<s>
Random	B-Algorithm
forests	I-Algorithm
or	O
random	O
decision	O
forests	O
is	O
an	O
ensemble	B-Algorithm
learning	I-Algorithm
method	O
for	O
classification	B-General_Concept
,	O
regression	O
and	O
other	O
tasks	O
that	O
operates	O
by	O
constructing	O
a	O
multitude	O
of	O
decision	B-Algorithm
trees	I-Algorithm
at	O
training	O
time	O
.	O
</s>
<s>
For	O
classification	B-General_Concept
tasks	O
,	O
the	O
output	O
of	O
the	O
random	B-Algorithm
forest	I-Algorithm
is	O
the	O
class	O
selected	O
by	O
most	O
trees	O
.	O
</s>
<s>
Random	O
decision	O
forests	O
correct	O
for	O
decision	B-Algorithm
trees	I-Algorithm
 '	O
habit	O
of	O
overfitting	B-Error_Name
to	O
their	O
training	O
set	O
.	O
</s>
<s>
Random	B-Algorithm
forests	I-Algorithm
generally	O
outperform	O
decision	B-Algorithm
trees	I-Algorithm
,	O
but	O
their	O
accuracy	O
is	O
lower	O
than	O
gradient	O
boosted	O
trees	O
.	O
</s>
<s>
The	O
first	O
algorithm	O
for	O
random	O
decision	O
forests	O
was	O
created	O
in	O
1995	O
by	O
Tin	O
Kam	O
Ho	O
using	O
the	O
random	B-Algorithm
subspace	I-Algorithm
method	I-Algorithm
,	O
which	O
,	O
in	O
Ho	O
's	O
formulation	O
,	O
is	O
a	O
way	O
to	O
implement	O
the	O
"	O
stochastic	B-General_Concept
discrimination	I-General_Concept
"	O
approach	O
to	O
classification	B-General_Concept
proposed	O
by	O
Eugene	O
Kleinberg	O
.	O
</s>
<s>
An	O
extension	O
of	O
the	O
algorithm	O
was	O
developed	O
by	O
Leo	O
Breiman	O
and	O
Adele	O
Cutler	O
,	O
who	O
registered	O
"	O
Random	B-Algorithm
Forests	I-Algorithm
"	O
as	O
a	O
trademark	O
in	O
2006	O
(	O
,	O
owned	O
by	O
Minitab	B-Application
,	I-Application
Inc	I-Application
.	I-Application
)	O
.	O
</s>
<s>
The	O
extension	O
combines	O
Breiman	O
's	O
"	O
bagging	B-Algorithm
"	O
idea	O
and	O
random	O
selection	O
of	O
features	B-Algorithm
,	O
introduced	O
first	O
by	O
Ho	O
and	O
later	O
independently	O
by	O
Amit	O
and	O
Geman	O
in	O
order	O
to	O
construct	O
a	O
collection	O
of	O
decision	B-Algorithm
trees	I-Algorithm
with	O
controlled	O
variance	B-General_Concept
.	O
</s>
<s>
Random	B-Algorithm
forests	I-Algorithm
are	O
frequently	O
used	O
as	O
black	B-Device
box	I-Device
models	O
in	O
businesses	O
,	O
as	O
they	O
generate	O
reasonable	O
predictions	O
across	O
a	O
wide	O
range	O
of	O
data	O
while	O
requiring	O
little	O
configuration	O
.	O
</s>
<s>
Ho	O
established	O
that	O
forests	O
of	O
trees	O
splitting	O
with	O
oblique	O
hyperplanes	O
can	O
gain	O
accuracy	O
as	O
they	O
grow	O
without	O
suffering	O
from	O
overtraining	O
,	O
as	O
long	O
as	O
the	O
forests	O
are	O
randomly	O
restricted	O
to	O
be	O
sensitive	O
to	O
only	O
selected	O
feature	B-Algorithm
dimensions	O
.	O
</s>
<s>
A	O
subsequent	O
work	O
along	O
the	O
same	O
lines	O
