Decision trees are a highly useful visual aid in analyzing a series of predicted outcomes for a particular model. As such, it is often used as a supplement (or even alternative to) regression analysis in determining how a series of explanatory variables will impact the dependent variable.

In this particular example, we analyse the impact of explanatory variables of age, gender, miles, debt, and income on the dependent variable car sales.

Classification Problems and Decision Trees

Firstly, we load our dataset and create a response variable (which is used for the classification tree since we need to convert sales from a numerical to categorical variable):

We then create the training and test data (i.e. the data that we will use to create our model and then the data we will test this data against):

#Create training and test data inputData <- fullData[1:770, ] # training data testData <- fullData[771:963, ] # test data

Then, our classification tree is created:

#Classification Tree library(rpart) formula=response~Age+Gender+Miles+Debt+Income dtree=rpart(formula,data=inputData,method="class",control=rpart.control(minsplit=30,cp=0.001)) plot(dtree) text(dtree) summary(dtree) printcp(dtree) plotcp(dtree) printcp(dtree)

Note that the cp value is what indicates our desired tree size – we see that our X-val relative error is minimized when our size of tree value is 4. Therefore, the decision tree is created using the dtree variable by taking into account this variable.

summary(dtree) Call: rpart(formula = formula, data = inputData, method = "class", control = rpart.control(minsplit = 30, cp = 0.001)) n= 770 CP nsplit 1 0.496598639 0 2 0.013605442 1 3 0.008503401 6 4 0.001000000 10 rel error xerror 1 1.0000000 1.0000000 2 0.5034014 0.5170068 3 0.4353741 0.5646259 4 0.4013605 0.5442177 xstd 1 0.07418908 2 0.05630200 3 0.05854027 4 0.05759793

Tree Pruning

The decision tree is then “pruned”, where inappropriate nodes are removed from the tree to prevent overfitting of the data:

#Prune the Tree and Plot pdtree<- prune(dtree, cp=dtree$cptable[which.min(dtree$cptable[,"xerror"]),"CP"]) plot(pdtree, uniform=TRUE, main="Pruned Classification Tree For Sales") text(pdtree, use.n=TRUE, all=TRUE, cex=.8)

The model is now tested against the test data, and we see that we have a misclassification percentage of 16.75%. Clearly, the lower the better, since this indicates our model is more accurate at predicting the “real” data:

#Model Testing out table(out[1:193],testData$response) response_predicted response_input mean(response_input != response_predicted) # misclassification % [1] 0.2844156

Solving Regression Problems With Decision Trees

When the dependent variable is numerical rather than categorical, we will want to set up a regression tree instead as follows:

#Regression Tree fitreg <- rpart(CarSales~Age+Gender+Miles+Debt+Income, method="anova", data=inputData) printcp(fitreg) plotcp(fitreg) summary(fitreg) par(mfrow=c(1,2)) rsq.rpart(fitreg) # cross-validation results





#Regression Tree fitreg printcp(fitreg) Regression tree: rpart(formula = CarSales ~ Age + Gender + Miles + Debt + Income, data = inputData, method = "anova") Variables actually used in tree construction: [1] Age Debt Income Root node error: 6.283e+10/770 = 81597576 n= 770 CP nsplit rel error 1 0.698021 0 1.00000 2 0.094038 1 0.30198 3 0.028161 2 0.20794 4 0.023332 4 0.15162 5 0.010000 5 0.12829 xerror xstd 1 1.00162 0.033055 2 0.30373 0.016490 3 0.21261 0.012890 4 0.18149 0.013298 5 0.14781 0.013068

plotcp(fitreg) summary(fitreg) Call: rpart(formula = CarSales ~ Age + Gender + Miles + Debt + Income, data = inputData, method = "anova") n= 770 CP nsplit rel error 1 0.69802077 0 1.0000000 2 0.09403824 1 0.3019792 3 0.02816107 2 0.2079410 4 0.02333197 4 0.1516189 5 0.01000000 5 0.1282869 xerror xstd 1 1.0016159 0.03305536 2 0.3037301 0.01649002 3 0.2126110 0.01289041 4 0.1814939 0.01329778 5 0.1478078 0.01306756 Variable importance Debt Miles Income Age 53 23 20 4

Now, we prune our regression tree:

#Prune the Tree pfitreg<- prune(fitreg, cp=fitreg$cptable[which.min(fitreg$cptable[,"xerror"]),"CP"]) # from cptable plot(pfitreg, uniform=TRUE, main="Pruned Regression Tree for Sales") text(pfitreg, use.n=TRUE, all=TRUE, cex=.8)

Random Forests

However, what if we have many decision trees that we wish to fit without preventing overfitting? A solution to this is to use a random forest.

A random forest allows us to determine the most important predictors across the explanatory variables by generating many decision trees and then ranking the variables by importance.

library(randomForest) fitregforest print(fitregforest) # view results Call: randomForest(formula = CarSales ~ Age + Gender + Miles + Debt + Income, data = inputData) Type of random forest: regression Number of trees: 500 No. of variables tried at each split: 1 Mean of squared residuals: 10341022 % Var explained: 87.33 > importance(fitregforest) # importance of each predictor IncNodePurity Age 5920357954 Gender 187391341 Miles 10811341575 Debt 21813952812 Income 12694331712

From the above, we see that debt is ranked as the most important factor, i.e. customers with high debt levels will be more likely to spend a greater amount on a car. We see that 87.33% of the variation is “explained” by our random forest, and our error is minimized at roughly 100 trees.