Decision tree learning

Decision tree learning is one of the predictive modelling approaches used in statistics, data mining and machine learning. It uses a decision tree to go from observations about an item to conclusions about the item's target value. Tree models where the target variable can take a discrete set of values are called classification trees; in these tree structures, leaves represent class labels and branches represent conjunctions of features that lead to those class labels. Decision trees where the target variable can take continuous values are called regression trees. Decision trees are among the most popular machine learning algorithms given their intelligibility and simplicity.
In decision analysis, a decision tree can be used to visually and explicitly represent decisions and decision making. In data mining, a decision tree describes data. This page deals with decision trees in data mining.


Decision tree learning is a method commonly used in data mining. The goal is to create a model that predicts the value of a target variable based on several input variables.
A decision tree is a simple representation for classifying examples. For this section, assume that all of the input features have finite discrete domains, and there is a single target feature called the "classification". Each element of the domain of the classification is called a class.
A decision tree or a classification tree is a tree in which each internal node is labeled with an input feature. The arcs coming from a node labeled with an input feature are labeled with each of the possible values of the target feature or the arc leads to a subordinate decision node on a different input feature. Each leaf of the tree is labeled with a class or a probability distribution over the classes, signifying that the data set has been classified by the tree into either a specific class, or into a particular probability distribution.
A tree is built by splitting the source set, constituting the root node of the tree, into subsets—which constitute the successor children. The splitting is based on a set of splitting rules based on classification features. This process is repeated on each derived subset in a recursive manner called recursive partitioning.
The recursion is completed when the subset at a node has all the same values of the target variable, or when splitting no longer adds value to the predictions. This process of top-down induction of decision trees is an example of a greedy algorithm, and it is by far the most common strategy for learning decision trees from data.
In data mining, decision trees can be described also as the combination of mathematical and computational techniques to aid the description, categorization and generalization of a given set of data.
Data comes in records of the form:
The dependent variable,, is the target variable that we are trying to understand, classify or generalize. The vector is composed of the features, etc., that are used for that task.
after surgery, given the age of the patient and the
vertebra at which surgery was started.
The same tree is shown in three different ways.
Left The colored leaves show the probability of kyphosis after surgery,
and percentage of patients in the leaf.
Middle The tree as a perspective plot.
Right Aerial view of the middle plot.
The probability of kyphosis after surgery is higher in the darker areas.

Decision tree types

Decision trees used in data mining are of two main types:
The term Classification And Regression Tree analysis is an umbrella term used to refer to both of the above procedures, first introduced by Breiman et al. in 1984. Trees used for regression and trees used for classification have some similarities - but also some differences, such as the procedure used to determine where to split.
Some techniques, often called ensemble methods, construct more than one decision tree:
A special case of a decision tree is a decision list, which is a one-sided decision tree, so that every internal node has exactly 1 leaf node and exactly 1 internal node as a child. While less expressive, decision lists are arguably easier to understand than general decision trees due to their added sparsity, permit non-greedy learning methods and monotonic constraints to be imposed.
Notable decision tree algorithms include:
ID3 and CART were invented independently at around the same time, yet follow a similar approach for learning a decision tree from training tuples.
It has also been proposed to leverage concepts of fuzzy set theory for the definition of a special version of decision tree, known as Fuzzy Decision Tree.
In this type of fuzzy classification, generally an input vector is associated with multiple classes, each with a different confidence value.
Boosted ensembles of FDTs have been recently investigated as well, and they have shown performances comparable to those of other very efficient fuzzy classifiers.


Algorithms for constructing decision trees usually work top-down, by choosing a variable at each step that best splits the set of items. Different algorithms use different metrics for measuring "best". These generally measure the homogeneity of the target variable within the subsets. Some examples are given below. These metrics are applied to each candidate subset, and the resulting values are combined to provide a measure of the quality of the split.

