<s>
In	O
data	B-Application
mining	I-Application
and	O
statistics	O
,	O
hierarchical	B-Algorithm
clustering	I-Algorithm
(	O
also	O
called	O
hierarchical	B-Algorithm
cluster	I-Algorithm
analysis	I-Algorithm
or	O
HCA	O
)	O
is	O
a	O
method	O
of	O
cluster	B-Algorithm
analysis	I-Algorithm
that	O
seeks	O
to	O
build	O
a	O
hierarchy	O
of	O
clusters	O
.	O
</s>
<s>
Strategies	O
for	O
hierarchical	B-Algorithm
clustering	I-Algorithm
generally	O
fall	O
into	O
two	O
categories	O
:	O
</s>
<s>
In	O
general	O
,	O
the	O
merges	O
and	O
splits	O
are	O
determined	O
in	O
a	O
greedy	B-Algorithm
manner	O
.	O
</s>
<s>
The	O
results	O
of	O
hierarchical	B-Algorithm
clustering	I-Algorithm
are	O
usually	O
presented	O
in	O
a	O
dendrogram	B-Application
.	O
</s>
<s>
The	O
standard	O
algorithm	O
for	O
hierarchical	O
agglomerative	B-Algorithm
clustering	I-Algorithm
(	O
HAC	O
)	O
has	O
a	O
time	O
complexity	O
of	O
and	O
requires	O
memory	O
,	O
which	O
makes	O
it	O
too	O
slow	O
for	O
even	O
medium	O
data	O
sets	O
.	O
</s>
<s>
However	O
,	O
for	O
some	O
special	O
cases	O
,	O
optimal	O
efficient	O
agglomerative	O
methods	O
(	O
of	O
complexity	O
)	O
are	O
known	O
:	O
SLINK	B-Algorithm
for	O
single-linkage	B-Algorithm
and	O
CLINK	O
for	O
complete-linkage	B-Algorithm
clustering	I-Algorithm
.	O
</s>
<s>
With	O
a	O
heap	B-Application
,	O
the	O
runtime	O
of	O
the	O
general	O
case	O
can	O
be	O
reduced	O
to	O
,	O
an	O
improvement	O
on	O
the	O
aforementioned	O
bound	O
of	O
,	O
at	O
the	O
cost	O
of	O
further	O
increasing	O
the	O
memory	O
requirements	O
.	O
</s>
<s>
Except	O
for	O
the	O
special	O
case	O
of	O
single-linkage	B-Algorithm
,	O
none	O
of	O
the	O
algorithms	O
(	O
except	O
exhaustive	O
search	O
in	O
)	O
can	O
be	O
guaranteed	O
to	O
find	O
the	O
optimum	O
solution	O
.	O
</s>
<s>
Divisive	O
clustering	O
with	O
an	O
exhaustive	O
search	O
is	O
,	O
but	O
it	O
is	O
common	O
to	O
use	O
faster	O
heuristics	O
to	O
choose	O
splits	O
,	O
such	O
as	O
k-means	B-Algorithm
.	O
</s>
<s>
Hierarchical	B-Algorithm
clustering	I-Algorithm
has	O
the	O
distinct	O
advantage	O
that	O
any	O
valid	O
measure	O
of	O
distance	O
can	O
be	O
used	O
.	O
</s>
<s>
In	O
most	O
methods	O
of	O
hierarchical	B-Algorithm
clustering	I-Algorithm
,	O
this	O
is	O
achieved	O
by	O
use	O
of	O
an	O
appropriate	O
distance	O
d	O
,	O
such	O
as	O
the	O
Euclidean	O
distance	O
,	O
between	O
single	O
observations	O
of	O
the	O
data	O
set	O
,	O
and	O
a	O
linkage	O
criterion	O
,	O
which	O
specifies	O
the	O
dissimilarity	O
of	O
sets	O
as	O
a	O
function	O
of	O
the	O
pairwise	O
distances	O
of	O
observations	O
in	O
the	O
sets	O
.	O
</s>
<s>
For	O
example	O
,	O
complete-linkage	O
tends	O
to	O
produce	O
more	O
spherical	O
clusters	O
than	O
single-linkage	B-Algorithm
.	O
</s>
<s>
Names	O
Formula	O
Maximum	O
or	O
complete-linkage	B-Algorithm
clustering	I-Algorithm
Minimum	O
or	O
single-linkage	B-Algorithm
clustering	I-Algorithm
Unweighted	O
average	O
linkage	O
clustering	O
(	O
or	O
UPGMA	B-Algorithm
)	O
Weighted	O
average	O
linkage	O
clustering	O
(	O
or	O
WPGMA	B-Algorithm
)	O
Centroid	O
linkage	O
clustering	O
,	O
or	O
UPGMC	O
where	O
and	O
are	O
the	O
centroids	O
of	O
A	O
resp	O
.	O
</s>
<s>
Some	O
of	O
these	O
can	O
only	O
be	O
recomputed	O
recursively	O
(	O
WPGMA	B-Algorithm
,	O
WPGMC	O
)	O
,	O
for	O
many	O
a	O
recursive	O
computation	O
with	O
Lance-Williams-equations	O
is	O
more	O
efficient	O
,	O
while	O
for	O
other	O
(	O
Mini-Max	O
,	O
Hausdorff	O
,	O
Medoid	O
)	O
the	O
distances	O
have	O
to	O
be	O
computed	O
with	O
the	O
slower	O
full	O
formula	O
.	O
</s>
<s>
The	O
hierarchical	B-Algorithm
clustering	I-Algorithm
dendrogram	B-Application
would	O
be	O
:	O
</s>
<s>
In	O
this	O
example	O
,	O
cutting	O
