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CART: Classification and Regression Trees Chris Franck LISA Short Course March 26, 2013

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Outline Overview of LISA Overview of CART Classification tree description – Examples – iris and skull data. Regression tree description – Examples – simulated and car data Going further – Mention cross validation, pruning, cost-complexity

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In addition to CART, these statistical and practical principals will be discussed R programming. Importance of exploratory data analysis. Use trees to predict outcomes for newly collected data. Graphical Comparison with regression. Performance assessment on simulated data. Importance of model validation (brief).

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Laboratory for Interdisciplinary Statistical Analysis Collaboration: Visit our website to request personalized statistical advice and assistance with: Experimental Design Data Analysis Interpreting Results Grant Proposals Software (R, SAS, JMP, SPSS...) LISA statistical collaborators aim to explain concepts in ways useful for your research. Great advice right now: Meet with LISA before collecting your data. All services are FREE for VT researchers. We assist with research—not class projects or homework. LISA helps VT researchers benefit from the use of Statistics www.lisa.stat.vt.edu LISA also offers: Educational Short Courses: Designed to help graduate students apply statistics in their research Walk-In Consulting: M-F 1-3 PM GLC Video Conference Room for questions requiring <30 mins Also 3-5 PM Port (Library/Torg Bridge) and 9-11 AM ICTAS Café X 4

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Tree-based methods

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The above idea is simple, although some of the language surrounding CART can sound technical.

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How can CART divide the data space?

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Example 1 iris data In Rstudio, type ‘?iris’ (no quotes) to open the help file on the iris data. Preceding built-in data objects or functions with ‘?’ in R opens the help file. Install the ‘tree’ package – Tools -> Install Packages… -> type ‘tree’ -> click install Open ‘CART course code’

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Iris data review

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Tree splits are chosen to minimize Deviance at each step

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Another example – Tibetan skulls Description from Hand et. al. (1996).

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Skull data review We grew another classification tree Predicted an outcome based on new data Looked at deviance calculation

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Under the hood CART uses a greedy algorithm. At each step the chosen split is the one which. maximizes classification/ minimizes error. Similar to forward variable selection in regression. The splitting continues until nodes become “too small” or deviance explained by a new split is small relative to starting deviance (see ‘?tree.control’ for more details)

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Final two examples: Regression trees Similar to classification trees but for continuous outcomes. Simulated example – when we know the correct answer, does the method work? Motor trend car data

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Advantages of CART Can be used to characterize outcomes as a function of many predictors Simple, yet powerful. Tree can be visualized easily in high dimension. Classification is highly similar to regression in CART. (more similar than orginary least squares versus logistic regression in my opinion).

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Caveats of CART Trees tend to overfit data – We saw low classification error rates and good deviance performance for the data used to construct the tree. – Would the trees we built necessarily predict new irises, skulls, or cars as well? Small changes to input data could result in major changes to tree structure (homework).

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Cross validation is typically used to assess overfitting K-fold cross validation: A technique which assesses the predictive value of a model (tree in this case) for new data. – Split the data into k (say 10) parts. – Withhold one part (validation set), grow the tree using other 9 parts (training set). – Assess predictive accuracy on the validation part using the tree. – Repeat, holding all 10 parts out in turn.

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How big should the tree be? Too big will overfit data Too small might miss important structures Generally, cost-complexity pruning can be used. – Make a big tree (say until no node has more than 5 observations) – Consider all subtrees which can be achieved by pruning the big tree. Choose tree which satisfies a cost-complexity criterion (see ?prune.tree in R, references for more detail).

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Random Forests – another tree-based technique Basic idea is to sample data with replacement (i.e. bootstrap sample). 1/3 of each sample is left out. 2/3 of data used to build a tree, then performance of tree determined based on hold- out data Grow a large number of trees, each of which “votes” for a certain classification. See http://www.stat.berkeley.edu/~breiman/Random Forests/cc_home.htm

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References The Elements of Statistical Learning: Data Mining, Inference, and Prediction. Hastie T, Tibshirani R, Friedman J. Second Edition 2009. Morant GM. A First Study of the Tibetan Skull Biometrika, Vol. 14, No. 3/4 (Mar., 1923), pp. 193-260 Discussion of “deviance” http://stats.stackexchange.com/questions/6581/what- is-deviance-specifically-in-cart-rpart http://stats.stackexchange.com/questions/6581/what- is-deviance-specifically-in-cart-rpart Hand DJ, Daly F, McConway K, Lunn D, Ostrowski E. A Handbook of Small Data Sets, Chapman and Hall 1996.

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Chapter 7 Classification and Regression Trees

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