Gini impurity

Used by the CART algorithm for classification trees, Gini impurity is a measure of how often a randomly chosen element from the set would be incorrectly labeled if it was randomly labeled according to the distribution of labels in the subset. The Gini impurity can be computed by summing the probability of an item with label being chosen times the probability of a mistake in categorizing that item. It reaches its minimum when all cases in the node fall into a single target category.
The Gini impurity is also an information theoretic measure and corresponds to Tsallis Entropy with deformation coefficient, which in Physics is associated with the lack of information in out-of-equlibrium, non-extensive, dissipative and quantum systems. For the limit one recovers the usual Boltzmann-Gibbs or Shannon entropy. In this sense, the Gini impurity is but a variation of the usual entropy measure for decision trees.
To compute Gini impurity for a set of items with classes, suppose, and let be the fraction of items labeled with class in the set.

Information gain

Used by the ID3, C4.5 and C5.0 tree-generation algorithms. Information gain is based on the concept of entropy and information content from information theory.
Entropy is defined as below
where are fractions that add up to 1 and represent the percentage of each class present in the child node that results from a split in the tree.
Averaging over the possible values of,
That is, the expected information gain is the mutual information, meaning that on average, the reduction in the entropy of T is the mutual information.
Information gain is used to decide which feature to split on at each step in building the tree. Simplicity is best, so we want to keep our tree small. To do so, at each step we should choose the split that results in the purest daughter nodes. A commonly used measure of purity is called information which is measured in bits. For each node of the tree, the information value "represents the expected amount of information that would be needed to specify whether a new instance should be classified yes or no, given that the example reached that node".
Consider an example data set with four attributes: outlook, temperature, humidity, and windy, with a binary target variable, play, and 14 data points. To construct a decision tree on this data, we need to compare the information gain of each of four trees, each split on one of the four features. The split with the highest information gain will be taken as the first split and the process will continue until all children nodes are pure, or until the information gain is 0.
The split using the feature windy results in two children nodes, one for a windy value of true and one for a windy value of false. In this data set, there are six data points with a true windy value, three of which have a play value of yes and three with a play value of no. The eight remaining data points with a windy value of false contain two no's and six yes's. The information of the windy=true node is calculated using the entropy equation above. Since there is an equal number of yes's and no's in this node, we have
For the node where windy=false there were eight data points, six yes's and two no's. Thus we have
To find the information of the split, we take the weighted average of these two numbers based on how many observations fell into which node.
To find the information gain of the split using windy, we must first calculate the information in the data before the split. The original data contained nine yes's and five no's.
Now we can calculate the information gain achieved by splitting on the windy feature.
To build the tree, the information gain of each possible first split would need to be calculated. The best first split is the one that provides the most information gain. This process is repeated for each impure node until the tree is complete. This example is adapted from the example appearing in Witten et al.

Variance reduction

Introduced in CART, variance reduction is often employed in cases where the target variable is continuous, meaning that use of many other metrics would first require discretization before being applied. The variance reduction of a node is defined as the total reduction of the variance of the target variable due to the split at this node:
where,, and are the set of presplit sample indices, set of sample indices for which the split test is true, and set of sample indices for which the split test is false, respectively. Each of the above summands are indeed variance estimates, though, written in a form without directly referring to the mean.



Amongst other data mining methods, decision trees have various advantages:
Many data mining software packages provide implementations of one or more decision tree algorithms.
Examples include Salford Systems CART, IBM SPSS Modeler, RapidMiner, SAS Enterprise Miner, Matlab, R, Weka, Orange, KNIME, Microsoft SQL Server , and scikit-learn.


Decision graphs

In a decision tree, all paths from the root node to the leaf node proceed by way of conjunction, or AND. In a decision graph, it is possible to use disjunctions to join two more paths together using minimum message length. Decision graphs have been further extended to allow for previously unstated new attributes to be learnt dynamically and used at different places within the graph. The more general coding scheme results in better predictive accuracy and log-loss probabilistic scoring. In general, decision graphs infer models with fewer leaves than decision trees.

Alternative search methods

Evolutionary algorithms have been used to avoid local optimal decisions and search the decision tree space with little a priori bias.
It is also possible for a tree to be sampled using MCMC.
The tree can be searched for in a bottom-up fashion.