Decision tree learning: Difference between revisions

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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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* 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 = 1–32 | 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>

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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