<s>
BIRCH	B-Algorithm
(	O
balanced	O
iterative	O
reducing	O
and	O
clustering	O
using	O
hierarchies	O
)	O
is	O
an	O
unsupervised	O
data	B-Application
mining	I-Application
algorithm	O
used	O
to	O
perform	O
hierarchical	B-Algorithm
clustering	I-Algorithm
over	O
particularly	O
large	O
data-sets	B-General_Concept
.	O
</s>
<s>
With	O
modifications	O
it	O
can	O
also	O
be	O
used	O
to	O
accelerate	O
k-means	B-Algorithm
clustering	I-Algorithm
and	O
Gaussian	O
mixture	O
modeling	O
with	O
the	O
expectation	B-Algorithm
–	I-Algorithm
maximization	I-Algorithm
algorithm	I-Algorithm
.	O
</s>
<s>
An	O
advantage	O
of	O
BIRCH	B-Algorithm
is	O
its	O
ability	O
to	O
incrementally	O
and	O
dynamically	O
cluster	O
incoming	O
,	O
multi-dimensional	O
metric	O
data	B-Application
points	I-Application
in	O
an	O
attempt	O
to	O
produce	O
the	O
best	O
quality	O
clustering	O
for	O
a	O
given	O
set	O
of	O
resources	O
(	O
memory	O
and	O
time	O
constraints	O
)	O
.	O
</s>
<s>
In	O
most	O
cases	O
,	O
BIRCH	B-Algorithm
only	O
requires	O
a	O
single	O
scan	O
of	O
the	O
database	O
.	O
</s>
<s>
Its	O
inventors	O
claim	O
BIRCH	B-Algorithm
to	O
be	O
the	O
"	O
first	O
clustering	B-Algorithm
algorithm	I-Algorithm
proposed	O
in	O
the	O
database	O
area	O
to	O
handle	O
'	O
noise	O
 '	O
(	O
data	B-Application
points	I-Application
that	O
are	O
not	O
part	O
of	O
the	O
underlying	O
pattern	O
)	O
effectively	O
"	O
,	O
beating	O
DBSCAN	B-Algorithm
by	O
two	O
months	O
.	O
</s>
<s>
The	O
BIRCH	B-Algorithm
algorithm	O
received	O
the	O
SIGMOD	O
10	O
year	O
test	O
of	O
time	O
award	O
in	O
2006	O
.	O
</s>
<s>
Previous	O
clustering	B-Algorithm
algorithms	I-Algorithm
performed	O
less	O
effectively	O
over	O
very	O
large	O
databases	O
and	O
did	O
not	O
adequately	O
consider	O
the	O
case	O
wherein	O
a	O
data-set	B-General_Concept
was	O
too	O
large	O
to	O
fit	O
in	O
main	O
memory	O
.	O
</s>
<s>
Furthermore	O
,	O
most	O
of	O
BIRCH	B-Algorithm
's	O
predecessors	O
inspect	O
all	O
data	B-Application
points	I-Application
(	O
or	O
all	O
currently	O
existing	O
clusters	O
)	O
equally	O
for	O
each	O
'	O
clustering	O
decision	O
 '	O
and	O
do	O
not	O
perform	O
heuristic	O
weighting	O
based	O
on	O
the	O
distance	O
between	O
these	O
data	B-Application
points	I-Application
.	O
</s>
<s>
It	O
is	O
local	O
in	O
that	O
each	O
clustering	O
decision	O
is	O
made	O
without	O
scanning	O
all	O
data	B-Application
points	I-Application
and	O
currently	O
existing	O
clusters	O
.	O
</s>
<s>
It	O
exploits	O
the	O
observation	O
that	O
the	O
data	O
space	O
is	O
not	O
usually	O
uniformly	O
occupied	O
and	O
not	O
every	O
data	B-Application
point	I-Application
is	O
equally	O
important	O
.	O
</s>
<s>
It	O
is	O
also	O
an	O
incremental	O
method	O
that	O
does	O
not	O
require	O
the	O
whole	O
data	B-General_Concept
set	I-General_Concept
in	O
advance	O
.	O
</s>
<s>
The	O
BIRCH	B-Algorithm
algorithm	O
takes	O
as	O
input	O
a	O
set	O
of	O
data	B-Application
points	I-Application
,	O
represented	O
as	O
real-valued	B-Algorithm
vectors	I-Algorithm
,	O
and	O
a	O
desired	O
number	O
of	O
clusters	O
.	O
</s>
<s>
The	O
first	O
phase	O
builds	O
a	O
