	O
intended	O
application	O
.	O
</s>
<s>
For	O
example	O
,	O
one	O
could	O
cluster	O
the	O
data	B-General_Concept
set	I-General_Concept
by	O
the	O
Silhouette	O
coefficient	O
;	O
except	O
that	O
there	O
is	O
no	O
known	O
efficient	O
algorithm	O
for	O
this	O
.	O
</s>
<s>
On	O
the	O
other	O
hand	O
,	O
the	O
labels	O
only	O
reflect	O
one	O
possible	O
partitioning	O
of	O
the	O
data	B-General_Concept
set	I-General_Concept
,	O
which	O
does	O
not	O
imply	O
that	O
there	O
does	O
not	O
exist	O
a	O
different	O
,	O
and	O
maybe	O
even	O
better	O
,	O
clustering	O
.	O
</s>
<s>
One	O
drawback	O
of	O
using	O
internal	O
criteria	O
in	O
cluster	O
evaluation	O
is	O
that	O
high	O
scores	O
on	O
an	O
internal	O
measure	O
do	O
not	O
necessarily	O
result	O
in	O
effective	O
information	B-Library
retrieval	I-Library
applications	O
.	O
</s>
<s>
For	O
example	O
,	O
k-means	B-Algorithm
clustering	I-Algorithm
naturally	O
optimizes	O
object	O
distances	O
,	O
and	O
a	O
distance-based	O
internal	O
criterion	O
will	O
likely	O
overrate	O
the	O
resulting	O
clustering	O
.	O
</s>
<s>
Validity	O
as	O
measured	O
by	O
such	O
an	O
index	O
depends	O
on	O
the	O
claim	O
that	O
this	O
kind	O
of	O
structure	O
exists	O
in	O
the	O
data	B-General_Concept
set	I-General_Concept
.	O
</s>
<s>
An	O
algorithm	O
designed	O
for	O
some	O
kind	O
of	O
models	O
has	O
no	O
chance	O
if	O
the	O
data	B-General_Concept
set	I-General_Concept
contains	O
a	O
radically	O
different	O
set	O
of	O
models	O
,	O
or	O
if	O
the	O
evaluation	O
measures	O
a	O
radically	O
different	O
criterion	O
.	O
</s>
<s>
For	O
example	O
,	O
k-means	B-Algorithm
clustering	I-Algorithm
can	O
only	O
find	O
convex	O
clusters	O
,	O
and	O
many	O
evaluation	O
indexes	O
assume	O
convex	O
clusters	O
.	O
</s>
<s>
On	O
a	O
data	B-General_Concept
set	I-General_Concept
with	O
non-convex	O
clusters	O
neither	O
the	O
use	O
of	O
k-means	B-Algorithm
,	O
nor	O
of	O
an	O
evaluation	O
criterion	O
that	O
assumes	O
convexity	O
,	O
is	O
sound	O
.	O
</s>
<s>
For	O
example	O
,	O
the	O
following	O
methods	O
can	O
be	O
used	O
to	O
assess	O
the	O
quality	O
of	O
clustering	B-Algorithm
algorithms	I-Algorithm
based	O
on	O
internal	O
criterion	O
:	O
</s>
<s>
where	O
n	O
is	O
the	O
number	B-Algorithm
of	I-Algorithm
clusters	I-Algorithm
,	O
is	O
the	O
centroid	O
of	O
cluster	O
,	O
is	O
the	O
average	O
distance	O
of	O
all	O
elements	O
in	O
cluster	O
to	O
centroid	O
,	O
and	O
is	O
the	O
distance	O
between	O
centroids	O
and	O
.	O
</s>
<s>
Since	O
algorithms	O
that	O
produce	O
clusters	O
with	O
low	O
intra-cluster	O
distances	O
(	O
high	O
intra-cluster	O
similarity	O
)	O
and	O
high	O
inter-cluster	O
distances	O
(	O
low	O
inter-cluster	O
similarity	O
)	O
will	O
have	O
a	O
low	O
Davies	O
–	O
Bouldin	O
index	O
,	O
the	O
clustering	B-Algorithm
algorithm	I-Algorithm
that	O
produces	O
a	O
collection	O
of	O
clusters	O
with	O
the	O
smallest	O
Davies	O
–	O
Bouldin	O
index	O
is	O
considered	O
the	O
best	O
algorithm	O
based	O
on	O
this	O
criterion	O
.	O
</s>
<s>
Objects	O
with	O
a	O
high	O
silhouette	O
value	O
are	O
considered	O
well	O
clustered	O
,	O
objects	O
with	O
a	O
low	O
value	O
may	O
be	O
outliers	B-Algorithm
.	O
</s>
<s>
This	O
index	O
works	O
well	O
with	O
k-means	B-Algorithm
