<s>
Consensus	B-Algorithm
clustering	I-Algorithm
is	O
a	O
method	O
of	O
aggregating	O
(	O
potentially	O
conflicting	O
)	O
results	O
from	O
multiple	O
clustering	B-Algorithm
algorithms	I-Algorithm
.	O
</s>
<s>
Consensus	B-Algorithm
clustering	I-Algorithm
is	O
thus	O
the	O
problem	O
of	O
reconciling	O
clustering	O
information	O
about	O
the	O
same	O
data	O
set	O
coming	O
from	O
different	O
sources	O
or	O
from	O
different	O
runs	O
of	O
the	O
same	O
algorithm	O
.	O
</s>
<s>
When	O
cast	O
as	O
an	O
optimization	O
problem	O
,	O
consensus	B-Algorithm
clustering	I-Algorithm
is	O
known	O
as	O
median	O
partition	O
,	O
and	O
has	O
been	O
shown	O
to	O
be	O
NP-complete	O
,	O
even	O
when	O
the	O
number	O
of	O
input	O
clusterings	O
is	O
three	O
.	O
</s>
<s>
Consensus	B-Algorithm
clustering	I-Algorithm
for	O
unsupervised	O
learning	O
is	O
analogous	O
to	O
ensemble	B-Algorithm
learning	I-Algorithm
in	O
supervised	O
learning	O
.	O
</s>
<s>
The	O
result	O
of	O
the	O
clustering	B-Algorithm
algorithm	I-Algorithm
(	O
that	O
,	O
in	O
many	O
cases	O
,	O
can	O
be	O
arbitrary	O
itself	O
)	O
can	O
be	O
interpreted	O
in	O
different	O
ways	O
.	O
</s>
<s>
An	O
extremely	O
important	O
issue	O
in	O
cluster	B-Algorithm
analysis	I-Algorithm
is	O
the	O
validation	O
of	O
the	O
clustering	O
results	O
,	O
that	O
is	O
,	O
how	O
to	O
gain	O
confidence	O
about	O
the	O
significance	O
of	O
the	O
clusters	O
provided	O
by	O
the	O
clustering	O
technique	O
(	O
cluster	O
numbers	O
and	O
cluster	O
assignments	O
)	O
.	O
</s>
<s>
Iterative	O
descent	O
clustering	O
methods	O
,	O
such	O
as	O
the	O
SOM	B-Algorithm
and	O
k-means	B-Algorithm
clustering	I-Algorithm
circumvent	O
some	O
of	O
the	O
shortcomings	O
of	O
hierarchical	B-Algorithm
clustering	I-Algorithm
by	O
providing	O
for	O
univocally	O
defined	O
clusters	O
and	O
cluster	O
boundaries	O
.	O
</s>
<s>
Consensus	B-Algorithm
clustering	I-Algorithm
provides	O
a	O
method	O
that	O
represents	O
the	O
consensus	O
across	O
multiple	O
runs	O
of	O
a	O
clustering	B-Algorithm
algorithm	I-Algorithm
,	O
to	O
determine	O
the	O
number	O
of	O
clusters	O
in	O
the	O
data	O
,	O
and	O
to	O
assess	O
the	O
stability	O
of	O
the	O
discovered	O
clusters	O
.	O
</s>
<s>
The	O
method	O
can	O
also	O
be	O
used	O
to	O
represent	O
the	O
consensus	O
over	O
multiple	O
runs	O
of	O
a	O
clustering	B-Algorithm
algorithm	I-Algorithm
with	O
random	O
restart	O
(	O
such	O
as	O
K-means	B-Algorithm
,	O
model-based	O
Bayesian	O
clustering	O
,	O
SOM	B-Algorithm
,	O
etc	O
.	O
</s>
<s>
However	O
,	O
they	O
lack	O
the	O
intuitive	O
and	O
visual	O
appeal	O
of	O
hierarchical	B-Algorithm
clustering	I-Algorithm
dendrograms	O
,	O
and	O
the	O
number	O
of	O
clusters	O
must	O
be	O
chosen	O
a	O
priori	O
.	O
</s>
<s>
The	O
Monti	O
consensus	B-Algorithm
clustering	I-Algorithm
algorithm	O
is	O
one	O
of	O
the	O
most	O
popular	O
consensus	B-Algorithm
clustering	I-Algorithm
algorithms	O
and	O
is	O
used	O
to	O
determine	O
the	O
number	O
of	O
clusters	O
,	O
.	O
</s>
<s>
More	O
specifically	O
,	O
given	O
a	O
set	O
of	O
points	O
to	O
cluster	O
,	O
,	O
let	O
be	O
the	O
list	O
of	O
perturbed	O
(	O
resampled	O
)	O
datasets	O
of	O
the	O
original	O
dataset	O
,	O
and	O
let	O
denote	O
the	O
connectivity	O
matrix	O
resulting	O
from	O
applying	O
a	O
clustering	B-Algorithm
algorithm	I-Algorithm
to	O
the	O
dataset	O
.	O
</s>
<s>
Monti	O
consensus	B-Algorithm
clustering	I-Algorithm
can	O
be	O
a	O
powerful	O
tool	O
for	O
identifying	O
clusters	O
,	O
but	O
it	O
needs	O
to	O
be	O
applied	O
with	O
caution	O
as	O
shown	O
by	O
Şenbabaoğlu	O
et	O
al	O
.	O
</s>
<s>
It	O
has	O
been	O
shown	O
that	O
the	O
Monti	O
consensus	B-Algorithm
clustering	I-Algorithm
algorithm	O
is	O
able	O
to	O
claim	O
apparent	O
stability	O
of	O
chance	O
partitioning	O
of	O
null	O
datasets	O
drawn	O
from	O
a	O
unimodal	O
distribution	O
,	O
and	O
thus	O
has	O
the	O
potential	O
to	O
lead	O
to	O
over-interpretation	O
of	O
cluster	O
stability	O
in	O
a	O
real	O
study	O
.	O
</s>
<s>
If	O
clusters	O
are	O
not	O
well	O
separated	O
,	O
consensus	B-Algorithm
clustering	I-Algorithm
could	O
lead	O
one	O
to	O
conclude	O
apparent	O
structure	O
when	O
there	O
is	O
none	O
,	O
or	O
declare	O
cluster	O
stability	O
when	O
it	O
is	O
subtle	O
.	O
</s>
<s>
Fred	O
and	O
Jain	O
:	O
They	O
proposed	O
to	O
use	O
a	O
single	O
linkage	O
algorithm	O
to	O
combine	O
multiple	O
runs	O
of	O
the	O
k-means	B-Algorithm
algorithm	I-Algorithm
.	O
</s>
<s>
They	O
proposed	O
information	O
theoretic	O
distance	O
measures	O
,	O
and	O
they	O
propose	O
genetic	B-Algorithm
algorithms	I-Algorithm
for	O
finding	O
the	O
best	O
aggregation	O
solution	O
.	O
</s>
<s>
:	O
They	O
defined	O
clustering	O
aggregation	O
as	O
a	O
maximum	O
likelihood	O
estimation	O
problem	O
,	O
and	O
they	O
proposed	O
an	O
EM	B-Algorithm
algorithm	I-Algorithm
for	O
finding	O
the	O
consensus	B-Algorithm
clustering	I-Algorithm
.	O
</s>
<s>
Hyper-graph	O
partitioning	O
algorithm	O
(	O
HGPA	O
)	O
:	O
The	O
HGPA	O
algorithm	O
takes	O
a	O
very	O
different	O
approach	O
to	O
finding	O
the	O
consensus	B-Algorithm
clustering	I-Algorithm
than	O
the	O
previous	O
method	O
.	O
</s>
<s>
Each	O
instance	O
in	O
a	O
soft	O
ensemble	O
is	O
represented	O
by	O
a	O
concatenation	O
of	O
r	O
posterior	O
membership	O
probability	O
distributions	O
obtained	O
from	O
the	O
constituent	O
clustering	B-Algorithm
algorithms	I-Algorithm
.	O
</s>
<s>
Bayesian	O
consensus	B-Algorithm
clustering	I-Algorithm
(	O
BCC	O
)	O
:	O
defines	O
a	O
fully	O
Bayesian	O
model	O
for	O
soft	O
consensus	B-Algorithm
clustering	I-Algorithm
in	O
which	O
multiple	O
source	O
clusterings	O
,	O
defined	O
by	O
different	O
input	O
data	O
or	O
different	O
probability	O
models	O
,	O
are	O
assumed	O
to	O
adhere	O
loosely	O
to	O
a	O
consensus	B-Algorithm
clustering	I-Algorithm
.	O
</s>
<s>
The	O
full	O
posterior	O
for	O
the	O
separate	O
clusterings	O
,	O
and	O
the	O
consensus	B-Algorithm
clustering	I-Algorithm
,	O
are	O
inferred	O
simultaneously	O
via	O
Gibbs	B-Algorithm
sampling	I-Algorithm
.	O
</s>
<s>
Ensemble	O
Clustering	O
Fuzzification	O
Means	O
(	O
ECF-Means	O
)	O
:	O
ECF-means	O
is	O
a	O
clustering	B-Algorithm
algorithm	I-Algorithm
,	O
which	O
combines	O
different	O
clustering	O
results	O
in	O
ensemble	O
,	O
achieved	O
by	O
different	O
runs	O
of	O
a	O
chosen	O
algorithm	O
(	O
k-means	B-Algorithm
)	O
,	O
into	O
a	O
single	O
final	O
clustering	O
configuration	O
.	O
</s>
