Decision tree learning

This article is about decision trees in machine learning. For the use of the term in decision analysis, see Decision tree.

Decision tree learning uses a decision tree as a predictive model which maps observations about an item to conclusions about the item's target value. It is one of the predictive modelling approaches used in statistics, data mining and machine learning. Tree models where the target variable can take a finite 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.

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 not decisions; rather the resulting classification tree can be an input for decision making. This page deals with decision trees in data mining.

General

Decision tree learning is a method commonly used in data mining.[1] The goal is to create a model that predicts the value of a target variable based on several input variables. An example is shown on the right. Each interior node corresponds to one of the input variables; there are edges to children for each of the possible values of that input variable. Each leaf represents a value of the target variable given the values of the input variables represented by the path from the root to the leaf.

A decision tree is a simple representation for classifying examples. Decision tree learning is one of the most successful techniques for supervised classification learning. For this section, assume that all of the 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 a feature are labeled with each of the possible values of the feature. Each leaf of the tree is labeled with a class or a probability distribution over the classes.

A tree can be "learned" by splitting the source set into subsets based on an attribute value test. 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 value of the target variable, or when splitting no longer adds value to the predictions. This process of top-down induction of decision trees (TDIDT) [2] 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, categorisation and generalisation of a given set of data.

Data comes in records of the form:

(\textbf{x},Y) = (x_1, x_2, x_3, ..., x_k, Y)

The dependent variable, Y, is the target variable that we are trying to understand, classify or generalize. The vector x is composed of the input variables, x1, x2, x3 etc., that are used for that task.

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 both of the above procedures, first introduced by Breiman et al.[3] 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.[3]

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

Decision tree learning is the construction of a decision tree from class-labeled training tuples. A decision tree is a flow-chart-like structure, where each internal (non-leaf) node denotes a test on an attribute, each branch represents the outcome of a test, and each leaf (or terminal) node holds a class label. The topmost node in a tree is the root node.

There are many specific decision-tree algorithms. Notable ones include:

ID3 and CART were invented independently at around the same time (between 1970 and 1980), yet follow a similar approach for learning decision tree from training tuples.

Metrics

Algorithms for constructing decision trees usually work top-down, by choosing a variable at each step that best splits the set of items.[11] 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.

Gini impurity

Not to be confused with Gini coefficient.

Used by the CART (classification and regression tree) algorithm, Gini impurity is a measure of how often a randomly chosen element from the set would be incorrectly labeled if it were randomly labeled according to the distribution of labels in the subset. Gini impurity can be computed by summing the probability of each item being chosen times the probability of a mistake in categorizing that item. It reaches its minimum (zero) when all cases in the node fall into a single target category.

To compute Gini impurity for a set of items, suppose i \in \{1, 2, ..., m\}, and let f_i be the fraction of items labeled with value i in the set.

I_{G}(f) = \sum_{i=1}^{m} f_i (1-f_i) = \sum_{i=1}^{m} (f_i - {f_i}^2) = \sum_{i=1}^m f_i - \sum_{i=1}^{m} {f_i}^2 = 1 - \sum^{m}_{i=1} {f_i}^{2}

Information gain

Used by the ID3, C4.5 and C5.0 tree-generation algorithms. Information gain is based on the concept of entropy from information theory.

I_{E}(f) = - \sum^{m}_{i=1} f_i \log^{}_2 f_i

Variance reduction

Introduced in CART,[3] 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 x due to the split at this node:

I_{V}(N) = \frac{1}{|S|}\sum_{i\in S} \sum_{j\in S} \frac{1}{2}(x_i - x_j)^2 - \left(\frac{1}{|S_t|}\sum_{i\in S_t} \sum_{j\in S_t} \frac{1}{2}(x_i - x_j)^2 + \frac{1}{|S_f|}\sum_{i\in S_f} \sum_{j\in S_f} \frac{1}{2}(x_i - x_j)^2\right)

where S, S_t, and S_f 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.

Decision tree advantages

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

Limitations

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).[18] 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.[19] 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.[20][21]

It is also possible for a tree to be sampled using MCMC.[22]

The tree can be searched for in a bottom-up fashion.[23]

See also

Implementations

Many data mining software packages provide implementations of one or more decision tree algorithms. Several examples include Salford Systems CART (which licensed the proprietary code of the original CART authors[3]), IBM SPSS Modeler, RapidMiner, SAS Enterprise Miner, Matlab, R (an open source software environment for statistical computing which includes several CART implementations such as rpart, party and randomForest packages), Weka (a free and open-source data mining suite, contains many decision tree algorithms), Orange (a free data mining software suite, which includes the tree module orngTree), KNIME, Microsoft SQL Server , and scikit-learn (a free and open-source machine learning library for the Python programming language).