concluded	O
that	O
other	O
splitting	O
methods	O
behave	O
similarly	O
,	O
as	O
long	O
as	O
they	O
are	O
randomly	O
forced	O
to	O
be	O
insensitive	O
to	O
some	O
feature	B-Algorithm
dimensions	O
.	O
</s>
<s>
Note	O
that	O
this	O
observation	O
of	O
a	O
more	O
complex	O
classifier	B-General_Concept
(	O
a	O
larger	O
forest	O
)	O
getting	O
more	O
accurate	O
nearly	O
monotonically	O
is	O
in	O
sharp	O
contrast	O
to	O
the	O
common	O
belief	O
that	O
the	O
complexity	O
of	O
a	O
classifier	B-General_Concept
can	O
only	O
grow	O
to	O
a	O
certain	O
level	O
of	O
accuracy	O
before	O
being	O
hurt	O
by	O
overfitting	B-Error_Name
.	O
</s>
<s>
The	O
explanation	O
of	O
the	O
forest	O
method	O
's	O
resistance	O
to	O
overtraining	O
can	O
be	O
found	O
in	O
Kleinberg	O
's	O
theory	O
of	O
stochastic	B-General_Concept
discrimination	I-General_Concept
.	O
</s>
<s>
tree	B-Algorithm
.	O
</s>
<s>
The	O
idea	O
of	O
random	O
subspace	O
selection	O
from	O
Ho	O
was	O
also	O
influential	O
in	O
the	O
design	O
of	O
random	B-Algorithm
forests	I-Algorithm
.	O
</s>
<s>
into	O
a	O
randomly	O
chosen	O
subspace	O
before	O
fitting	O
each	O
tree	B-Algorithm
or	O
each	O
node	O
.	O
</s>
<s>
optimization	O
and	O
bagging	B-Algorithm
.	O
</s>
<s>
modern	O
practice	O
of	O
random	B-Algorithm
forests	I-Algorithm
,	O
in	O
particular	O
:	O
</s>
<s>
Using	O
out-of-bag	B-Algorithm
error	I-Algorithm
as	O
an	O
estimate	O
of	O
the	O
generalization	B-Algorithm
error	I-Algorithm
.	O
</s>
<s>
Decision	B-Algorithm
trees	I-Algorithm
are	O
a	O
popular	O
method	O
for	O
various	O
machine	O
learning	O
tasks	O
.	O
</s>
<s>
Tree	B-Algorithm
learning	O
"come[s]	O
closest	O
to	O
meeting	O
the	O
requirements	O
for	O
serving	O
as	O
an	O
off-the-shelf	O
procedure	O
for	O
data	O
mining	O
"	O
,	O
say	O
Hastie	O
et	O
al.	O
,	O
"	O
because	O
it	O
is	O
invariant	O
under	O
scaling	O
and	O
various	O
other	O
transformations	O
of	O
feature	B-Algorithm
values	O
,	O
is	O
robust	O
to	O
inclusion	O
of	O
irrelevant	O
features	B-Algorithm
,	O
and	O
produces	O
inspectable	O
models	O
.	O
</s>
<s>
In	O
particular	O
,	O
trees	O
that	O
are	O
grown	O
very	O
deep	O
tend	O
to	O
learn	O
highly	O
irregular	O
patterns	O
:	O
they	O
overfit	B-Error_Name
their	O
training	O
sets	O
,	O
i.e.	O
</s>
<s>
have	O
low	B-General_Concept
bias	I-General_Concept
,	I-General_Concept
but	I-General_Concept
very	I-General_Concept
high	I-General_Concept
variance	I-General_Concept
.	O
</s>
<s>
Random	B-Algorithm
forests	I-Algorithm
are	O
a	O
way	O
of	O
averaging	O
multiple	O
deep	O
decision	B-Algorithm
trees	I-Algorithm
,	O
trained	O
on	O
different	O
parts	O
of	O
the	O
same	O
training	O
set	O
,	O
with	O
the	O
goal	O
of	O
reducing	O
the	O
variance	B-General_Concept
.	O
