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Applications of Random Forest Algorithm

Rosie Zou1 Matthias Schonlau, Ph.D.2

1Department of Computer Science University of Waterloo

2Professor, Department of Statistics University of Waterloo

Rosie Zou, Matthias Schonlau, Ph.D. (Universities of Waterloo)Applications of Random Forest Algorithm 1 / 33

Outline

1 Mathematical Background Decision Trees Random Forest

2 Stata Syntax

3 Classification Example: Credit Card Default

4 Regression Example: Consumer Finance Survey

Rosie Zou, Matthias Schonlau, Ph.D. (Universities of Waterloo)Applications of Random Forest Algorithm 2 / 33

Decision Trees

A versitile way of solving both regression and classification problems

Non-parametric model that involves recursively partitioning dataset

Partitioning criterion and stopping condition are pre-determined, usually based on entropy

Entropy, or information entropy, is a representation of how much information is encoded by given data

Rosie Zou, Matthias Schonlau, Ph.D. (Universities of Waterloo)Applications of Random Forest Algorithm 3 / 33

Decision Trees

In information theory, we are interested in the quantity of information, which can be measured by the length of the binary representation of given data points and values

Hence it is intuitive to determine when/how to split a decision tree based on whether the split maximizes information gain

At each node of a decision tree, the entropy is given by the formula:

E = − c∑

i=1

pi × log(pi )

where pi represents the proportion of observations with class label i , i = {1 · · · c} The information gain of a split can be measured by:

IG = E (parent)− weighted sum of E (child nodes)

Rosie Zou, Matthias Schonlau, Ph.D. (Universities of Waterloo)Applications of Random Forest Algorithm 4 / 33

Decision Trees

Figure: Recursive Binary Partition of a 2-Dimensional Subspace

Rosie Zou, Matthias Schonlau, Ph.D. (Universities of Waterloo)Applications of Random Forest Algorithm 5 / 33

Decision Trees

Figure: A Graphical Representation of the Decision Tree in Previous Slide

Rosie Zou, Matthias Schonlau, Ph.D. (Universities of Waterloo)Applications of Random Forest Algorithm 6 / 33

Random Forest

A drawback of decision trees is that they are prone to over-fitting

Over-fitting = model becomes too catered towards the training data set and performs poorly on testing data

This will lead to low general predictive accuracy, herein referred to as generalization accuracy

Rosie Zou, Matthias Schonlau, Ph.D. (Universities of Waterloo)Applications of Random Forest Algorithm 7 / 33

Random Forest

One way to increase generalization accuracy is to only consider a subset of the samples and build many individual trees

Random Forest model is an ensemble tree-based learning algorithm; that is the algorithms averages predictions over many individual trees

The algorithm also utilizes bootstrap aggregating, also known as bagging, to reduce overfitting and improve generalization accuracy

Bagging refers to fitting each tree on a bootstrap sample rather than on the original sample

Rosie Zou, Matthias Schonlau, Ph.D. (Universities of Waterloo)Applications of Random Forest Algorithm 8 / 33

Random Forest

for i ← 1 to B by 1 do Draw a bootstrap sample with size N from the training data; while node size != minimum node size do

randomly select a subset of m predictor variables from total p; for j ← 1 to m do

if j-th predictor optimizes splitting criterion then split internal node into two child nodes; break;

end

end

end

end return the ensemble tree of all B sub-trees grown in the for loop;

Algorithm 1: Random Forest Algorithm

Rosie Zou, Matthias Schonlau, Ph.D. (Universities of Waterloo)Applications of Random Forest Algorithm 9 / 33

Random Forest

Prediction for a classification problem:

f̂ (x) = majority vote of all predicted classes over B trees

Prediction for a regression problem:

f̂ (x) = sum of all sub-tree predictions divided over B trees

Rosie Zou, Matthias Schonlau, Ph.D. (Universities of Waterloo)Applications of Random Forest Algorithm 10 / 33

Random Forest

The random forest algorithm gives a more accurate estimate of the error rate, as compared with decision trees

Error rate has been mathematically proven to always converge (Breiman, 2001)

Error rate for classification problems (or root mean-squared error for regression problems) is approximated by the out-of-bag error during the training process

In each tree of the random forest, the out-of-bag error is calculated based on predictions for observations that were not in the bootstrap sample for that tree