clustering	I-Algorithm
,	O
and	O
is	O
also	O
used	O
to	O
determine	O
the	O
optimal	O
number	B-Algorithm
of	I-Algorithm
clusters	I-Algorithm
.	O
</s>
<s>
However	O
,	O
it	O
has	O
recently	O
been	O
discussed	O
whether	O
this	O
is	O
adequate	O
for	O
real	O
data	O
,	O
or	O
only	O
on	O
synthetic	O
data	B-General_Concept
sets	I-General_Concept
with	O
a	O
factual	O
ground	O
truth	O
,	O
since	O
classes	O
can	O
contain	O
internal	O
structure	O
,	O
the	O
attributes	O
present	O
may	O
not	O
allow	O
separation	O
of	O
clusters	O
or	O
the	O
classes	O
may	O
contain	O
anomalies	B-Algorithm
.	O
</s>
<s>
In	O
the	O
special	O
scenario	O
of	O
constrained	B-Algorithm
clustering	I-Algorithm
,	O
where	O
meta	O
information	O
(	O
such	O
as	O
class	O
labels	O
)	O
is	O
used	O
already	O
in	O
the	O
clustering	O
process	O
,	O
the	O
hold-out	O
of	O
information	O
for	O
evaluation	O
purposes	O
is	O
non-trivial	O
.	O
</s>
<s>
A	O
number	O
of	O
measures	O
are	O
adapted	O
from	O
variants	O
used	O
to	O
evaluate	O
classification	B-General_Concept
tasks	O
.	O
</s>
<s>
Also	O
,	O
purity	O
does	O
n't	O
work	O
well	O
for	O
imbalanced	O
data	O
,	O
where	O
even	O
poorly	O
performing	O
clustering	B-Algorithm
algorithms	I-Algorithm
will	O
give	O
a	O
high	O
purity	O
value	O
.	O
</s>
<s>
For	O
example	O
,	O
if	O
a	O
size	O
1000	O
dataset	B-General_Concept
consists	O
of	O
two	O
classes	O
,	O
one	O
containing	O
999	O
points	O
and	O
the	O
other	O
containing	O
1	O
point	O
,	O
then	O
every	O
possible	O
partition	O
will	O
have	O
a	O
purity	O
of	O
at	O
least	O
99.9	O
%	O
.	O
</s>
<s>
The	O
Rand	B-General_Concept
index	I-General_Concept
computes	O
how	O
similar	O
the	O
clusters	O
(	O
returned	O
by	O
the	O
clustering	B-Algorithm
algorithm	I-Algorithm
)	O
are	O
to	O
the	O
benchmark	O
classifications	O
.	O
</s>
<s>
If	O
the	O
dataset	B-General_Concept
is	O
of	O
size	O
N	O
,	O
then	O
.	O
</s>
<s>
One	O
issue	O
with	O
the	O
Rand	B-General_Concept
index	I-General_Concept
is	O
that	O
false	O
positives	O
and	O
false	O
negatives	O
are	O
equally	O
weighted	O
.	O
</s>
<s>
The	O
F-measure	B-General_Concept
addresses	O
this	O
concern	O
,	O
as	O
does	O
the	O
chance-corrected	O
adjusted	O
Rand	B-General_Concept
index	I-General_Concept
.	O
</s>
<s>
The	O
F-measure	B-General_Concept
can	O
be	O
used	O
to	O
balance	O
the	O
contribution	O
of	O
false	O
negatives	O
by	O
weighting	O
recall	O
through	O
a	O
parameter	O
.	O
</s>
<s>
We	O
can	O
calculate	O
the	O
F-measure	B-General_Concept
by	O
using	O
the	O
following	O
formula	O
:	O
</s>
<s>
In	O
other	O
words	O
,	O
recall	O
has	O
no	O
impact	O
on	O
the	O
F-measure	B-General_Concept
when	O
,	O
and	O
increasing	O
allocates	O
an	O
increasing	O
amount	O
of	O
weight	O
to	O
recall	O
in	O
the	O
final	O
F-measure	B-General_Concept
.	O
</s>
<s>
The	O
Jaccard	O
index	O
is	O
used	O
to	O
quantify	O
the	O
similarity	O
between	O
two	O
datasets	B-General_Concept
.	O
</s>
<s>
An	O
index	O
of	O
1	O
means	O
that	O
the	O
two	O
dataset	B-General_Concept
are	O
identical	O
,	O
and	O
an	O
index	O
of	O
0	O
indicates	O
that	O
the	O
datasets	B-General_Concept
have	O
no	O
common	O
elements	O
.	O
</s>
<s>
The	O
Fowlkes	O
–	O
Mallows	O
index	O
computes	O
the	O
similarity	O
between	O