</s>
<s>
Forests	O
are	O
like	O
the	O
pulling	O
together	O
of	O
decision	B-Algorithm
tree	I-Algorithm
algorithm	O
efforts	O
.	O
</s>
<s>
Taking	O
the	O
teamwork	O
of	O
many	O
trees	O
thus	O
improving	O
the	O
performance	O
of	O
a	O
single	O
random	O
tree	B-Algorithm
.	O
</s>
<s>
The	O
training	O
algorithm	O
for	O
random	B-Algorithm
forests	I-Algorithm
applies	O
the	O
general	O
technique	O
of	O
bootstrap	B-Algorithm
aggregating	I-Algorithm
,	O
or	O
bagging	B-Algorithm
,	O
to	O
tree	B-Algorithm
learners	O
.	O
</s>
<s>
Given	O
a	O
training	O
set	O
=	O
,	O
...	O
,	O
with	O
responses	O
=	O
,	O
...	O
,	O
,	O
bagging	B-Algorithm
repeatedly	O
(	O
B	O
times	O
)	O
selects	O
a	O
random	O
sample	O
with	O
replacement	O
of	O
the	O
training	O
set	O
and	O
fits	O
trees	O
to	O
these	O
samples	O
:	O
</s>
<s>
Train	O
a	O
classification	B-General_Concept
or	O
regression	B-Algorithm
tree	I-Algorithm
on	O
,	O
.	O
</s>
<s>
After	O
training	O
,	O
predictions	O
for	O
unseen	O
samples	O
can	O
be	O
made	O
by	O
averaging	O
the	O
predictions	O
from	O
all	O
the	O
individual	O
regression	B-Algorithm
trees	I-Algorithm
on	O
:	O
</s>
<s>
or	O
by	O
taking	O
the	O
in	O
the	O
case	O
of	O
classification	B-Algorithm
trees	I-Algorithm
.	O
</s>
<s>
This	O
bootstrapping	B-Algorithm
procedure	O
leads	O
to	O
better	O
model	O
performance	O
because	O
it	O
decreases	O
the	O
variance	B-General_Concept
of	O
the	O
model	O
,	O
without	O
increasing	O
the	O
bias	O
.	O
</s>
<s>
This	O
means	O
that	O
while	O
the	O
predictions	O
of	O
a	O
single	O
tree	B-Algorithm
are	O
highly	O
sensitive	O
to	O
noise	O
in	O
its	O
training	O
set	O
,	O
the	O
average	O
of	O
many	O
trees	O
is	O
not	O
,	O
as	O
long	O
as	O
the	O
trees	O
are	O
not	O
correlated	O
.	O
</s>
<s>
Simply	O
training	O
many	O
trees	O
on	O
a	O
single	O
training	O
set	O
would	O
give	O
strongly	O
correlated	O
trees	O
(	O
or	O
even	O
the	O
same	O
tree	B-Algorithm
many	O
times	O
,	O
if	O
the	O
training	O
algorithm	O
is	O
deterministic	O
)	O
;	O
bootstrap	O
sampling	O
is	O
a	O
way	O
of	O
de-correlating	O
the	O
trees	O
by	O
showing	O
them	O
different	O
training	O
sets	O
.	O
</s>
<s>
Additionally	O
,	O
an	O
estimate	O
of	O
the	O
uncertainty	O
of	O
the	O
prediction	O
can	O
be	O
made	O
as	O
the	O
standard	O
deviation	O
of	O
the	O
predictions	O
from	O
all	O
the	O
individual	O
regression	B-Algorithm
trees	I-Algorithm
on	O
:	O
</s>
<s>
An	O
optimal	O
number	O
of	O
trees	O
can	O
be	O
found	O
using	O
cross-validation	B-Application
,	O
or	O
by	O
observing	O
the	O
out-of-bag	B-Algorithm
error	I-Algorithm
:	O
the	O
mean	O
prediction	O
error	O
on	O
each	O
training	O
sample	O
,	O
using	O
only	O
the	O
trees	O
that	O
did	O
not	O
have	O
in	O
their	O
bootstrap	O
sample	O
