Decision tree learning: Difference between revisions

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Tree models where the target variable can take a discrete set of values are called '''[[Statistical classification|classification]] [[decision tree|trees]]'''; in these tree structures, [[leaf node|leaves]] represent class labels and branches represent [[Logical conjunction|conjunction]]s of features that lead to those class labels. Decision trees where the target variable can take continuous values (typically [[real numbers]]) are called '''[[regression analysis|regression]] [[decision tree|trees]]'''. More generally, the concept of regression tree can be extended to any kind of object equipped with pairwise dissimilarities such as categorical sequences.<ref name=":1">{{Cite journal |last1=Studer |first1=Matthias |last2=Ritschard |first2=Gilbert |last3=Gabadinho |first3=Alexis |last4=Müller |first4=Nicolas S. |date=2011 |title=Discrepancy Analysis of State Sequences |url=http://journals.sagepub.com/doi/10.1177/0049124111415372 |journal=Sociological Methods & Research |language=en |volume=40 |issue=3 |pages=471–510 |doi=10.1177/0049124111415372 |s2cid=13307797 |issn=0049-1241}}</ref>

Decision trees are among the most popular machine learning algorithms given their intelligibility and simplicity because they produce algorithms that are easy to interpret and visualize, even for users without a statistical background.<ref>{{Cite journal|last1=Wu|first1=Xindong|last2=Kumar|first2=Vipin|last3=Ross Quinlan|first3=J.|last4=Ghosh|first4=Joydeep|last5=Yang|first5=Qiang|last6=Motoda|first6=Hiroshi|last7=McLachlan|first7=Geoffrey J.|last8=Ng|first8=Angus|last9=Liu|first9=Bing|last10=Yu|first10=Philip S.|last11=Zhou|first11=Zhi-Hua|date=2008-01-01|title=Top 10 algorithms in data mining|journal=Knowledge and Information Systems|language=en|volume=14|issue=1|pages=1–37|doi=10.1007/s10115-007-0114-2|s2cid=2367747|issn=0219-3116|hdl=10983/15329|hdl-access=free}}</ref>

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).

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|isbn=978-9814590075

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}}</ref> The goal is to create aan modelalgorithm 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 [[Feature (machine learning)|feature]]s 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''.

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The dependent variable, <math>Y</math>, is the target variable that we are trying to understand, classify or generalize. The vector <math>\textbf{x}</math> is composed of the features, <math>x_1, x_2, x_3</math> etc., that are used for that task.

[[File:Cart tree kyphosis.png|thumb|800px500x500px|

alt=Three different representations of a regression tree of kyphosis data|

An example tree which estimates the probability of

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]]

== Decision tree types ==

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

* '''[[Classification tree]]''' analysis is when the predicted outcome is the class (discrete) to which the data belongs.

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

* '''[[Gradient boosted trees|Boosted trees]]''' Incrementally building an ensemble by training each new instance to emphasize the training instances previously mis-modeled. A typical example is [[AdaBoost]]. These can be used for regression-type and classification-type problems.<ref>Friedman, J. H. (1999). ''[https://astro.temple.edu/~msobel/courses_files/StochasticBoosting(gradient).pdf Stochastic gradient boosting] {{Webarchive|url=https://web.archive.org/web/20181128041212/https://astro.temple.edu/~msobel/courses_files/StochasticBoosting(gradient).pdf |date=2018-11-28 }}.'' Stanford University.</ref><ref>Hastie, T., Tibshirani, R., Friedman, J. H. (2001). ''The elements of statistical learning : Data mining, inference, and prediction.'' New York: Springer Verlag.</ref>

* '''Committees of decision trees''' (also called k-DT<ref>Heath, D., Kasif, S. and Salzberg, S. (1993). ''k-DT: A multi-tree learning method.'' In <em>''Proceedings of the Second Intl. Workshop on Multistrategy Learning</em>'', pp. 138-149.</ref>), an early method that used randomized decision tree algorithms to generate multiple different trees from the training data, and then combine them using majority voting to generate output.<ref>Heath, D., Kasif, S., and Salzberg, S. L. (1996). ''Committees of decision trees.'' In B. Gorayska and J. Mey (Eds.), <em>''Cognitive Technology: In Search of a Humane Interface</em>'' (pp. 305–317). Amsterdam: Elsevier Science B.V.</ref>

* '''[[Bootstrap aggregating|Bootstrap aggregated]]''' (or bagged) decision trees, an early ensemble method, builds multiple decision trees by repeatedly [[Bootstrapping (statistics)|resampling training data with replacement]], and voting the trees for a consensus prediction.<ref>{{cite journal |last=Breiman |first=L. |year=1996 |title=Bagging Predictors |journal=Machine Learning |volume=24 |issue=2 |pages=123–140 |doi=10.1007/BF00058655 |doi-access=free }}</ref>

