Fitting Classification and Regression Trees Using Statgraphics and R. Presented by Dr. Neil W. Polhemus
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1 Fitting Classification and Regression Trees Using Statgraphics and R Presented by Dr. Neil W. Polhemus
2 Classification and Regression Trees Machine learning methods used to construct predictive models from data. Recursively partitions the data space using simple binary decisions. Commonly portrayed as a tree with a split at each decision node.
3 Example Fisher s Iris Data Species Petal.length<2.45 setosa (p=1.0) Petal.width<1.75 Petal.length<4.95 Petal.length<4.95 Sepal.length<5.15 virginica (p= ) virginica (p= ) virginica (p=1.0) versicolor (p=0.8) versicolor (p=1.0)
4 Basic Model Structure Y: variable to be predicted If categorical, we construct a classification tree. If continuous, we construct a regression tree. X 1, X 2, X p : predictor variables May be either categorical or continuous.
5 Start at the root node. Partitioning Algorithm Amongst all variables X j, find the split that minimizes the resulting average within-node deviance. For a continuous variable, the split is of the form X j < c. For a discrete variable, the split divides the possible values into 2 distinct groups. If one or more stopping criteria are met after the split, stop. Otherwise consider splitting the child nodes.
6 RMS Titanic
7 Sample Data File n=1,309 observations (passengers only) Source: Frank Harrell and Thomas Cason, University of Virginia
8 Data Input
9 Analysis Options
10 Partitioning Options
11 de Impurity Measure the impurity in a tree using the residual mean deviance (RMD). n = number of observations in training set k = number of leaves p i,j = proportion of data of same type as i at its assigned leaf j Y i = predicted value of observation i Classification trees Regression trees
12 Analysis Window
13 Decision Tree survived sex=female pclass=3 age<9.5 fare< sibsp=3,4,5 pclass=2,3
14 Decision Tree Options
15 Tree Structure * te: based on complete cases only.
16 de Probabilities * te: based on complete cases only.
17 Classification Table * te: based on both complete and partial cases.
18 Training and Validation Sets May separate the data into 2 sets: Training set used to build the tree. Test set used to estimate the tree misclassification percentages.
19 Compare Results
20 Pruning Options Reduces the complexity of the tree by removing branches.
21 Pruning by Cross-validation Runs a 10-fold cross-validation experiment. Builds 10 trees, leaving out 10% of the data each time, and averages the results. Uses all of the observations for both training and validation. Can be used to determine the optimal size for the tree by increasing the number of leaves until you see warnings such as:
22 Pruning Example First reduce within-node deviance to fit a complex tree.
23 Pruning Example surv iv ed sex=fem pclass=3 age<9.5 fare< fare< sibsp=0,1,2 pclass=1 embarked=q,sfare< parch=2,3 fare< age<3.5 fare< age<32.25 age<54.5 age<27.5 fare< age<17.5 fare< sibsp=1,2,3 age<29.5 fare< sibsp=0,2,3 sibsp=1 fare< sibsp=0,1,3 fare< fare< fare<86.35 age<31.5 fare< age<43.0 fare< fare<29.85 fare< parch=0 fare<15.7 fare< age<42.5 embarked=q age<31.5 fare< fare< fare< age<21.5 fare< fare<22.0 age<25.0 fare<13.25 age<37.0 embarked=q,s age<36.25 fare< age<42.5 fare< fare< fare< age<39.5 age<36.5 fare< age<33.5 fare< age<20.75 fare< pclass=3 fare<13.25 age<33.5 fare< fare< fare< age<25.75 fare<11.0 age<46.0 fare< fare< fare< age<23.5 age<25.5 age<21.5 age<59.0 fare< fare< age<32.5 fare< age<27.5 age<19.5 parch=1 fare< fare< fare< age<21.5
24 Pruning Example w reduce the number of leaves to 10 and select cross-validation. surv iv ed sex=female pclass=3 age<9.5 fare< fare< sibsp=3,4,5 pclass=2,3 age<32.25 age<54.5
25 10 Leaves is Too Complex
26 Finding Optimal Number of Leaves Step 1: Copy script from Statgraphics to Word and modify last line.
27 Finding Optimal Number of Leaves Step 2: Copy modified script to R and run it. Look for size with minimum deviance.
28 Prune Tree
29 Final Tree survived sex=female pclass=3 age<9.5 sibsp=3,4,5
30 Predict Additional Cases To make predictions for additional cases, add them to the bottom of the original data, leaving the cell for Y blank.
31 Predictions and Residuals
32 Example 2: World Bank Demographics
33 Decision Tree Life Expectancy Fertility.Rate<3.3 GDP.per.Capita< Fertility.Rate<4.585 GDP.per.Capita< Fem ale.percentage<49.97 Pop..Density< Pop..Density<40.37 Age.Dependency.Ratio<72.925
34 References StatFolios and data files are at: R Package tree (2015) Classic text: Brieman, L., Friedman, J., Stone, C.J. and Olshen, R.A. (1998) Classification and Regression Trees. Wadsworth.
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