.	O
</s>
<s>
The	O
above	O
procedure	O
describes	O
the	O
original	O
bagging	B-Algorithm
algorithm	O
for	O
trees	O
.	O
</s>
<s>
Random	B-Algorithm
forests	I-Algorithm
also	O
include	O
another	O
type	O
of	O
bagging	B-Algorithm
scheme	O
:	O
they	O
use	O
a	O
modified	O
tree	B-Algorithm
learning	O
algorithm	O
that	O
selects	O
,	O
at	O
each	O
candidate	O
split	O
in	O
the	O
learning	O
process	O
,	O
a	O
random	B-Algorithm
subset	I-Algorithm
of	I-Algorithm
the	I-Algorithm
features	I-Algorithm
.	O
</s>
<s>
This	O
process	O
is	O
sometimes	O
called	O
"	O
feature	B-Algorithm
bagging	B-Algorithm
"	O
.	O
</s>
<s>
The	O
reason	O
for	O
doing	O
this	O
is	O
the	O
correlation	O
of	O
the	O
trees	O
in	O
an	O
ordinary	O
bootstrap	O
sample	O
:	O
if	O
one	O
or	O
a	O
few	O
features	B-Algorithm
are	O
very	O
strong	O
predictors	O
for	O
the	O
response	O
variable	O
(	O
target	O
output	O
)	O
,	O
these	O
features	B-Algorithm
will	O
be	O
selected	O
in	O
many	O
of	O
the	O
trees	O
,	O
causing	O
them	O
to	O
become	O
correlated	O
.	O
</s>
<s>
An	O
analysis	O
of	O
how	O
bagging	B-Algorithm
and	O
random	O
subspace	O
projection	O
contribute	O
to	O
accuracy	O
gains	O
under	O
different	O
conditions	O
is	O
given	O
by	O
Ho	O
.	O
</s>
<s>
Typically	O
,	O
for	O
a	O
classification	B-General_Concept
problem	O
with	O
features	B-Algorithm
,	O
(	O
rounded	O
down	O
)	O
features	B-Algorithm
are	O
used	O
in	O
each	O
split	O
.	O
</s>
<s>
While	O
similar	O
to	O
ordinary	O
random	B-Algorithm
forests	I-Algorithm
in	O
that	O
they	O
are	O
an	O
ensemble	O
of	O
individual	O
trees	O
,	O
there	O
are	O
two	O
main	O
differences	O
:	O
first	O
,	O
each	O
tree	B-Algorithm
is	O
trained	O
using	O
the	O
whole	O
learning	O
sample	O
(	O
rather	O
than	O
a	O
bootstrap	O
sample	O
)	O
,	O
and	O
second	O
,	O
the	O
top-down	O
splitting	O
in	O
the	O
tree	B-Algorithm
learner	O
is	O
randomized	O
.	O
</s>
<s>
Instead	O
of	O
computing	O
the	O
locally	O
optimal	O
cut-point	O
for	O
each	O
feature	B-Algorithm
under	O
consideration	O
(	O
based	O
on	O
,	O
e.g.	O
,	O
information	B-Algorithm
gain	I-Algorithm
or	O
the	O
Gini	O
impurity	O
)	O
,	O
a	O
random	O
cut-point	O
is	O
selected	O
.	O
</s>
<s>
This	O
value	O
is	O
selected	O
from	O
a	O
uniform	O
distribution	O
within	O
the	O
feature	B-Algorithm
's	O
empirical	O
range	O
(	O
in	O
the	O
tree	B-Algorithm
's	O
training	O
set	O
)	O
.	O
</s>
<s>
Similar	O
to	O
ordinary	O
random	B-Algorithm
forests	I-Algorithm
,	O
the	O
number	O
of	O
randomly	O
selected	O
features	B-Algorithm
to	O
be	O
considered	O
at	O
each	O
node	O
can	O
be	O
specified	O
.	O
</s>
<s>
Default	O
values	O
for	O
this	O
parameter	O
are	O
for	O
classification	B-General_Concept
and	O
for	O
regression	O
,	O
where	O
is	O
the	O
