<s>
In	O
cluster	B-Algorithm
analysis	I-Algorithm
,	O
the	O
elbow	B-General_Concept
method	I-General_Concept
is	O
a	O
heuristic	B-Algorithm
used	O
in	O
determining	B-Algorithm
the	I-Algorithm
number	I-Algorithm
of	I-Algorithm
clusters	I-Algorithm
in	I-Algorithm
a	I-Algorithm
data	I-Algorithm
set	I-Algorithm
.	O
</s>
<s>
The	O
same	O
method	O
can	O
be	O
used	O
to	O
choose	O
the	O
number	O
of	O
parameters	O
in	O
other	O
data-driven	O
models	O
,	O
such	O
as	O
the	O
number	O
of	O
principal	B-Application
components	I-Application
to	O
describe	O
a	O
data	O
set	O
.	O
</s>
<s>
Using	O
the	O
"	O
elbow	O
"	O
or	O
"	O
knee	O
of	O
a	O
curve	O
"	O
as	O
a	O
cutoff	O
point	O
is	O
a	O
common	O
heuristic	B-Algorithm
in	O
mathematical	O
optimization	O
to	O
choose	O
a	O
point	O
where	O
diminishing	O
returns	O
are	O
no	O
longer	O
worth	O
the	O
additional	O
cost	O
.	O
</s>
<s>
The	O
intuition	O
is	O
that	O
increasing	O
the	O
number	O
of	O
clusters	O
will	O
naturally	O
improve	O
the	O
fit	O
(	O
explain	O
more	O
of	O
the	O
variation	O
)	O
,	O
since	O
there	O
are	O
more	O
parameters	O
(	O
more	O
clusters	O
)	O
to	O
use	O
,	O
but	O
that	O
at	O
some	O
point	O
this	O
is	O
over-fitting	B-Error_Name
,	O
and	O
the	O
elbow	O
reflects	O
this	O
.	O
</s>
<s>
For	O
example	O
,	O
given	O
data	O
that	O
actually	O
consist	O
of	O
k	O
labeled	O
groups	O
–	O
for	O
example	O
,	O
k	O
points	O
sampled	O
with	O
noise	O
–	O
clustering	O
with	O
more	O
than	O
k	O
clusters	O
will	O
"	O
explain	O
"	O
more	O
of	O
the	O
variation	O
(	O
since	O
it	O
can	O
use	O
smaller	O
,	O
tighter	O
clusters	O
)	O
,	O
but	O
this	O
is	O
over-fitting	B-Error_Name
,	O
since	O
it	O
is	O
subdividing	O
the	O
labeled	O
groups	O
into	O
multiple	O
clusters	O
.	O
</s>
<s>
Assuming	O
this	O
happens	O
,	O
there	O
will	O
be	O
a	O
sharp	O
elbow	O
in	O
the	O
graph	O
of	O
explained	O
variation	O
versus	O
clusters	O
:	O
increasing	O
rapidly	O
up	O
to	O
k	O
(	O
under-fitting	B-Error_Name
region	O
)	O
,	O
and	O
then	O
increasing	O
slowly	O
after	O
k	O
(	O
over-fitting	B-Error_Name
region	O
)	O
.	O
</s>
<s>
The	O
elbow	B-General_Concept
method	I-General_Concept
is	O
considered	O
both	O
subjective	O
and	O
unreliable	O
.	O
</s>
<s>
This	O
can	O
even	O
hold	O
in	O
cases	O
where	O
all	O
other	O
methods	O
for	O
determining	B-Algorithm
the	I-Algorithm
number	I-Algorithm
of	I-Algorithm
clusters	I-Algorithm
in	I-Algorithm
a	I-Algorithm
data	I-Algorithm
set	I-Algorithm
(	O
as	O
mentioned	O
in	O
that	O
article	O
)	O
agree	O
on	O
the	O
number	O
of	O
clusters	O
.	O
</s>
<s>
There	O
are	O
various	O
measures	O
of	O
"	O
explained	O
variation	O
"	O
used	O
in	O
the	O
elbow	B-General_Concept
method	I-General_Concept
.	O
</s>
<s>
Alternatively	O
,	O
one	O
uses	O
the	O
ratio	O
of	O
between-group	O
variance	O
to	O
within-group	O
variance	O
,	O
which	O
is	O
the	O
one-way	O
ANOVA	B-General_Concept
F-test	B-General_Concept
statistic	I-General_Concept
.	O
</s>