after	O
the	O
second	O
row	O
(	O
from	O
the	O
top	O
)	O
of	O
the	O
dendrogram	B-Application
will	O
yield	O
clusters	O
 { a } 	O
{	O
b	O
c}	O
{	O
d	O
e}	O
 { f } 	O
.	O
</s>
<s>
A	O
simple	O
agglomerative	B-Algorithm
clustering	I-Algorithm
algorithm	O
is	O
described	O
in	O
the	O
single-linkage	B-Algorithm
clustering	I-Algorithm
page	O
;	O
it	O
can	O
easily	O
be	O
adapted	O
to	O
different	O
types	O
of	O
linkage	O
(	O
see	O
below	O
)	O
.	O
</s>
<s>
The	O
maximum	O
distance	O
between	O
elements	O
of	O
each	O
cluster	O
(	O
also	O
called	O
complete-linkage	B-Algorithm
clustering	I-Algorithm
)	O
:	O
</s>
<s>
The	O
minimum	O
distance	O
between	O
elements	O
of	O
each	O
cluster	O
(	O
also	O
called	O
single-linkage	B-Algorithm
clustering	I-Algorithm
)	O
:	O
</s>
<s>
in	O
UPGMA	B-Algorithm
)	O
:	O
</s>
<s>
In	O
case	O
of	O
tied	O
minimum	O
distances	O
,	O
a	O
pair	O
is	O
randomly	O
chosen	O
,	O
thus	O
being	O
able	O
to	O
generate	O
several	O
structurally	O
different	O
dendrograms	B-Application
.	O
</s>
<s>
Alternatively	O
,	O
all	O
tied	O
pairs	O
may	O
be	O
joined	O
at	O
the	O
same	O
time	O
,	O
generating	O
a	O
unique	O
dendrogram	B-Application
.	O
</s>
<s>
ALGLIB	B-Library
implements	O
several	O
hierarchical	B-Algorithm
clustering	I-Algorithm
algorithms	O
(	O
single-link	O
,	O
complete-link	O
,	O
Ward	O
)	O
in	O
C++	O
and	O
C#	O
with	O
O(n² )	O
memory	O
and	O
O(n³ )	O
run	O
time	O
.	O
</s>
<s>
ELKI	B-Language
includes	O
multiple	O
hierarchical	B-Algorithm
clustering	I-Algorithm
algorithms	O
,	O
various	O
linkage	O
strategies	O
and	O
also	O
includes	O
the	O
efficient	O
SLINK	B-Algorithm
,	O
CLINK	O
and	O
Anderberg	O
algorithms	O
,	O
flexible	O
cluster	O
extraction	O
from	O
dendrograms	B-Application
and	O
various	O
other	O
cluster	B-Algorithm
analysis	I-Algorithm
algorithms	O
.	O
</s>
<s>
Julia	B-Application
has	O
an	O
implementation	O
inside	O
the	O
Clustering.jl	O
package	O
.	O
</s>
<s>
Octave	B-Language
,	O
the	O
GNU	B-Operating_System
analog	O
to	O
MATLAB	B-Language
implements	O
hierarchical	B-Algorithm
clustering	I-Algorithm
in	O
function	O
"	O
linkage	O
"	O
.	O
</s>
<s>
Orange	B-Application
,	O
a	O
data	B-Application
mining	I-Application
software	O
suite	O
,	O
includes	O
hierarchical	B-Algorithm
clustering	I-Algorithm
with	O
interactive	O
dendrogram	B-Application
visualisation	O
.	O
</s>
<s>
R	B-Language
has	O
built-in	O
functions	O
and	O
packages	O
that	O
provide	O
functions	O
for	O
hierarchical	B-Algorithm
clustering	I-Algorithm
.	O
</s>
<s>
SciPy	B-Application
implements	O
hierarchical	B-Algorithm
clustering	I-Algorithm
in	O
Python	O
,	O
including	O
the	O
efficient	O
SLINK	B-Algorithm
algorithm	O
.	O
</s>
<s>
scikit-learn	B-Application
also	O
implements	O
hierarchical	B-Algorithm
clustering	I-Algorithm
in	O
Python	O
.	O
</s>
<s>
Weka	B-Language
includes	O
hierarchical	B-Algorithm
cluster	I-Algorithm
analysis	I-Algorithm
.	O
</s>
<s>
MATLAB	B-Language
includes	O
hierarchical	B-Algorithm
cluster	I-Algorithm
analysis	I-Algorithm
.	O
</s>
<s>
SAS	B-Language
includes	O
hierarchical	B-Algorithm
cluster	I-Algorithm
analysis	I-Algorithm
in	O
PROC	O
CLUSTER	O
.	O
</s>
<s>
Mathematica	B-Language
includes	O
a	O
Hierarchical	B-Algorithm
Clustering	I-Algorithm
Package	O
.	O
</s>
<s>
NCSS	O
includes	O
hierarchical	B-Algorithm
cluster	I-Algorithm
analysis	I-Algorithm
.	O
</s>
<s>
SPSS	B-Algorithm
includes	O
hierarchical	B-Algorithm
cluster	I-Algorithm
analysis	I-Algorithm
.	O
</s>
<s>
Qlucore	B-Application
Omics	O
Explorer	O
includes	O
hierarchical	B-Algorithm
cluster	I-Algorithm
analysis	I-Algorithm
.	O
</s>
<s>
Stata	B-Algorithm
includes	O
hierarchical	B-Algorithm
cluster	I-Algorithm
analysis	I-Algorithm
.	O
</s>