number	O
of	O
features	B-Algorithm
in	O
the	O
model	O
.	O
</s>
<s>
Random	B-Algorithm
forests	I-Algorithm
can	O
be	O
used	O
to	O
rank	O
the	O
importance	O
of	O
variables	O
in	O
a	O
regression	O
or	O
classification	B-General_Concept
problem	O
in	O
a	O
natural	O
way	O
.	O
</s>
<s>
The	O
following	O
technique	O
was	O
described	O
in	O
Breiman	O
's	O
original	O
paper	O
and	O
is	O
implemented	O
in	O
the	O
R	B-Language
package	O
randomForest	O
.	O
</s>
<s>
The	O
first	O
step	O
in	O
measuring	O
the	O
variable	O
importance	O
in	O
a	O
data	O
set	O
is	O
to	O
fit	O
a	O
random	B-Algorithm
forest	I-Algorithm
to	O
the	O
data	O
.	O
</s>
<s>
During	O
the	O
fitting	O
process	O
the	O
out-of-bag	B-Algorithm
error	I-Algorithm
for	O
each	O
data	O
point	O
is	O
recorded	O
and	O
averaged	O
over	O
the	O
forest	O
(	O
errors	O
on	O
an	O
independent	O
test	O
set	O
can	O
be	O
substituted	O
if	O
bagging	B-Algorithm
is	O
not	O
used	O
during	O
training	O
)	O
.	O
</s>
<s>
To	O
measure	O
the	O
importance	O
of	O
the	O
-th	O
feature	B-Algorithm
after	O
training	O
,	O
the	O
values	O
of	O
the	O
-th	O
feature	B-Algorithm
are	O
permuted	O
among	O
the	O
training	O
data	O
and	O
the	O
out-of-bag	B-Algorithm
error	I-Algorithm
is	O
again	O
computed	O
on	O
this	O
perturbed	O
data	O
set	O
.	O
</s>
<s>
The	O
importance	O
score	O
for	O
the	O
-th	O
feature	B-Algorithm
is	O
computed	O
by	O
averaging	O
the	O
difference	O
in	O
out-of-bag	B-Algorithm
error	I-Algorithm
before	O
and	O
after	O
the	O
permutation	O
over	O
all	O
trees	O
.	O
</s>
<s>
Features	B-Algorithm
which	O
produce	O
large	O
values	O
for	O
this	O
score	O
are	O
ranked	O
as	O
more	O
important	O
than	O
features	B-Algorithm
which	O
produce	O
small	O
values	O
.	O
</s>
<s>
For	O
data	O
including	O
categorical	O
variables	O
with	O
different	O
number	O
of	O
levels	O
,	O
random	B-Algorithm
forests	I-Algorithm
are	O
biased	O
in	O
favor	O
of	O
those	O
attributes	O
with	O
more	O
levels	O
.	O
</s>
<s>
If	O
the	O
data	O
contain	O
groups	O
of	O
correlated	O
features	B-Algorithm
of	O
similar	O
relevance	O
for	O
the	O
output	O
,	O
then	O
smaller	O
groups	O
are	O
favored	O
over	O
larger	O
groups	O
.	O
</s>
<s>
A	O
relationship	O
between	O
random	B-Algorithm
forests	I-Algorithm
and	O
the	O
-nearest	O
neighbor	O
algorithm	O
(	O
-NN	O
)	O
was	O
pointed	O
out	O
by	O
Lin	O
and	O
Jeon	O
in	O
2002	O
.	O
</s>
<s>
Here	O
,	O
is	O
the	O
non-negative	O
weight	O
of	O
the	O
'	O
th	O
training	O
point	O
relative	O
to	O
the	O
new	O
point	O
in	O
the	O
same	O
tree	B-Algorithm
.	O
</s>
<s>
In	O
a	O
tree	B-Algorithm
,	O
if	O
is	O
one	O
of	O
the	O
points	O
in	O
the	O
same	O
leaf	O
as	O
,	O
and	O
zero	O
otherwise	O
.	O
</s>
<s>
The	O
neighbors	O
of	O
in	O
this	O
interpretation	O
