1 Peter Fox Data Analytics – ITWS-4963/ITWS-6965 Week 11a, April 7, 2014 Support Vector Machines, Decision Trees, Cross- validation

Reading? 2

Probabilities… library(kernlab) data(promotergene) ## create test and training set ind <- sample(1:dim(promotergene)[1],20) genetrain <- promotergene[-ind, ] genetest <- promotergene[ind, ] ## train a support vector machine gene <- ksvm(Class~.,data=genetrain,kernel="rbfdot",\ kpar=list(sigma=0.015),C=70,cross=4,prob.model=TRUE) ## predict gene type probabilities on the test set genetype <- predict(gene,genetest,type="probabilities") 3

Result > genetype + - [1,] [2,] [3,] [4,] [5,] [6,] [7,] [8,] [9,] [10,] [11,] [12,] [13,] [14,] [15,] [16,] [17,] [18,] [19,] [20,]

Glass library(e1071) library(rpart) data(Glass, package="mlbench") index <- 1:nrow(Glass) testindex <- sample(index, trunc(length(index)/3)) testset <- Glass[testindex,] trainset <- Glass[-testindex,] svm.model <- svm(Type ~., data = trainset, cost = 100, gamma = 1) svm.pred <- predict(svm.model, testset[,-10]) 5

> table(pred = svm.pred, true = testset[,10]) true pred

Now what? # now what happens? > rpart.model <- rpart(Type ~., data = trainset) > rpart.pred <- predict(rpart.model, testset[,-10], type = "class”) 7

General idea behind trees Although the basic philosophy of all the classifiers based on decision trees is identical, there are many possibilities for its construction. Among all the key points in the selection of an algorithm to build decision trees some of them should be highlighted for their importance: –Criteria for the choice of feature to be used in each node –How to calculate the partition of the set of examples –When you decide that a node is a leaf –What is the criterion to select the class to assign to each leaf 8

Some important advantages can be pointed to the decision trees, including: –Can be applied to any type of data –The final structure of the classifier is quite simple and can be stored and handled in a graceful manner –Handles very efficiently conditional information, subdividing the space into sub-spaces that are handled individually –Reveal normally robust and insensitive to misclassification in the training set –The resulting trees are usually quite understandable and can be easily used to obtain a better understanding of the phenomenon in question. This is perhaps the most important of all the advantages listed 9

Stopping – leaves on the tree A number of stopping conditions can be used to stop the recursive process. The algorithm stops when any one of the conditions is true: –All the samples belong to the same class, i.e. have the same label since the sample is already "pure" –Stop if most of the points are already of the same class. This is a generalization of the first approach, with some error threshold –There are no remaining attributes on which the samples may be further partitioned –There are no samples for the branch test attribute 10

Recursive partitioning Recursive partitioning is a fundamental tool in data mining. It helps us explore the structure of a set of data, while developing easy to visualize decision rules for predicting a categorical (classification tree) or continuous (regression tree) outcome. The rpart programs build classification or regression models of a very general structure using a two stage procedure; the resulting models can be represented as binary trees. 11

Recursive partitioning The tree is built by the following process: –first the single variable is found which best splits the data into two groups ('best' will be defined later). The data is separated, and then this process is applied separately to each sub-group, and so on recursively until the subgroups either reach a minimum size or until no improvement can be made. –second stage of the procedure consists of using cross-validation to trim back the full tree. 12

Why are we careful doing this? Because we will USE these trees, i.e. apply them to make decisions about what things are and what to do with them! 13

> printcp(rpart.model) Classification tree: rpart(formula = Type ~., data = trainset) Variables actually used in tree construction: [1] Al Ba Mg RI Root node error: 92/143 = n= 143 CP nsplit rel error xerror xstd

plotcp(rpart.model) 15

> rsq.rpart(rpart.model) Classification tree: rpart(formula = Type ~., data = trainset) Variables actually used in tree construction: [1] Al Ba Mg RI Root node error: 92/143 = n= 143 CP nsplit rel error xerror xstd Warning message: In rsq.rpart(rpart.model) : may not be applicable for this method 16

rsq.rpart 17

> print(rpart.model) n= 143 node), split, n, loss, yval, (yprob) * denotes terminal node 1) root ( ) 2) Ba< ( ) 4) Al< ( ) 8) RI>= ( ) 16) RI< ( ) * 17) RI>= ( ) 34) RI>= ( ) 68) Mg>= ( ) * 69) Mg< ( ) * 35) RI< ( ) * 9) RI< ( ) * 5) Al>= ( ) 10) Mg>= ( ) * 11) Mg< ( ) * 3) Ba>= ( ) * 18

