Decision tree learning

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Decision tree learning is a supervised learning approach used in statistics, data mining and machine learning. In this formalism, a classification or regression decision tree is used as a predictive model to draw conclusions about a set of observations.

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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 (typically real numbers) are called regression trees . More generally, the concept of regression tree can be extended to any kind of object equipped with pairwise dissimilarities such as categorical sequences. [1]

Decision trees are among the most popular machine learning algorithms given their intelligibility and simplicity. [2]

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 (but the resulting classification tree can be an input for decision making).

General

A tree showing survival of passengers on the Titanic ("sibsp" is the number of spouses or siblings aboard). The figures under the leaves show the probability of survival and the percentage of observations in the leaf. Summarizing: Your chances of survival were good if you were (i) a female or (ii) a male at most 9.5 years old with strictly fewer than 3 siblings. Decision Tree.jpg
A tree showing survival of passengers on the Titanic ("sibsp" is the number of spouses or siblings aboard). The figures under the leaves show the probability of survival and the percentage of observations in the leaf. Summarizing: Your chances of survival were good if you were (i) a female or (ii) a male at most 9.5 years old with strictly fewer than 3 siblings.

Decision tree learning is a method commonly used in data mining. [3] 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 (non-leaf) 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 (which, if the decision tree is well-constructed, is skewed towards certain subsets of classes).

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. [4] 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 (TDIDT) [5] is an example of a greedy algorithm, and it is by far the most common strategy for learning decision trees from data. [6]

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.

An example tree which estimates the probability of kyphosis after spinal 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 spinal 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. (Note: The treatment of kyphosis has advanced considerably since this rather small set of data was collected.
) Cart tree kyphosis.png
An example tree which estimates the probability of kyphosis after spinal 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 spinal 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. (Note: The treatment of kyphosis has advanced considerably since this rather small set of data was collected. )

Decision tree types

Decision trees used in data mining are of two main types:

The term classification and regression tree (CART) analysis is an umbrella term used to refer to either of the above procedures, first introduced by Breiman et al. in 1984. [7] 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. [7]

Some techniques, often called ensemble methods, construct more than one decision tree:

A special case of a decision tree is a decision list, [14] 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 (except for the bottommost node, whose only child is a single leaf node). While less expressive, decision lists are arguably easier to understand than general decision trees due to their added sparsity[ citation needed ], permit non-greedy learning methods [15] and monotonic constraints to be imposed. [16]

Notable decision tree algorithms include:

ID3 and CART were invented independently at around the same time (between 1970 and 1980)[ citation needed ], 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 (FDT). [23] 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. [24]

Metrics

Algorithms for constructing decision trees usually work top-down, by choosing a variable at each step that best splits the set of items. [6] 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 (e.g., averaged) to provide a measure of the quality of the split. Depending on the underlying metric, the performance of various heuristic algorithms for decision tree learning may vary significantly. [25]

Estimate of Positive Correctness

A simple and effective metric can be used to identify the degree to which true positives outweigh false positives (see Confusion matrix). This metric, "Estimate of Positive Correctness" is defined below:

In this equation, the total false positives (FP) are subtracted from the total true positives (TP). The resulting number gives an estimate on how many positive examples the feature could correctly identify within the data, with higher numbers meaning that the feature could correctly classify more positive samples. Below is an example of how to use the metric when the full confusion matrix of a certain feature is given:

Feature A Confusion Matrix

Predicted
Class
Actual Class
CancerNon-cancer
Cancer83
Non-cancer25

Here we can see that the TP value would be 8 and the FP value would be 2 (the underlined numbers in the table). When we plug these numbers in the equation we are able to calculate the estimate: . This means that using the estimate on this feature would have it receive a score of 6.