the	O
clusters	O
returned	O
by	O
the	O
clustering	B-Algorithm
algorithm	I-Algorithm
and	O
the	O
benchmark	O
classifications	O
.	O
</s>
<s>
The	O
index	O
is	O
the	O
geometric	O
mean	O
of	O
the	O
precision	O
and	O
recall	O
and	O
,	O
and	O
is	O
thus	O
also	O
known	O
as	O
the	O
G-measure	O
,	O
while	O
the	O
F-measure	B-General_Concept
is	O
their	O
harmonic	O
mean	O
.	O
</s>
<s>
Chance	O
normalized	O
versions	O
of	O
recall	O
,	O
precision	O
and	O
G-measure	O
correspond	O
to	O
Informedness	B-General_Concept
,	O
Markedness	O
and	O
Matthews	B-General_Concept
Correlation	I-General_Concept
and	O
relate	O
strongly	O
to	O
Kappa	B-General_Concept
.	O
</s>
<s>
Chi	O
Index	O
is	O
a	O
external	O
validation	O
index	O
that	O
measure	O
the	O
clustering	O
results	O
by	O
applying	O
the	O
chi-squared	B-General_Concept
statistic	I-General_Concept
.	O
</s>
<s>
The	O
mutual	O
information	O
is	O
an	O
information	O
theoretic	O
measure	O
of	O
how	O
much	O
information	O
is	O
shared	O
between	O
a	O
clustering	O
and	O
a	O
ground-truth	O
classification	B-General_Concept
that	O
can	O
detect	O
a	O
non-linear	O
similarity	O
between	O
two	O
clusterings	O
.	O
</s>
<s>
A	O
confusion	B-General_Concept
matrix	I-General_Concept
can	O
be	O
used	O
to	O
quickly	O
visualize	O
the	O
results	O
of	O
a	O
classification	B-General_Concept
(	O
or	O
clustering	O
)	O
algorithm	O
.	O
</s>
<s>
With	O
this	O
definition	O
,	O
uniform	O
random	O
data	O
should	O
tend	O
to	O
have	O
values	O
near	O
to	O
0.5	O
,	O
and	O
clustered	B-Algorithm
data	I-Algorithm
should	O
tend	O
to	O
have	O
values	O
nearer	O
to	O
1	O
.	O
</s>
<s>
Cluster	B-Algorithm
analysis	I-Algorithm
is	O
used	O
to	O
describe	O
and	O
to	O
make	O
spatial	O
and	O
temporal	O
comparisons	O
of	O
communities	O
(	O
assemblages	O
)	O
of	O
organisms	O
in	O
heterogeneous	O
environments	O
.	O
</s>
<s>
Clustering	O
is	O
used	O
to	O
build	O
groups	O
of	O
genes	O
with	O
related	O
expression	O
patterns	O
(	O
also	O
known	O
as	O
coexpressed	O
genes	O
)	O
as	O
in	O
HCS	B-Algorithm
clustering	I-Algorithm
algorithm	I-Algorithm
.	O
</s>
<s>
Clustering	B-Algorithm
algorithms	I-Algorithm
are	O
used	O
to	O
automatically	O
assign	O
genotypes	O
.	O
</s>
<s>
On	O
PET	B-Application
scans	I-Application
,	O
cluster	B-Algorithm
analysis	I-Algorithm
can	O
be	O
used	O
to	O
differentiate	O
between	O
different	O
types	O
of	O
tissue	O
in	O
a	O
three-dimensional	O
image	O
for	O
many	O
different	O
purposes	O
.	O
</s>
<s>
Cluster	B-Algorithm
analysis	I-Algorithm
can	O
be	O
used	O
to	O
analyse	O
patterns	O
of	O
antibiotic	O
resistance	O
,	O
to	O
classify	O
antimicrobial	O
compounds	O
according	O
to	O
their	O
mechanism	O
of	O
action	O
,	O
to	O
classify	O
antibiotics	O
according	O
to	O
their	O
antibacterial	O
activity	O
.	O
</s>
<s>
Cluster	B-Algorithm
analysis	I-Algorithm
is	O
widely	O
used	O
in	O
market	O
research	O
when	O
working	O
with	O
multivariate	O
data	O
from	O
surveys	O
and	O
test	O
panels	O
.	O
</s>
<s>
Market	O
researchers	O
use	O
cluster	B-Algorithm
analysis	I-Algorithm
to	O
partition	O
the	O
general	O
population	O
of	O
consumers	O
into	O
market	O
segments	O
and	O
to	O
better	O
understand	O
the	O
relationships	O
between	O
different	O
groups	O
of	O