Finding parameters that would produce a low out-of-bag error is often a key consideration in parameter tuning

Rosie Zou, Matthias Schonlau, Ph.D. (Universities of Waterloo)Applications of Random Forest Algorithm 11 / 33

Stata Syntax

The Stata syntax to fit a random forest model is:

randomforest depvar indepvars [if] [in] , [ options ]

Post-estimation command:

predict newvar | varlist | stub* [if] [in] , [ pr ]

Rosie Zou, Matthias Schonlau, Ph.D. (Universities of Waterloo)Applications of Random Forest Algorithm 12 / 33

Classification Example

The following sample data comes from a 2009 research project on data mining techniques for the predictive accuracy of probability of default of credit card clients (Yeh & Lien, 2009)

Data Description:

30,000 observations, 23 variables, no missing values 11 out of 23 variables are categorical response variable = whether the card holder will default on his/her debt predictor variables contain demographic information and banking information such as credit limit and monthly bill amount

Rosie Zou, Matthias Schonlau, Ph.D. (Universities of Waterloo)Applications of Random Forest Algorithm 13 / 33

Classification Example

First we need to randomize the data. Then we can tune the iterations hyper parameter.

set seed 201807

gen u=uniform()

sort u

gen out_of_bag_error1 = .

gen validation_error = .

gen iter1 = .

local j = 0

Rosie Zou, Matthias Schonlau, Ph.D. (Universities of Waterloo)Applications of Random Forest Algorithm 14 / 33

Classification Example

forvalues i = 10(5)500{

local j = ‘j’ + 1

randomforest defaultpaymentnextmonth limit_bal sex ///

education marriage_enum* age pay* bill* in 1/15000, ///

type(class) iter(‘i’) numvars(1)

replace iter1 = ‘i’ in ‘j’

replace out_of_bag_error1 = ‘e(OOB_Error)’ in ‘j’

predict p in 15001/30000

replace validation_error = ‘e(error_rate)’ in ‘j’

drop p

}

Rosie Zou, Matthias Schonlau, Ph.D. (Universities of Waterloo)Applications of Random Forest Algorithm 15 / 33

Classification Example

Figure: Out of Bag Error and Validation Error vs. Number of Iterations

Rosie Zou, Matthias Schonlau, Ph.D. (Universities of Waterloo)Applications of Random Forest Algorithm 16 / 33

Classification Example

Upon a first look at the graph, we can see the OOB error and validation error level off after 400 iterations and stabilize at somewhere below 0.19. Next we can tune the hyper parameter numvars, and fix the number of iterations to be 500.

Rosie Zou, Matthias Schonlau, Ph.D. (Universities of Waterloo)Applications of Random Forest Algorithm 17 / 33

Classification Example

gen oob_error = .

gen nvars = .

gen val_error = .

local j = 0

forvalues i = 1(1)26{

local j = ‘j’ + 1

randomforest defaultpaymentnextmonth limit_bal sex ///

education marriage_enum* age pay* bill* ///

in 1/15000, type(class) iter(500) numvars(‘i’)

replace nvars = ‘i’ in ‘j’

replace oob_error = ‘e(OOB_Error)’ in ‘j’

predict p in 15001/30000

replace val_error = ‘e(error_rate)’ in ‘j’

drop p

}

Rosie Zou, Matthias Schonlau, Ph.D. (Universities of Waterloo)Applications of Random Forest Algorithm 18 / 33

Classification Example

Figure: Out of Bag Error and Validation Error vs. Number of Variables

Rosie Zou, Matthias Schonlau, Ph.D. (Universities of Waterloo)Applications of Random Forest Algorithm 19 / 33

Classification Example

Finalized model is:

randomforest defaultpaymentnextmonth limit_bal sex ///

education marriage_enum* age pay* bill* ///

in 1/15000, type(class) iter(1000) numvars(14)

Rosie Zou, Matthias Schonlau, Ph.D. (Universities of Waterloo)Applications of Random Forest Algorithm 20 / 33

Classification Example

Figure: Variable Importance Plot

Rosie Zou, Matthias Schonlau, Ph.D. (Universities of Waterloo)Applications of Random Forest Algorithm 21 / 33

Classification Example

Using the finalized model to predict against the test data (observations 15001 to 30000), the error rate is e ≈ 0.18053 In comparison, the testing error for