However, it should be worth noting that this number is only an estimate. For example, if two features both had a FP value of 2 while one of the features had a higher TP value, that feature would be ranked higher than the other because the resulting estimate when using the equation would give a higher value. This could lead to some inaccuracies when using the metric if some features have more positive samples than others. To combat this, one could use a more powerful metric known as Sensitivity that takes into account the proportions of the values from the confusion matrix to give the actual true positive rate (TPR). The difference between these metrics is shown in the example below:

Feature A Confusion Matrix
Predicted
Class
Actual Class
CancerNon-cancer
Cancer83
Non-cancer25
Feature B Confusion Matrix
Predicted
Class
Actual Class
CancerNon-cancer
Cancer62
Non-cancer28

In this example, Feature A had an estimate of 6 and a TPR of approximately 0.73 while Feature B had an estimate of 4 and a TPR of 0.75. This shows that although the positive estimate for some feature may be higher, the more accurate TPR value for that feature may be lower when compared to other features that have a lower positive estimate. Depending on the situation and knowledge of the data and decision trees, one may opt to use the positive estimate for a quick and easy solution to their problem. On the other hand, a more experienced user would most likely prefer to use the TPR value to rank the features because it takes into account the proportions of the data and all the samples that should have been classified as positive.

Gini impurity

Gini impurity, Gini's diversity index, [26] or Gini-Simpson Index in biodiversity research, is named after Italian mathematician Corrado Gini and used by the CART (classification and regression tree) algorithm for classification trees. Gini impurity measures how often a randomly chosen element of a set would be incorrectly labeled if it were labeled randomly and independently according to the distribution of labels in the set. It reaches its minimum (zero) when all cases in the node fall into a single target category.

For a set of items with classes and relative frequencies , , the probability of choosing an item with label is , and the probability of miscategorizing that item is . The Gini impurity is computed by summing pairwise products of these probabilities for each class label:

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-equilibrium, 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 nothing but a variation of the usual entropy measure for decision trees.

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. [27]

Averaging over the possible values of ,

Where weighted sum of entropies is given by,

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 most consistent child nodes. A commonly used measure of consistency 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". [27]

Consider an example data set with four attributes: outlook (sunny, overcast, rainy), temperature (hot, mild, cool), humidity (high, normal), and windy (true, false), with a binary (yes or no) 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 each have consistent data, or until the information gain is 0.

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.

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 (where play is the target variable) 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.

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. [27]

Information gain is also known as Shannon index in bio diversity research.

Variance reduction

Introduced in CART, [7] variance reduction is often employed in cases where the target variable is continuous (regression tree), meaning that use of many other metrics would first require discretization before being applied. The variance reduction of a node N is defined as the total reduction of the variance of the target variable Y 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.

By replacing in the formula above with the dissimilarity between two objects and , the variance reduction criterion applies to any kind of object for which pairwise dissimilarities can be computed. [1]

Measure of "goodness"

Used by CART in 1984, [28] the measure of "goodness" is a function that seeks to optimize the balance of a candidate split's capacity to create pure children with its capacity to create equally-sized children. This process is repeated for each impure node until the tree is complete. The function , where is a candidate split at node , is defined as below

where and are the left and right children of node using split , respectively; and are the proportions of records in in and , respectively; and and are the proportions of class records in and , respectively.

Consider an example data set with three attributes: savings(low, medium, high), assets(low, medium, high), income(numerical value), and a binary target variable credit risk(good, bad) and 8 data points. [28] The full data is presented in the table below. To start a decision tree, we will calculate the maximum value of using each feature to find which one will split the root node. This process will continue until all children are pure or all values are below a set threshold.

CustomerSavingsAssetsIncome ($1000s)Credit risk
1MediumHigh75Good
2LowLow50Bad
3HighMedium25Bad
4MediumMedium50Good
5LowMedium100Good
6HighHigh25Good
7LowLow25Bad
8MediumMedium75Good

To find of the feature savings, we need to note the quantity of each value. The original data contained three low's, three medium's, and two high's. Out of the low's, one had a good credit risk while out of the medium's and high's, 4 had a good credit risk. Assume a candidate split such that records with a low savings will be put in the left child and all other records will be put into the right child.

To build the tree, the "goodness" of all candidate splits for the root node need to be calculated. The candidate with the maximum value will split the root node, and the process will continue for each impure node until the tree is complete.