consumers/potential	O
customers	O
,	O
and	O
for	O
use	O
in	O
market	O
segmentation	B-Algorithm
,	O
product	O
positioning	O
,	O
new	O
product	O
development	O
and	O
selecting	O
test	O
markets	O
.	O
</s>
<s>
In	O
the	O
process	O
of	O
intelligent	O
grouping	O
of	O
the	O
files	O
and	O
websites	O
,	O
clustering	O
may	O
be	O
used	O
to	O
create	O
a	O
more	O
relevant	O
set	O
of	O
search	O
results	O
compared	O
to	O
normal	O
search	O
engines	O
like	O
Google	B-Application
.	I-Application
</s>
<s>
There	O
are	O
currently	O
a	O
number	O
of	O
web-based	O
clustering	O
tools	O
such	O
as	O
Clusty	B-Application
.	O
</s>
<s>
Flickr	B-Algorithm
's	O
map	O
of	O
photos	O
and	O
other	O
map	O
sites	O
use	O
clustering	O
to	O
reduce	O
the	O
number	O
of	O
markers	O
on	O
a	O
map	O
.	O
</s>
<s>
Clustering	O
can	O
be	O
used	O
to	O
divide	O
a	O
digital	B-General_Concept
image	O
into	O
distinct	O
regions	O
for	O
border	B-Algorithm
detection	I-Algorithm
or	O
object	O
recognition	O
.	O
</s>
<s>
Clustering	O
may	O
be	O
used	O
to	O
identify	O
different	O
niches	O
within	O
the	O
population	O
of	O
an	O
evolutionary	B-Algorithm
algorithm	I-Algorithm
so	O
that	O
reproductive	O
opportunity	O
can	O
be	O
distributed	O
more	O
evenly	O
amongst	O
the	O
evolving	O
species	O
or	O
subspecies	O
.	O
</s>
<s>
Recommender	B-Application
systems	I-Application
are	O
designed	O
to	O
recommend	O
new	O
items	O
based	O
on	O
a	O
user	O
's	O
tastes	O
.	O
</s>
<s>
They	O
sometimes	O
use	O
clustering	B-Algorithm
algorithms	I-Algorithm
to	O
predict	O
a	O
user	O
's	O
preferences	O
based	O
on	O
the	O
preferences	O
of	O
other	O
users	O
in	O
the	O
user	O
's	O
cluster	O
.	O
</s>
<s>
Anomalies/outliers	O
are	O
typically	O
–	O
be	O
it	O
explicitly	O
or	O
implicitly	O
–	O
defined	O
with	O
respect	O
to	O
clustering	O
structure	O
in	O
data	O
.	O
</s>
<s>
Cluster	B-Algorithm
analysis	I-Algorithm
is	O
used	O
to	O
identify	O
patterns	O
of	O
family	O
life	O
trajectories	O
,	O
professional	O
careers	O
,	O
and	O
daily	O
or	O
weekly	O
time	O
use	O
for	O
example	O
.	O
</s>
<s>
Cluster	B-Algorithm
analysis	I-Algorithm
can	O
be	O
used	O
to	O
identify	O
areas	O
where	O
there	O
are	O
greater	O
incidences	O
of	O
particular	O
types	O
of	O
crime	O
.	O
</s>
<s>
Cluster	B-Algorithm
analysis	I-Algorithm
is	O
for	O
example	O
used	O
to	O
identify	O
groups	O
of	O
schools	O
or	O
students	O
with	O
similar	O
properties	O
.	O
</s>
<s>
From	O
poll	O
data	O
,	O
projects	O
such	O
as	O
those	O
undertaken	O
by	O
the	O
Pew	O
Research	O
Center	O
use	O
cluster	B-Algorithm
analysis	I-Algorithm
to	O
discern	O
typologies	O
of	O
opinions	O
,	O
habits	O
,	O
and	O
demographics	O
that	O
may	O
be	O
useful	O
in	O
politics	O
and	O
marketing	O
.	O
</s>
<s>
Clustering	B-Algorithm
algorithms	I-Algorithm
are	O
used	O
for	O
robotic	O
situational	O
awareness	O
to	O
track	O
objects	O
and	O
detect	O
outliers	B-Algorithm
in	O
sensor	O
data	O
.	O
</s>
<s>
Cluster	B-Algorithm
analysis	I-Algorithm
has	O
been	O
used	O
to	O
cluster	O
stocks	O
into	O
sectors	O
.	O
</s>
<s>
Cluster	B-Algorithm
analysis	I-Algorithm
is	O
used	O
to	O
reconstruct	O
missing	O
bottom	O
hole	O
core	O
data	O
or	O
missing	O
log	O
curves	O
in	O
order	O
to	O
evaluate	O
reservoir	O
properties	O
.	O
</s>