Tree plot 19 plot(object, uniform=FALSE, branch=1, compress=FALSE, nspace, margin=0, minbranch=.3, args) > plot(rpart.model,compress=TRUE) > text(rpart.model, use.n=TRUE)

And if you are brave summary(rpart.model) … pages…. 20

Wait, did anyone LOOK at the data? > names(Glass) [1] "RI" "Na" "Mg" "Al" "Si" "K" "Ca" "Ba" "Fe" "Type" > head(Glass) RI Na Mg Al Si K Ca Ba Fe Type

rpart.pred > rpart.pred Levels:

plot(rpart.pred) 23

Hierarchical clustering 24 > dswiss <- dist(as.matrix(swiss)) > hs <- hclust(dswiss) > plot(hs)

ctree 25 require(party) swiss_ctree <- ctree(Fertility ~ Agriculture + Education + Catholic, data = swiss) plot(swiss_ctree)

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hclust for iris 27

plot(iris_ctree) 28

Ctree > iris_ctree <- ctree(Species ~ Sepal.Length + Sepal.Width + Petal.Length + Petal.Width, data=iris) > print(iris_ctree) Conditional inference tree with 4 terminal nodes Response: Species Inputs: Sepal.Length, Sepal.Width, Petal.Length, Petal.Width Number of observations: 150 1) Petal.Length <= 1.9; criterion = 1, statistic = )* weights = 50 1) Petal.Length > 1.9 3) Petal.Width <= 1.7; criterion = 1, statistic = ) Petal.Length <= 4.8; criterion = 0.999, statistic = )* weights = 46 4) Petal.Length > 4.8 6)* weights = 8 3) Petal.Width > 1.7 7)* weights = 46 29

> plot(iris_ctree, type="simple”) 30

New dataset to work with trees fitK <- rpart(Kyphosis ~ Age + Number + Start, method="class", data=kyphosis) printcp(fitK) # display the results plotcp(fitK) # visualize cross-validation results summary(fitK) # detailed summary of splits # plot tree plot(fitK, uniform=TRUE, main="Classification Tree for Kyphosis") text(fitK, use.n=TRUE, all=TRUE, cex=.8) # create attractive postscript plot of tree post(fitK, file = “kyphosistree.ps", title = "Classification Tree for Kyphosis") # might need to convert to PDF (distill) 31

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33 > pfitK<- prune(fitK, cp= fitK$cptable[which.min(fitK$cptable[,"xerror"]),"CP"]) > plot(pfitK, uniform=TRUE, main="Pruned Classification Tree for Kyphosis") > text(pfitK, use.n=TRUE, all=TRUE, cex=.8) > post(pfitK, file = “ptree.ps", title = "Pruned Classification Tree for Kyphosis”)

34 > fitK <- ctree(Kyphosis ~ Age + Number + Start, data=kyphosis) > plot(fitK, main="Conditional Inference Tree for Kyphosis”)

35 > plot(fitK, main="Conditional Inference Tree for Kyphosis",type="simple")

randomForest > require(randomForest) > fitKF <- randomForest(Kyphosis ~ Age + Number + Start, data=kyphosis) > print(fitKF) # view results Call: randomForest(formula = Kyphosis ~ Age + Number + Start, data = kyphosis) Type of random forest: classification Number of trees: 500 No. of variables tried at each split: 1 OOB estimate of error rate: 20.99% Confusion matrix: absent present class.error absent present > importance(fitKF) # importance of each predictor MeanDecreaseGini Age Number Start Random forests improve predictive accuracy by generating a large number of bootstrapped trees (based on random samples of variables), classifying a case using each tree in this new "forest", and deciding a final predicted outcome by combining the results across all of the trees (an average in regression, a majority vote in classification).