are	O
the	O
points	O
sharing	O
the	O
same	O
leaf	O
in	O
any	O
tree	B-Algorithm
.	O
</s>
<s>
Lin	O
and	O
Jeon	O
show	O
that	O
the	O
shape	O
of	O
the	O
neighborhood	O
used	O
by	O
a	O
random	B-Algorithm
forest	I-Algorithm
adapts	O
to	O
the	O
local	O
importance	O
of	O
each	O
feature	B-Algorithm
.	O
</s>
<s>
As	O
part	O
of	O
their	O
construction	O
,	O
random	B-Algorithm
forest	I-Algorithm
predictors	O
naturally	O
lead	O
to	O
a	O
dissimilarity	O
measure	O
among	O
the	O
observations	O
.	O
</s>
<s>
One	O
can	O
also	O
define	O
a	O
random	B-Algorithm
forest	I-Algorithm
dissimilarity	O
measure	O
between	O
unlabeled	O
data	O
:	O
the	O
idea	O
is	O
to	O
construct	O
a	O
random	B-Algorithm
forest	I-Algorithm
predictor	O
that	O
distinguishes	O
the	O
"	O
observed	O
"	O
data	O
from	O
suitably	O
generated	O
synthetic	O
data	O
.	O
</s>
<s>
A	O
random	B-Algorithm
forest	I-Algorithm
dissimilarity	O
can	O
be	O
attractive	O
because	O
it	O
handles	O
mixed	O
variable	O
types	O
very	O
well	O
,	O
is	O
invariant	O
to	O
monotonic	O
transformations	O
of	O
the	O
input	O
variables	O
,	O
and	O
is	O
robust	O
to	O
outlying	O
observations	O
.	O
</s>
<s>
The	O
random	B-Algorithm
forest	I-Algorithm
dissimilarity	O
easily	O
deals	O
with	O
a	O
large	O
number	O
of	O
semi-continuous	O
variables	O
due	O
to	O
its	O
intrinsic	O
variable	O
selection	O
;	O
for	O
example	O
,	O
the	O
"	O
Addcl	O
1	O
"	O
random	B-Algorithm
forest	I-Algorithm
dissimilarity	O
weighs	O
the	O
contribution	O
of	O
each	O
variable	O
according	O
to	O
how	O
dependent	O
it	O
is	O
on	O
other	O
variables	O
.	O
</s>
<s>
The	O
random	B-Algorithm
forest	I-Algorithm
dissimilarity	O
has	O
been	O
used	O
in	O
a	O
variety	O
of	O
applications	O
,	O
e.g.	O
</s>
<s>
Instead	O
of	O
decision	B-Algorithm
trees	I-Algorithm
,	O
linear	O
models	O
have	O
been	O
proposed	O
and	O
evaluated	O
as	O
base	O
estimators	O
in	O
random	B-Algorithm
forests	I-Algorithm
,	O
in	O
particular	O
multinomial	O
logistic	O
regression	O
and	O
naive	B-General_Concept
Bayes	I-General_Concept
classifiers	I-General_Concept
.	O
</s>
<s>
In	O
machine	O
learning	O
,	O
kernel	O
random	B-Algorithm
forests	I-Algorithm
(	O
KeRF	O
)	O
establish	O
the	O
connection	O
between	O
random	B-Algorithm
forests	I-Algorithm
and	O
kernel	B-Algorithm
methods	I-Algorithm
.	O
</s>
<s>
By	O
slightly	O
modifying	O
their	O
definition	O
,	O
random	B-Algorithm
forests	I-Algorithm
can	O
be	O
rewritten	O
as	O
kernel	B-Algorithm
methods	I-Algorithm
,	O
which	O
are	O
more	O
interpretable	O
and	O
easier	O
to	O
analyze	O
.	O
</s>
<s>
Leo	O
Breiman	O
was	O
the	O
first	O
person	O
to	O
notice	O
the	O
link	O
between	O
random	B-Algorithm
forest	I-Algorithm
and	O
kernel	B-Algorithm
methods	I-Algorithm