Compared to other metrics such as information gain, the measure of "goodness" will attempt to create a more balanced tree, leading to more-consistent decision time. However, it sacrifices some priority for creating pure children which can lead to additional splits that are not present with other metrics.

Uses

Advantages

Amongst other data mining methods, decision trees have various advantages:

Limitations

Implementations

Many data mining software packages provide implementations of one or more decision tree algorithms (e.g. random forest).

Open source examples include:

Notable commercial software:

Extensions

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 (ORs) to join two more paths together using minimum message length (MML). [43] 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. [44] The more general coding scheme results in better predictive accuracy and log-loss probabilistic scoring.[ citation needed ] 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. [45] [46]

It is also possible for a tree to be sampled using MCMC. [47]

The tree can be searched for in a bottom-up fashion. [48] Or several trees can be constructed parallelly to reduce the expected number of tests till classification. [38]

See also

Related Research Articles

<span class="mw-page-title-main">Entropy (information theory)</span> Expected amount of information needed to specify the output of a stochastic data source

In information theory, the entropy of a random variable is the average level of "information", "surprise", or "uncertainty" inherent to the variable's possible outcomes. Given a discrete random variable , which takes values in the set and is distributed according to , the entropy is where denotes the sum over the variable's possible values. The choice of base for , the logarithm, varies for different applications. Base 2 gives the unit of bits, while base e gives "natural units" nat, and base 10 gives units of "dits", "bans", or "hartleys". An equivalent definition of entropy is the expected value of the self-information of a variable.

<span class="mw-page-title-main">Supervised learning</span> Paradigm in machine learning

Supervised learning (SL) is a paradigm in machine learning where input objects and a desired output value train a model. The training data is processed, building a function that maps new data on expected output values. An optimal scenario will allow for the algorithm to correctly determine output values for unseen instances. This requires the learning algorithm to generalize from the training data to unseen situations in a "reasonable" way. This statistical quality of an algorithm is measured through the so-called generalization error.

A Bayesian network is a probabilistic graphical model that represents a set of variables and their conditional dependencies via a directed acyclic graph (DAG). While it is one of several forms of causal notation, causal networks are special cases of Bayesian networks. Bayesian networks are ideal for taking an event that occurred and predicting the likelihood that any one of several possible known causes was the contributing factor. For example, a Bayesian network could represent the probabilistic relationships between diseases and symptoms. Given symptoms, the network can be used to compute the probabilities of the presence of various diseases.

<span class="mw-page-title-main">Decision tree</span> Decision support tool

A decision tree is a decision support hierarchical model that uses a tree-like model of decisions and their possible consequences, including chance event outcomes, resource costs, and utility. It is one way to display an algorithm that only contains conditional control statements.

<span class="mw-page-title-main">Mutual information</span> Measure of dependence between two variables

In probability theory and information theory, the mutual information (MI) of two random variables is a measure of the mutual dependence between the two variables. More specifically, it quantifies the "amount of information" obtained about one random variable by observing the other random variable. The concept of mutual information is intimately linked to that of entropy of a random variable, a fundamental notion in information theory that quantifies the expected "amount of information" held in a random variable.

In statistics, Gibbs sampling or a Gibbs sampler is a Markov chain Monte Carlo (MCMC) algorithm for sampling from a specified multivariate probability distribution when direct sampling from the joint distribution is difficult, but sampling from the conditional distribution is more practical. This sequence can be used to approximate the joint distribution ; to approximate the marginal distribution of one of the variables, or some subset of the variables ; or to compute an integral. Typically, some of the variables correspond to observations whose values are known, and hence do not need to be sampled.

Belief propagation, also known as sum–product message passing, is a message-passing algorithm for performing inference on graphical models, such as Bayesian networks and Markov random fields. It calculates the marginal distribution for each unobserved node, conditional on any observed nodes. Belief propagation is commonly used in artificial intelligence and information theory, and has demonstrated empirical success in numerous applications, including low-density parity-check codes, turbo codes, free energy approximation, and satisfiability.