References

  1. ↑ Rokach, Lior; Maimon, O. (2008). Data mining with decision trees: theory and applications. World Scientific Pub Co Inc. ISBN 978-9812771711.
  2. ↑ Quinlan, J. R., (1986). Induction of Decision Trees. Machine Learning 1: 81-106, Kluwer Academic Publishers
  3. ↑ 3.0 3.1 3.2 3.3 Breiman, Leo; Friedman, J. H.; Olshen, R. A.; Stone, C. J. (1984). Classification and regression trees. Monterey, CA: Wadsworth & Brooks/Cole Advanced Books & Software. ISBN 978-0-412-04841-8.
  4. ↑ Breiman, L. (1996). Bagging Predictors. "Machine Learning, 24": pp. 123-140.
  5. ↑ Friedman, J. H. (1999). Stochastic gradient boosting. Stanford University.
  6. ↑ Hastie, T., Tibshirani, R., Friedman, J. H. (2001). The elements of statistical learning : Data mining, inference, and prediction. New York: Springer Verlag.
  7. ↑ Rodriguez, J.J. and Kuncheva, L.I. and Alonso, C.J. (2006), Rotation forest: A new classifier ensemble method, IEEE Transactions on Pattern Analysis and Machine Intelligence, 28(10):1619-1630.
  8. ↑ Kass, G. V. (1980). "An exploratory technique for investigating large quantities of categorical data". Applied Statistics 29 (2): 119–127. doi:10.2307/2986296. JSTOR 2986296.
  9. ↑ 9.0 9.1 9.2 Hothorn, T.; Hornik, A.; Zeileis (2006). "Unbiased Recursive Partitioning: A Conditional Inference Framework". Journal of Computational and Graphical Statistics 15 (3): 651–674. doi:10.1198/106186006X133933. JSTOR 27594202.
  10. ↑ 10.0 10.1 Strobl, C.; Malley, G.; Tutz (2009). "An Introduction to Recursive Partitioning: Rationale, Application and Characteristics of Classification and Regression Trees, Bagging and Random Forests". Psychological Methods 14 (4): 323–348. doi:10.1037/a0016973.
  11. ↑ Rokach, L.; Maimon, O. (2005). "Top-down induction of decision trees classifiers-a survey". IEEE Transactions on Systems, Man, and Cybernetics, Part C 35 (4): 476–487. doi:10.1109/TSMCC.2004.843247.
  12. ↑ Hyafil, Laurent; Rivest, RL (1976). "Constructing Optimal Binary Decision Trees is NP-complete". Information Processing Letters 5 (1): 15–17. doi:10.1016/0020-0190(76)90095-8.
  13. ↑ Murthy S. (1998). Automatic construction of decision trees from data: A multidisciplinary survey. Data Mining and Knowledge Discovery
  14. ↑ Ben-Gal I. Dana A., Shkolnik N. and Singer (20). "Efficient Construction of Decision Trees by the Dual Information Distance Method". Quality Technology & Quantitative Management (QTQM), 11( 1), 133-147. Check date values in: |date= (help)
  15. ↑ "Principles of Data Mining". 2007. doi:10.1007/978-1-84628-766-4. ISBN 978-1-84628-765-7.
  16. ↑ Horváth, Tamás; Yamamoto, Akihiro, eds. (2003). "Inductive Logic Programming". Lecture Notes in Computer Science 2835. doi:10.1007/b13700. ISBN 978-3-540-20144-1.
  17. ↑ Deng,H.; Runger, G.; Tuv, E. (2011). Bias of importance measures for multi-valued attributes and solutions. Proceedings of the 21st International Conference on Artificial Neural Networks (ICANN). pp. 293–300.
  18. ↑ http://citeseer.ist.psu.edu/oliver93decision.html
  19. ↑ Tan & Dowe (2003)
  20. ↑ Papagelis A., Kalles D.(2001). Breeding Decision Trees Using Evolutionary Techniques, Proceedings of the Eighteenth International Conference on Machine Learning, p.393-400, June 28-July 01, 2001
  21. ↑ Barros, Rodrigo C., Basgalupp, M. P., Carvalho, A. C. P. L. F., Freitas, Alex A. (2011). A Survey of Evolutionary Algorithms for Decision-Tree Induction. IEEE Transactions on Systems, Man and Cybernetics, Part C: Applications and Reviews, vol. 42, n. 3, p. 291-312, May 2012.
  22. ↑ Chipman, Hugh A., Edward I. George, and Robert E. McCulloch. "Bayesian CART model search." Journal of the American Statistical Association 93.443 (1998): 935-948.
  23. ↑ Barros R. C., Cerri R., Jaskowiak P. A., Carvalho, A. C. P. L. F., A bottom-up oblique decision tree induction algorithm. Proceedings of the 11th International Conference on Intelligent Systems Design and Applications (ISDA 2011).

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