More on another dataset. # Regression Tree Example library(rpart) # build the tree fitM <- rpart(Mileage~Price + Country + Reliability + Type, method="anova", data=cu.summary) printcp(fitM) # display the results …. Root node error: /60 = n=60 (57 observations deleted due to missingness) CP nsplit rel error xerror xstd

Mileage… plotcp(fitM) # visualize cross-validation results summary(fitM) # detailed summary of splits 38

39 par(mfrow=c(1,2)) rsq.rpart(fitM) # visualize cross-validation results

# plot tree plot(fitM, uniform=TRUE, main="Regression Tree for Mileage ") text(fitM, use.n=TRUE, all=TRUE, cex=.8) # prune the tree pfitM<- prune(fitM, cp= ) # from cptable # plot the pruned tree plot(pfitM, uniform=TRUE, main="Pruned Regression Tree for Mileage") text(pfitM, use.n=TRUE, all=TRUE, cex=.8) 40

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# Conditional Inference Tree for Mileage fit2M <- ctree(Mileage~Price + Country + Reliability + Type, data=na.omit(cu.summary)) 42

Example n <- 150 # number of data points p <- 2 # dimension sigma <- 1 # variance of the distribution meanpos <- 0 # centre of the distribution of positive examples meanneg <- 3 # centre of the distribution of negative examples npos <- round(n/2) # number of positive examples nneg <- n-npos # number of negative examples # Generate the positive and negative examples xpos <- matrix(rnorm(npos*p,mean=meanpos,sd=sigma),npos,p) xneg <- matrix(rnorm(nneg*p,mean=meanneg,sd=sigma),npos,p) x <- rbind(xpos,xneg) # Generate the labels y <- matrix(c(rep(1,npos),rep(-1,nneg))) 43

Train/ test ntrain <- round(n*0.8) # number of training examples tindex <- sample(n,ntrain) # indices of training samples xtrain <- x[tindex,] xtest <- x[-tindex,] ytrain <- y[tindex] ytest <- y[-tindex] istrain=rep(0,n) istrain[tindex]=1 44

Example svp <- ksvm(xtrain,ytrain,type="C-svc", kernel='vanilladot', C=100,scaled=c()) > alpha(svp) [[1]] [1] > svp Support Vector Machine object of class "ksvm” SV type: C-svc (classification) parameter : cost C = 100 Linear (vanilla) kernel function. Number of Support Vectors : 8 Objective Function Value : Training error :

Example > alphaindex(svp) [[1]] [1] > b(svp) [1] plot(svp,data=xtrain) 46

Cross-validation Cross-validation is a model validation technique for assessing how the results of a statistical analysis will generalize to an independent data set. It is mainly used in settings where the goal is prediction, and one wants to estimate how accurately a predictive model will perform in practice. I.e. predictive and prescriptive analytics… 47

Cross-validation In a prediction problem, a model is usually given a dataset of known data on which training is run (training dataset), and a dataset of unknown data (or first seen data) against which the model is tested (testing dataset). Sound familiar? 48

Cross-validation The goal of cross validation is to define a dataset to "test" the model in the training phase (i.e., the validation dataset), in order to limit problems like overfitting And, give an insight on how the model will generalize to an independent data set (i.e., an unknown dataset, for instance from a real problem), etc. 49

Common type of x-validation K-fold 2-fold (do you know this one?) Rep-random-subsample Leave out-subsample Lab on Friday… to try these out 50

Admin info (keep/ print this slide) Class: ITWS-4963/ITWS 6965 Hours: 12:00pm-1:50pm Tuesday/ Friday Location: SAGE 3101 Instructor: Peter Fox Instructor contact: (do not leave a Contact hours: Monday** 3:00-4:00pm (or by appt) Contact location: Winslow 2120 (sometimes Lally 207A announced by ) TA: Lakshmi Chenicheri Web site: –Schedule, lectures, syllabus, reading, assignments, etc. 51

Ozone > library(e1071) > library(rpart) > data(Ozone, package=“mlbench”) # beware of “()” > ## split data into a train and test set > index <- 1:nrow(Ozone) > testindex <- sample(index, trunc(length(index)/3)) > testset <- na.omit(Ozone[testindex,-3]) > trainset <- na.omit(Ozone[-testindex,-3]) > svm.model <- svm(V4 ~., data = trainset, cost = 1000, gamma = ) > svm.pred <- predict(svm.model, testset[,-3]) > crossprod(svm.pred - testset[,3]) / length(testindex) See: 52