<span class="mw-page-title-main">Boltzmann machine</span> Type of stochastic recurrent neural network

A Boltzmann machine, named after Ludwig Boltzmann is a stochastic spin-glass model with an external field, i.e., a Sherrington–Kirkpatrick model, that is a stochastic Ising model. It is a statistical physics technique applied in the context of cognitive science. It is also classified as a Markov random field.

Feature selection is the process of selecting a subset of relevant features for use in model construction. Stylometry and DNA microarray analysis are two cases where feature selection is used. It should be distinguished from feature extraction.

In information theory, Shannon's source coding theorem establishes the statistical limits to possible data compression for data whose source is an independent identically-distributed random variable, and the operational meaning of the Shannon entropy.

Random forests or random decision forests is an ensemble learning method for classification, regression and other tasks that operates by constructing a multitude of decision trees at training time. For classification tasks, the output of the random forest is the class selected by most trees. For regression tasks, the mean or average prediction of the individual trees is returned. Random decision forests correct for decision trees' habit of overfitting to their training set.

In information theory, redundancy measures the fractional difference between the entropy H(X) of an ensemble X, and its maximum possible value . Informally, it is the amount of wasted "space" used to transmit certain data. Data compression is a way to reduce or eliminate unwanted redundancy, while forward error correction is a way of adding desired redundancy for purposes of error detection and correction when communicating over a noisy channel of limited capacity.

<span class="mw-page-title-main">ID3 algorithm</span> Decision tree algorithm

In decision tree learning, ID3 is an algorithm invented by Ross Quinlan used to generate a decision tree from a dataset. ID3 is the precursor to the C4.5 algorithm, and is typically used in the machine learning and natural language processing domains.

<span class="mw-page-title-main">C4.5 algorithm</span> Algorithm for making decision trees

C4.5 is an algorithm used to generate a decision tree developed by Ross Quinlan. C4.5 is an extension of Quinlan's earlier ID3 algorithm. The decision trees generated by C4.5 can be used for classification, and for this reason, C4.5 is often referred to as a statistical classifier. In 2011, authors of the Weka machine learning software described the C4.5 algorithm as "a landmark decision tree program that is probably the machine learning workhorse most widely used in practice to date".

Variational message passing (VMP) is an approximate inference technique for continuous- or discrete-valued Bayesian networks, with conjugate-exponential parents, developed by John Winn. VMP was developed as a means of generalizing the approximate variational methods used by such techniques as latent Dirichlet allocation, and works by updating an approximate distribution at each node through messages in the node's Markov blanket.

In information theory and machine learning, information gain is a synonym for Kullback–Leibler divergence; the amount of information gained about a random variable or signal from observing another random variable. However, in the context of decision trees, the term is sometimes used synonymously with mutual information, which is the conditional expected value of the Kullback–Leibler divergence of the univariate probability distribution of one variable from the conditional distribution of this variable given the other one.

An alternating decision tree (ADTree) is a machine learning method for classification. It generalizes decision trees and has connections to boosting.

<span class="mw-page-title-main">Information gain ratio</span> Decision tree training concept

In decision tree learning, information gain ratio is a ratio of information gain to the intrinsic information. It was proposed by Ross Quinlan, to reduce a bias towards multi-valued attributes by taking the number and size of branches into account when choosing an attribute.

Gradient boosting is a machine learning technique based on boosting in a functional space, where the target is pseudo-residuals rather than the typical residuals used in traditional boosting. It gives a prediction model in the form of an ensemble of weak prediction models, i.e., models that make very few assumptions about the data, which are typically simple decision trees. When a decision tree is the weak learner, the resulting algorithm is called gradient-boosted trees; it usually outperforms random forest. A gradient-boosted trees model is built in a stage-wise fashion as in other boosting methods, but it generalizes the other methods by allowing optimization of an arbitrary differentiable loss function.

In computer science, an optimal binary search tree (Optimal BST), sometimes called a weight-balanced binary tree, is a binary search tree which provides the smallest possible search time (or expected search time) for a given sequence of accesses (or access probabilities). Optimal BSTs are generally divided into two types: static and dynamic.

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Further reading