How to grow a Decision Tree

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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     select m variables at random out of the M variables     For j = 1 .. m         If j‘th attribute is categorical             IG{}{}{~}j{~} = IG(Y|X{}{}{~}j{~}) (see Information Gain)                     Else (j‘th attribute is real-valued)             IG{}{}{~}j{~} = IG(Y|X{}{}{~}j{~}) (see Information Gain)     Let *j* = argmax{~}j~ IG{}{}{~}j{~} (this is the splitting attribute we’ll use)     If j* is categorical then         For each value v of the j‘th attribute             Let X{}{}{^}v{^} = subset of rows of X in which X{}{}{~}ij{~} = v. Let Y{}{}{^}v{^} = corresponding subset of Y             Let Child{}{}{^}v{^} = LearnUnprunedTree(X{}{}{^}v{^},Y{}{}{^}v{^})         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{^}     Else j* is real-valued and let t be the best split threshold         Let X{}{}{^}LO{^} = subset of rows of X in which X{}{}{~}ij{~} <= t. Let Y{}{}{^}LO{^} = corresponding subset of Y         Let Child{}{}{^}LO{^} = LearnUnprunedTree(X{}{}{^}LO{^},Y{}{}{^}LO{^})         Let X{}{}{^}HI{^} = subset of rows of X in which X{}{}{~}ij{~} > t. Let Y{}{}{^}HI{^} = corresponding subset of Y         Let Child{}{}{^}HI{^} = LearnUnprunedTree(X{}{}{^}HI{^},Y{}{}{^}HI{^})         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  

Information gain

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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~   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)

How to grow a Random Forest

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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.
2. 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.
3. each tree is grown to its large extent possible. There is no pruning.

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.  

How to predict the label of a case

Classify(node,V)     Input: node from the decision tree, if node.attribute = j then the split is done on the j‘th attribute

    Input: V a vector of M columns where V{}{}{~}j{~} = the value of the j‘th attribute.     Output: label of V

    If node is a Leaf then             Return the value predicted by node

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

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

The out of bag (oob) error estimation

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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 most 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.

Other RF uses

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• 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

References

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