Title: Random Forests
<a name="RandomForests-HowtogrowaDecisionTree"></a>
### How to grow a Decision Tree

source : \[3\](3\.html)

LearnUnprunedTree(*X*,*Y*)

Input: *X* a matrix of *R* rows and *M* columns where *X{*}{*}{~}ij{~}* =
the value of the *j*'th attribute in the *i*'th input datapoint. Each
column consists of either all real values or all categorical values.
Input: *Y* a vector of *R* elements, where *Y{*}{*}{~}i{~}* = the output
class of the *i*'th datapoint. The *Y{*}{*}{~}i{~}* values are categorical.
Output: An Unpruned decision tree


If all records in *X* have identical values in all their attributes (this
includes the case where *R<2*), return a Leaf Node predicting the majority
output, breaking ties randomly. This case also includes
If all values in *Y* are the same, return a Leaf Node predicting this value
as the output
Else
&nbsp;&nbsp;&nbsp; select *m* variables at random out of the *M* variables
&nbsp;&nbsp;&nbsp; For *j* = 1 .. *m*
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; If *j*'th attribute is
categorical
*&nbsp; &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
IG{*}{*}{~}j{~}* = IG(*Y*\|*X{*}{*}{~}j{~}*) (see Information
Gain)&nbsp;&nbsp; &nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp; &nbsp;
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; Else (*j*'th attribute is
real-valued)
*&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
IG{*}{*}{~}j{~}* = IG*(*Y*\|*X{*}{*}{~}j{~}*) (see Information Gain)
&nbsp;&nbsp;&nbsp; Let *j\** = argmax{~}j~ *IG{*}{*}{~}j{~}* (this is the
splitting attribute we'll use)
&nbsp;&nbsp;&nbsp; If *j\** is categorical then
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; For each value *v* of the *j*'th
attribute
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; Let
*X{*}{*}{^}v{^}* = subset of rows of *X* in which *X{*}{*}{~}ij{~}* = *v*.
Let *Y{*}{*}{^}v{^}* = corresponding subset of *Y*
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; Let *Child{*}{*}{^}v{^}* =
LearnUnprunedTree(*X{*}{*}{^}v{^}*,*Y{*}{*}{^}v{^}*)
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; Return a decision tree node,
splitting on *j*'th attribute. The number of children equals the number of
values of the *j*'th attribute, and the *v*'th child is
*Child{*}{*}{^}v{^}*
&nbsp;&nbsp;&nbsp; Else *j\** is real-valued and let *t* be the best split
threshold
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; Let *X{*}{*}{^}LO{^}* = subset
of rows of *X* in which *X{*}{*}{~}ij{~}* *<= t*. Let *Y{*}{*}{^}LO{^}* =
corresponding subset of *Y*
&nbsp; &nbsp; &nbsp; &nbsp; Let *Child{*}{*}{^}LO{^}* =
LearnUnprunedTree(*X{*}{*}{^}LO{^}*,*Y{*}{*}{^}LO{^}*)
&nbsp; &nbsp; &nbsp; &nbsp; Let *X{*}{*}{^}HI{^}* = subset of rows of *X*
in which *X{*}{*}{~}ij{~}* *> t*. Let *Y{*}{*}{^}HI{^}* = corresponding
subset of *Y*
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; Let *Child{*}{*}{^}HI{^}* =
LearnUnprunedTree(*X{*}{*}{^}HI{^}*,*Y{*}{*}{^}HI{^}*)
&nbsp; &nbsp; &nbsp; &nbsp; Return a decision tree node, splitting on
*j*'th attribute. It has two children corresponding to whether the *j*'th
attribute is above or below the given threshold.

*Note*: There are alternatives to Information Gain for splitting nodes
&nbsp;

<a name="RandomForests-Informationgain"></a>
### Information gain

source : \[3\](3\.html)
1. h4. nominal attributes

suppose X can have one of m values V{~}1~,V{~}2~,...,V{~}m~
P(X=V{~}1~)=p{~}1~, P(X=V{~}2~)=p{~}2~,...,P(X=V{~}m~)=p{~}m~
&nbsp;
H(X)= \-sum{~}j=1{~}{^}m^ p{~}j~ log{~}2~ p{~}j~ (The entropy of X)
H(Y\|X=v) = the entropy of Y among only those records in which X has value
v
H(Y\|X) = sum{~}j~ p{~}j~ H(Y\|X=v{~}j~)
IG(Y\|X) = H(Y) - H(Y\|X)
1. h4. real-valued attributes

suppose X is real valued
define IG(Y\|X:t) as H(Y) - H(Y\|X:t)
define H(Y\|X:t) = H(Y\|X<t) P(X<t) + H(Y\|X>=t) P(X>=t)
define IG*(Y\|X) = max{~}t~ IG(Y\|X:t)

<a name="RandomForests-HowtogrowaRandomForest"></a>
### How to grow a Random Forest

source : \[1\](1\.html)

Each tree is grown as follows:
1. if the number of cases in the training set is *N*, sample *N* cases at
random \-but with replacement, from the original data. This sample will be
the training set for the growing tree.
1. if there are *M* input variables, a number *m << M* is specified such
that at each node, *m* variables are selected at random out of the *M* and
the best split on these *m* is used to split the node. The value of *m* is
held constant during the forest growing.
1. each tree is grown to its large extent possible. There is no pruning.