** A '''[[random forest]]''' classifier is a specific type of [[bootstrap aggregating]]

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* [[C4.5 algorithm|C4.5]] (successor of ID3)

* CART (Classification And Regression Tree)<ref name="bfos" />

* OC1 (Oblique classifier 1). First method that created multivariate splits at each node.<ref>{{Cite journal | doi = 10.1613/jair.63 | last1 = Murthy | first1 = S. K. | year = 1994 | title = A System for Induction of Oblique Decision Trees | journal = Journal of Artificial Intelligence Research | volume = 2 | issue = 1 | pages = 1-321–32 | doi-access = free }}</ref>

* [[Chi-square automatic interaction detection]] (CHAID). Performs multi-level splits when computing classification trees.<ref>{{Cite journal | doi = 10.2307/2986296 | last1 = Kass | first1 = G. V. | year = 1980 | title = An exploratory technique for investigating large quantities of categorical data | jstor = 2986296| journal = Applied Statistics | volume = 29 | issue = 2| pages = 119–127 }}</ref><ref>{{Cite journal|last1=Biggs|first1=David|last2=De Ville|first2=Barry|last3=Suen|first3=Ed|date=1991|title=A method of choosing multiway partitions for classification and decision trees|url=https://doi.org/10.1080/02664769100000005|journal=Journal of Applied Statistics|volume=18|issue=1|pages=49–62|doi=10.1080/02664769100000005|bibcode=1991JApSt..18...49B |issn=0266-4763|url-access=subscription}}</ref><ref>Ritschard, G. (2013), '''"'''CHAID and Earlier Supervised Tree Methods", in J.J. McArdle and G. Ritschard (eds), ''Contemporary Issues in Exploratory Data Mining in the Behavioral Sciences'', Quantitative Methodology Series, New York: Routledge, pages 48-74. [https://www.researchgate.net/publication/315476407_CHAID_and_Earlier_Supervised_Tree_Methods Preprint]</ref>

* [[Multivariate adaptive regression splines|MARS]]: extends decision trees to handle numerical data better.

* Conditional Inference Trees. Statistics-based approach that uses non-parametric tests as splitting criteria, corrected for multiple testing to avoid overfitting. This approach results in unbiased predictor selection and does not require pruning.<ref name="Hothorn2006">{{Cite journal | doi = 10.1198/106186006X133933 | last1 = Hothorn | first1 = T.| last2 = Hornik | first2 = K.| last3 = Zeileis | first3 = A.| year = 2006 | title = Unbiased Recursive Partitioning: A Conditional Inference Framework | jstor = 27594202| journal = Journal of Computational and Graphical Statistics | volume = 15 | issue = 3| pages = 651–674 | citeseerx = 10.1.1.527.2935 | s2cid = 6074128 }}</ref><ref name="Strobl2009">{{Cite journal | doi = 10.1037/a0016973 | pmid = 19968396 | pmc = 2927982 | last1 = Strobl | first1 = C.| last2 = Malley | first2 = J.| last3 = Tutz | first3 = G.| year = 2009 | title = An Introduction to Recursive Partitioning: Rationale, Application and Characteristics of Classification and Regression Trees, Bagging and Random Forests | journal = Psychological Methods | volume = 14 | issue = 4| pages = 323–348 }}</ref>

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Boosted ensembles of FDTs have been recently investigated as well, and they have shown performances comparable to those of other very efficient fuzzy classifiers.<ref name="Barsacchi2020">{{Cite journal | url=http://www.sciencedirect.com/science/article/pii/S0957417420302608 | doi=10.1016/j.eswa.2020.113436|

title=An analysis of boosted ensembles of binary fuzzy decision trees| journal=Expert Systems with Applications|

volume=154| year=2020| last1=Barsacchi| first1=M.|last2=Bechini| first2=A.|last3=Marcelloni| first3=F.| page=113436| s2cid=216369273| url-access=subscription}}</ref>

==Metrics==

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===Variance reduction===

Introduced in CART,<ref name="bfos"/> variance reduction is ofte

Introduced in CART,<ref name="bfos"/> 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 {{mvar|N}} is defined as the total reduction of the variance of the target variable {{mvar|Y}} due to the split at this node:

:<math>

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By replacing <math>(y_i - y_j)^2</math> in the formula above with the dissimilarity <math>d_{ij}</math> between two objects <math>i</math> and <math>j</math>, the variance reduction criterion applies to any kind of object for which pairwise dissimilarities can be computed.<ref name=":1" />

===Measure of "goodness"===

Used by CART in 1984,<ref name="ll">{{Cite book

|last=Larose