clustering	O
feature	O
(	O
)	O
tree	O
out	O
of	O
the	O
data	B-Application
points	I-Application
,	O
a	O
height-balanced	O
tree	B-Application
data	I-Application
structure	I-Application
,	O
defined	O
as	O
follows	O
:	O
</s>
<s>
is	O
the	O
square	O
sum	O
of	O
data	B-Application
points	I-Application
.	O
</s>
<s>
Clustering	O
features	O
are	O
organized	O
in	O
a	O
CF	O
tree	O
,	O
a	O
height-balanced	O
tree	O
with	O
two	O
parameters	O
:	O
branching	B-Data_Structure
factor	I-Data_Structure
and	O
threshold	O
.	O
</s>
<s>
Each	O
non-leaf	O
node	O
contains	O
at	O
most	O
entries	O
of	O
the	O
form	O
,	O
where	O
is	O
a	O
pointer	O
to	O
its	O
th	O
child	B-Application
node	I-Application
and	O
the	O
clustering	O
feature	O
representing	O
the	O
associated	O
subcluster	O
.	O
</s>
<s>
A	O
leaf	B-Data_Structure
node	I-Data_Structure
contains	O
at	O
most	O
entries	O
each	O
of	O
the	O
form	O
.	O
</s>
<s>
It	O
also	O
has	O
two	O
pointers	O
prev	O
and	O
next	O
which	O
are	O
used	O
to	O
chain	O
all	O
leaf	B-Data_Structure
nodes	I-Data_Structure
together	O
.	O
</s>
<s>
So	O
can	O
be	O
varied	O
for	O
performance	B-Device
tuning	I-Device
.	O
</s>
<s>
It	O
is	O
a	O
very	O
compact	O
representation	O
of	O
the	O
dataset	B-General_Concept
because	O
each	O
entry	O
in	O
a	O
leaf	B-Data_Structure
node	I-Data_Structure
is	O
not	O
a	O
single	O
data	B-Application
point	I-Application
but	O
a	O
subcluster	O
.	O
</s>
<s>
This	O
step	O
is	O
marked	O
optional	O
in	O
the	O
original	O
presentation	O
of	O
BIRCH	B-Algorithm
.	O
</s>
<s>
In	O
step	O
three	O
an	O
existing	O
clustering	B-Algorithm
algorithm	I-Algorithm
is	O
used	O
to	O
cluster	O
all	O
leaf	O
entries	O
.	O
</s>
<s>
Here	O
an	O
agglomerative	B-Algorithm
hierarchical	I-Algorithm
clustering	I-Algorithm
algorithm	O
is	O
applied	O
directly	O
to	O
the	O
subclusters	O
represented	O
by	O
their	O
vectors	O
.	O
</s>
<s>
In	O
step	O
4	O
the	O
centroids	O
of	O
the	O
clusters	O
produced	O
in	O
step	O
3	O
are	O
used	O
as	O
seeds	O
and	O
redistribute	O
the	O
data	B-Application
points	I-Application
to	O
its	O
closest	O
seeds	O
to	O
obtain	O
a	O
new	O
set	O
of	O
clusters	O
.	O
</s>
<s>
Unfortunately	O
,	O
there	O
are	O
numerical	O
issues	O
associated	O
with	O
the	O
use	O
of	O
the	O
term	O
in	O
BIRCH	B-Algorithm
.	O
</s>
<s>
When	O
subtracting	O
or	O
similar	O
in	O
the	O
other	O
distances	O
such	O
as	O
,	O
catastrophic	B-Algorithm
cancellation	I-Algorithm
can	O
occur	O
and	O
yield	O
a	O
poor	O
precision	O
,	O
and	O
which	O
can	O
in	O
some	O
cases	O
even	O
cause	O
the	O
result	O
to	O
be	O
negative	O
(	O
and	O
the	O
square	O
root	O
then	O
become	O
undefined	O
)	O
.	O
</s>
<s>
When	O
storing	O
a	O
vector	O
respectively	O
a	O
matrix	O
for	O
the	O
squared	O
deviations	O
,	O
the	O
resulting	O
BIRCH	B-Algorithm
CF-tree	O
can	O
also	O
be	O
used	O
to	O
accelerate	O
Gaussian	O
Mixture	O
Modeling	O
with	O
the	O
expectation	B-Algorithm
–	I-Algorithm
maximization	I-Algorithm
algorithm	I-Algorithm
,	O
besides	O
k-means	B-Algorithm
clustering	I-Algorithm
and	O
hierarchical	B-Algorithm
agglomerative	I-Algorithm
clustering	I-Algorithm
.	O
</s>