.	O
</s>
<s>
He	O
pointed	O
out	O
that	O
random	B-Algorithm
forests	I-Algorithm
which	O
are	O
grown	O
using	O
i.i.d.	O
</s>
<s>
random	O
vectors	O
in	O
the	O
tree	B-Algorithm
construction	O
are	O
equivalent	O
to	O
a	O
kernel	O
acting	O
on	O
the	O
true	O
margin	O
.	O
</s>
<s>
Lin	O
and	O
Jeon	O
established	O
the	O
connection	O
between	O
random	B-Algorithm
forests	I-Algorithm
and	O
adaptive	O
nearest	B-General_Concept
neighbor	I-General_Concept
,	O
implying	O
that	O
random	B-Algorithm
forests	I-Algorithm
can	O
be	O
seen	O
as	O
adaptive	O
kernel	O
estimates	O
.	O
</s>
<s>
Davies	O
and	O
Ghahramani	O
proposed	O
Random	B-Algorithm
Forest	I-Algorithm
Kernel	O
and	O
show	O
that	O
it	O
can	O
empirically	O
outperform	O
state-of-art	O
kernel	B-Algorithm
methods	I-Algorithm
.	O
</s>
<s>
Scornet	O
first	O
defined	O
KeRF	O
estimates	O
and	O
gave	O
the	O
explicit	O
link	O
between	O
KeRF	O
estimates	O
and	O
random	B-Algorithm
forest	I-Algorithm
.	O
</s>
<s>
He	O
also	O
gave	O
explicit	O
expressions	O
for	O
kernels	O
based	O
on	O
centered	O
random	B-Algorithm
forest	I-Algorithm
and	O
uniform	O
random	B-Algorithm
forest	I-Algorithm
,	O
two	O
simplified	O
models	O
of	O
random	B-Algorithm
forest	I-Algorithm
.	O
</s>
<s>
Centered	O
forest	O
is	O
a	O
simplified	O
model	O
for	O
Breiman	O
's	O
original	O
random	B-Algorithm
forest	I-Algorithm
,	O
which	O
uniformly	O
selects	O
an	O
attribute	O
among	O
all	O
attributes	O
and	O
performs	O
splits	O
at	O
the	O
center	O
of	O
the	O
cell	O
along	O
the	O
pre-chosen	O
attribute	O
.	O
</s>
<s>
The	O
algorithm	O
stops	O
when	O
a	O
fully	O
binary	O
tree	B-Algorithm
of	O
level	O
is	O
built	O
,	O
where	O
is	O
a	O
parameter	O
of	O
the	O
algorithm	O
.	O
</s>
<s>
Uniform	O
forest	O
is	O
another	O
simplified	O
model	O
for	O
Breiman	O
's	O
original	O
random	B-Algorithm
forest	I-Algorithm
,	O
which	O
uniformly	O
selects	O
a	O
feature	B-Algorithm
among	O
all	O
features	B-Algorithm
and	O
performs	O
splits	O
at	O
a	O
point	O
uniformly	O
drawn	O
on	O
the	O
side	O
of	O
the	O
cell	O
,	O
along	O
the	O
preselected	O
feature	B-Algorithm
.	O
</s>
<s>
A	O
random	O
regression	O
forest	O
is	O
an	O
ensemble	O
of	O
randomized	O
regression	B-Algorithm
trees	I-Algorithm
.	O
</s>
<s>
Denote	O
the	O
predicted	O
value	O
at	O
point	O
by	O
the	O
-th	O
tree	B-Algorithm
,	O
where	O
are	O
independent	O
random	O
variables	O
,	O
distributed	O
as	O
a	O
generic	O
random	O
variable	O
,	O
independent	O
of	O
the	O
sample	O
.	O
</s>
<s>
This	O
random	O
variable	O
can	O
be	O
used	O
to	O
describe	O
the	O
randomness	O
induced	O
by	O
node	O
splitting	O
and	O
the	O
sampling	O
procedure	O
for	O
tree	B-Algorithm
construction	O
.	O
</s>
<s>
For	O
regression	B-Algorithm
