# Homework #5 Solution

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## Description

Notes:

You may work in groups of up to two persons on this homework. Each group only needs to submit one copy of the homework. Check the following link on how to submit group as-signments on Gradescope. Make sure you follow the instructions and assign team members correctly. https://www.youtube.com/watch?v=a6DERS94qPY&feature=youtu.be&t=32s

Please check the submission instructions for Gradescope provided on the course website. You must follow those instructions exactly.

Homework is due by 11:59 PM on the due date. Remember that you may not use more than 2 late days on any one homework, and you only have a budget of 5 in total.

Please keep in mind the collaboration policy as specified in the course syllabus. If you dis-cuss questions with others you must write their names on your submission, and if you use any outside resources you must reference them. Do not look at each others’ writeups, in-cluding code.

Please do not directly post your answers on Piazza even if you think they might be wrong. Please try to frame the question such that you dont give the answers away. If there is specific information you want to ask about your answers, try the office hours or private posts on Piazza.

There are 2 problems on 2 pages in this homework.

Problems:

1. (50 points) The purpose of this problem is to write code for bagging decision trees and com-puting the out-of-bag error. You may use matlab’s inbuilt fitctree function, which learns decision trees using the CART algorithm (read the documentation carefully), but do not use the inbuilt functions for producing bagged ensembles. In order to do this, you should complete the stub BaggedTrees function. Note that it only returns the out-of-bag error. You may want to use other functions that actually construct and maintain the ensemble. You may assume that all the x vectors in the input are vectors of real numbers, and there are no categorical variables/features. You will compare the performance of the bagging

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method with plain decision trees on the handwritten digit recognition problem (the dataset is in zip.train and zip.test, available from http://amlbook.com/support.html. 1) We will focus on two specific problems distinguishing between the digit one and the digit three, and distinguishing between the digit three and the digit five. Here are the steps for this problem:

• Complete the implementation of BaggedTrees. You may choose any reasonable rep-resentation that you wish; the two strict requirements are that you plot the out-of-bag error as a function of the number of bags from 1 to the number specified as in-put (numBags), and that you return the out-of-bag error for the whole ensemble of numBags trees. Include the plots (with clearly labeled axes) in your writeup, and, of course, submit your code.

• Run the provided OneThreeFive script, which creates training datasets based on the one-vs-three and three-vs-five cases we are interested in, and calls both the in-built decision tree routine and your bagging code, printing out the cross-validation error for decision trees and the OOB error for your bagging implementation. Report the results in your writeup.

• Now, learn a single decision tree model for each of the two specified problems (one-vs-three and three-vs-five) on the training data, and test their performance on zip.test what is the test error? Similarly, learn a single ensemble of 200 trees on the training data for each of the two specified problems and test the performance of the ensembles on the test data. Report your results.

• Summarize and interpret your results in one or two concise paragraphs as part of your writeup.

1. (50 points) Implement AdaBoost using decision stumps learned using information gain as the weak learners (you may use the fitctree function to implement the weak learner. Look at the ”deviance” split criterion), and apply this to one-vs-three and three-vs-five prob-lems (as described in Question 1) on the zip.train and zip.test data. In order to do this, you should complete the stub AdaBoost function. Graphically report the training set error and the test set error as a function of the number of weak hypotheses, and summarize and interpret your results.

• Check the links to “training set” and “test set”.

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