<a name="RandomForests-RandomForestparameters"></a>
### Random Forest parameters

source : \[2\](2\.html)
Random Forests are easy to use, the only 2 parameters a user of the
technique has to determine are the number of trees to be used and the
number of variables (*m*) to be randomly selected from the available set of
variables.
Breinman's recommendations are to pick a large number of trees, as well as
the square root of the number of variables for *m*.
&nbsp;

<a name="RandomForests-Howtopredictthelabelofacase"></a>
### How to predict the label of a case

Classify(*node*,*V*)
&nbsp;&nbsp;&nbsp; Input: *node* from the decision tree, if *node.attribute
= j* then the split is done on the *j*'th attribute

&nbsp;&nbsp; &nbsp;Input: *V* a vector of *M* columns where
*V{*}{*}{~}j{~}* = the value of the *j*'th attribute.
&nbsp;&nbsp;&nbsp; Output: label of *V*

&nbsp;&nbsp;&nbsp; If *node* is a Leaf then
&nbsp;&nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp; Return the value predicted
by *node*

&nbsp;&nbsp; &nbsp;Else
&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; Let *j =
node.attribute*
&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; If *j* is
categorical then
&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
Let *v* = *V{*}{*}{~}j{~}*
&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
Let *child{*}{*}{^}v{^}* = child node corresponding to the attribute's
value *v*
&nbsp; &nbsp; &nbsp; &nbsp;&nbsp; &nbsp; &nbsp;&nbsp;&nbsp;
&nbsp;&nbsp;&nbsp;&nbsp; Return Classify(*child{*}{*}{^}v{^}*,*V*)

&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; Else *j* is
real-valued
&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
Let *t = node.threshold* (split threshold)
&nbsp; &nbsp; &nbsp; &nbsp;&nbsp; &nbsp; &nbsp;&nbsp;&nbsp;
&nbsp;&nbsp;&nbsp;&nbsp; If Vj < t then
&nbsp;&nbsp; &nbsp;&nbsp;&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;
&nbsp; &nbsp; &nbsp;&nbsp;&nbsp;&nbsp; Let *child{*}{*}{^}LO{^}* = child
node corresponding to (*<t*)
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp; &nbsp; &nbsp;&nbsp;&nbsp;
&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp; Return
Classify(*child{*}{*}{^}LO{^}*,*V*)
&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
Else
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp; &nbsp; &nbsp;&nbsp;&nbsp;
&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;&nbsp; Let *child{*}{*}{^}HI{^}* =
child node corresponding to (*>=t*)
&nbsp;&nbsp; &nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;&nbsp; &nbsp; &nbsp;
&nbsp; &nbsp; &nbsp; &nbsp;&nbsp; &nbsp;&nbsp; Return
Classify(*child{*}{*}{^}HI{^}*,*V*)
&nbsp;

<a name="RandomForests-Theoutofbag(oob)errorestimation"></a>
### The out of bag (oob) error estimation

source : \[1\](1\.html)

in random forests, there is no need for cross-validation or a separate test
set to get an unbiased estimate of the test set error. It is estimated
internally, during the run, as follows:
* each tree is constructed using a different bootstrap sample from the
original data. About one-third of the cases left of the bootstrap sample
and not used in the construction of the _kth_ tree.
* put each case left out in the construction of the _kth_ tree down the
_kth{_}tree to get a classification. In this way, a test set classification
is obtained for each case in about one-thrid of the trees. At the end of
the run, take *j* to be the class that got mort of the the votes every time
case *n* was _oob_. The proportion of times that *j* is not equal to the
true class of *n* averaged over all cases is the _oob error estimate_. This
has proven to be unbiased in many tests.

<a name="RandomForests-OtherRFuses"></a>
### Other RF uses

source : \[1\](1\.html)
* variable importance
* gini importance
* proximities
* scaling
* prototypes
* missing values replacement for the training set
* missing values replacement for the test set
* detecting mislabeled cases
* detecting outliers
* detecting novelties
* unsupervised learning
* balancing prediction error
Please refer to \[1\](1\.html)
 for a detailed description

<a name="RandomForests-References"></a>
### References

\[1\](1\.html)
&nbsp; Random Forests - Classification Description
&nbsp;&nbsp; &nbsp;&nbsp;&nbsp; &nbsp;[http://www.stat.berkeley.edu/~breiman/RandomForests/cc_home.htm](http://www.stat.berkeley.edu/~breiman/RandomForests/cc_home.htm)
\[2\](2\.html)
&nbsp; B. Larivière & D. Van Den Poel, 2004. "Predicting Customer Retention
and Profitability by Using Random Forests and Regression Forests
Techniques,"
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; Working Papers of Faculty of
Economics and Business Administration, Ghent University, Belgium 04/282,
Ghent University,
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; Faculty of Economics and
Business Administration.
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; Available online : [http://ideas.repec.org/p/rug/rugwps/04-282.html](http://ideas.repec.org/p/rug/rugwps/04-282.html)
\[3\](3\.html)
&nbsp; Decision Trees - Andrew W. Moore\[4\]
&nbsp; &nbsp; &nbsp; &nbsp; http://www.cs.cmu.edu/~awm/tutorials\[1\](1\.html)
\[4\](4\.html)
&nbsp; Information Gain - Andrew W. Moore
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; [http://www.cs.cmu.edu/~awm/tutorials](http://www.cs.cmu.edu/~awm/tutorials)