trees	I-Algorithm
,	O
we	O
have	O
,	O
where	O
is	O
the	O
cell	O
containing	O
,	O
designed	O
with	O
randomness	O
and	O
dataset	O
,	O
and	O
.	O
</s>
<s>
Thus	O
random	B-Algorithm
forest	I-Algorithm
estimates	O
satisfy	O
,	O
for	O
all	O
,	O
.	O
</s>
<s>
Random	O
regression	O
forest	O
has	O
two	O
levels	O
of	O
averaging	O
,	O
first	O
over	O
the	O
samples	O
in	O
the	O
target	O
cell	O
of	O
a	O
tree	B-Algorithm
,	O
then	O
over	O
all	O
trees	O
.	O
</s>
<s>
Predictions	O
given	O
by	O
KeRF	O
and	O
random	B-Algorithm
forests	I-Algorithm
are	O
close	O
if	O
the	O
number	O
of	O
points	O
in	O
each	O
cell	O
is	O
controlled	O
:	O
</s>
<s>
When	O
the	O
number	O
of	O
trees	O
goes	O
to	O
infinity	O
,	O
then	O
we	O
have	O
infinite	O
random	B-Algorithm
forest	I-Algorithm
and	O
infinite	O
KeRF	O
.	O
</s>
<s>
Assume	O
that	O
,	O
where	O
is	O
a	O
centered	O
Gaussian	O
noise	O
,	O
independent	O
of	O
,	O
with	O
finite	O
variance	B-General_Concept
.	O
</s>
<s>
While	O
random	B-Algorithm
forests	I-Algorithm
often	O
achieve	O
higher	O
accuracy	O
than	O
a	O
single	O
decision	B-Algorithm
tree	I-Algorithm
,	O
they	O
sacrifice	O
the	O
intrinsic	O
interpretability	O
present	O
in	O
decision	B-Algorithm
trees	I-Algorithm
.	O
</s>
<s>
Decision	B-Algorithm
trees	I-Algorithm
are	O
among	O
a	O
fairly	O
small	O
family	O
of	O
machine	O
learning	O
models	O
that	O
are	O
easily	O
interpretable	O
along	O
with	O
linear	O
models	O
,	O
rule-based	B-Algorithm
models	O
,	O
and	O
attention-based	O
models	O
.	O
</s>
<s>
This	O
interpretability	O
is	O
one	O
of	O
the	O
most	O
desirable	O
qualities	O
of	O
decision	B-Algorithm
trees	I-Algorithm
.	O
</s>
<s>
For	O
example	O
,	O
following	O
the	O
path	O
that	O
a	O
decision	B-Algorithm
tree	I-Algorithm
takes	O
to	O
make	O
its	O
decision	O
is	O
quite	O
trivial	O
,	O
but	O
following	O
the	O
paths	O
of	O
tens	O
or	O
hundreds	O
of	O
trees	O
is	O
much	O
harder	O
.	O
</s>
<s>
To	O
achieve	O
both	O
performance	O
and	O
interpretability	O
,	O
some	O
model	O
compression	O
techniques	O
allow	O
transforming	O
a	O
random	B-Algorithm
forest	I-Algorithm
into	O
a	O
minimal	O
"	O
born-again	O
"	O
decision	B-Algorithm
tree	I-Algorithm
that	O
faithfully	O
reproduces	O
the	O
same	O
decision	O
function	O
.	O
</s>
<s>
If	O
it	O
is	O
established	O
that	O
the	O
predictive	O
attributes	O
are	O
linearly	O
correlated	O
with	O
the	O
target	O
variable	O
,	O
using	O
random	B-Algorithm
forest	I-Algorithm
may	O
not	O
enhance	O
the	O
accuracy	O
of	O
the	O
base	O
learner	O
.	O
</s>
<s>
Furthermore	O
,	O
in	O
problems	O
with	O
multiple	O
categorical	O
variables	O
,	O
random	B-Algorithm
forest	I-Algorithm
may	O
not	O
be	O
able	O
to	O
increase	O
the	O
accuracy	O
of	O
the	O
base	O
learner	O
.	O
</s>
