Classification Decision Tree

Classification decision tree is a kind of a decision tree described in Decision Tree.

Details

Given:

  • n feature vectors \(x_1 = (x_{11}, \ldots, x_{1p}), \ldots, x_n = (x_{n1}, \ldots, x_{np})\) of size \(p\)

  • The vector of class labels \(y = (y_1, \ldots, y_n)\) that describes the class to which the feature vector \(x_i\) belongs, where \(y_i \in \{0, 1, \ldots, C-1\}\) and C is the number of classes.

The problem is to build a decision tree classifier.

Split Criteria

The library provides the decision tree classification algorithm based on split criteria Gini index [Breiman84] and Information gain [Quinlan86], [Mitchell97]:

  1. Gini index

    \[{I}_{Gini}\left(D\right)=1-\sum _{i=0}^{C-1}{p}_{i}^{2}\]

    where

    • \(D\) is a set of observations that reach the node

    • \(p_i\) is the observed fraction of observations with class \(i\) in \(D\)

    To find the best test using Gini index, each possible test is examined using

    \[\text{Δ}{I}_{Gini}\left(D,\text{ τ}\right)={I}_{Gini}\left(D\right)-\sum _{v\in O\left(\text{τ}\right)}\frac{|{D}_{v}|}{|D|}{I}_{Gini}\left({D}_{v}\right)\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\]

    where

    • \(O(\tau)\) is the set of all possible outcomes of test \(\tau\)

    • \(D_v\) is the subset of \(D\), for which outcome of \(\tau\) is \(v\), for example \({D}_{v}=\left\{d\in D\text{|τ}\left(d\right)=v\right\}\)

    The test to be used in the node is selected as \(\underset{\tau }{\text{argmax}}\text{Δ}{I}_{Gini}\left(D,\tau \right)\). For binary decision tree with ‘true’ and ‘false’ branches, \(\text{Δ}{I}_{Gini}\left(D, \text{τ}\right)={I}_{Gini}\left(D\right)-\frac{|{D}_{true}|}{|D|}{I}_{Gini}\left({D}_{true}\right)-\frac{|{D}_{false}|}{|D|}{I}_{Gini}\left({D}_{false}\right)\)

  2. Information gain

\[\text{Δ}{I}_{Gini}\left(D, \text{τ}\right)={I}_{Gini}\left(D\right)-\frac{|{D}_{true}|}{|D|}{I}_{Gini}\left({D}_{true}\right)-\frac{|{D}_{false}|}{|D|}{I}_{Gini}\left({D}_{false}\right)\]

where

  • \(O(\tau)\), \(D\), \(D_v\) are defined above

  • \({I}_{Entropy}\left(D\right)=-\sum _{i=0}^{C-1}{p}_{i}\mathrm{log}{p}_{i}\), with \(p_i\) defined above in Gini index.

    Similarly to Gini index, the test to be used in the node is selected as \(\underset{\tau }{\text{argmax}}InfoGain\left(D,\tau \right)\). For binary decision tree with ‘true’ and ‘false’ branches, \(\text{Δ}{I}_{Gini}\left(D, \text{τ}\right)={I}_{Gini}\left(D\right)-\frac{|{D}_{true}|}{|D|}{I}_{Gini}\left({D}_{true}\right)-\frac{|{D}_{false}|}{|D|}{I}_{Gini}\left({D}_{false}\right)\)

Training Stage

The classification decision tree follows the algorithmic framework of decision tree training described in Decision Tree.

Prediction Stage

The classification decision tree follows the algorithmic framework of decision tree prediction described in Decision Tree.

Given decision tree and vectors \(x_i, \ldots, x_r\), the problem is to calculate the responses for those vectors.

Batch Processing

Decision tree classification follows the general workflow described in Classification Usage Model.

Training

In addition to common input for a classifier, decision trees can accept the following inputs that are used for post-pruning:

Input ID

Input

dataForPruning

Pointer to the \(m \times p\) numeric table with the pruning data set. This table can be an object of any class derived from NumericTable.

labelsForPruning

Pointer to the \(m \times 1\) numeric table with class labels. This table can be an object of any class derived from NumericTable except PackedSymmetricMatrix and PackedTriangularMatrix.

At the training stage, decision tree classifier has the following parameters:

Parameter

Default Value

Description

algorithmFPType

float

The floating-point type that the algorithm uses for intermediate computations. Can be float or double.

method

defaultDense

The computation method used by the decision tree classification. The only training method supported so far is the default dense method.

nClasses

Not applicable

The number of classes. A required parameter.

splitCriterion

infoGain

Split criterion to choose the best test for split nodes. Available split criteria for decision trees:

  • gini - the Gini index

  • infoGain - the information gain

pruning

reducedErrorPruning

Method to perform post-pruning. Available options for the pruning parameter:

  • reducedErrorPruning - reduced error pruning. Provide dataForPruning and labelsForPruning inputs, if you use pruning.

  • none - do not prune.

maxTreeDepth

\(0\)

Maximum tree depth. Zero value means unlimited depth. Can be any non-negative number.

minObservationsInLeafNodes

\(1\)

Minimum number of observations in the leaf node. Can be any positive number.

Prediction

At the prediction stage, decision tree classifier has the following parameters:

Parameter

Default Value

Description

algorithmFPType

float

The floating-point type that the algorithm uses for intermediate computations. Can be float or double.

method

defaultDense

The computation method used by the decision tree classification. The only training method supported so far is the default dense method.

Examples

Batch Processing:

Note

There is no support for Java on GPU.

Batch Processing: