Homework 4

Description

Disclaimer: This set of homework applies SMOTE to a seriously imbalanced dataset with a large number of features and data points. SMOTE is essentially a time consuming method. You need to start doing this homework early, so that you have enough time to run SMOTE on the full dataset.

1. 1. The LASSO and Boosting for Regression

1. 1. 1. Download the Communities and Crime data¹ from https://archive.ics.uci. edu/ml/datasets/Communities+and+Crime. Use the rst 1495 rows of data as the training set and the rest as the test set.

1. 1. 2. The data set has missing values. Use a data imputation technique to deal with the missing values in the data set. The data description mentions some features are nonpredictive. Ignore those features.

1. 1. 3. Plot a correlation matrix for the features in the data set.

1. 1. 4. Calculate the Coe cient of Variation CV for each feature, where CV = _m^s , in which s is sample standard deviation and m is sample mean..

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1. 1. 5. Pick b 128c features with highest CV , and make scatter plots and box plots for them. Can you draw conclusions about signi cance of those features, just by the scatter plots?

1. 1. 6. Fit a linear model using least squares to the training set and report the test error.

1. 1. 7. Fit a ridge regression model on the training set, with chosen by cross-validation. Report the test error obtained.

1. 1. 8. Fit a LASSO model on the training set, with chosen by cross-validation. Report the test error obtained, along with a list of the variables selected by the model. Repeat with standardized² features. Report the test error for both cases and compare them.

1. 1. 9. Fit a PCR model on the training set, with M (the number of principal compo-nents) chosen by cross-validation. Report the test error obtained.

1. 1. 10. In this section, we would like to t a boosting tree to the data. As in classi cation trees, one can use any type of regression at each node to build a multivariate regression tree. Because the number of variables is large in this problem, one can use L₁-penalized regression at each node. Such a tree is called L₁ penalized gradient boosting tree. You can use XGBoost³ to t the model tree. Determine (the regularization term) using cross-validation.

¹Question you may encounter: I tried opening the dataset and download it but the le is not readable. How to download the le? Just change .data to .csv. .

²In this data set, features are already normalized.

³Some hints on installing XGBoost on Windows: http://www.picnet.com.au/blogs/guido/2016/09/ 22/xgboost-windows-x64-binaries-for-download/.

Homework 4 DSCI 552, Instructor: Mohammad Reza Rajati

2. Tree-Based Methods

(a) Download the APS Failure data from: https://archive.ics.uci.edu/ml/datasets/ APS+Failure+at+Scania+Trucks . The dataset contains a training set and a test set. The training set contains 60,000 rows, of which 1,000 belong to the positive class and 171 columns, of which one is the class column. All attributes are nu-meric.

(b) Data Preparation

This data set has missing values. When the number of data with missing values is signi cant, discarding them is not a good idea. ⁴

i. Research what types of techniques are usually used for dealing with data with missing values.⁵ Pick at least one of them and apply it to this data in the next steps.⁶

ii. For each of the 170 features, calculate the coe cient of variation CV = _m^s , where s is sample standard deviation and m is sample mean.

iii. Plot a correlation matrix for your features using pandas or any other tool. p

1. 4. Pick b 170c features with highest CV , and make scatter plots and box plots for them, similar to those on p. 129 of ISLR. Can you draw conclusions about signi cance of those features, just by the scatter plots? This does not mean that you will only use those features in the following questions. We picked them only for visualization.

1. 4. Determine the number of positive and negative data. Is this data set imbal-anced?

3. Train a random forest to classify the data set. Do NOT compensate for class imbalance in the data set. Calculate the confusion matrix, ROC, AUC, and misclassi cation for training and test sets and report them (You may use pROC package). Calculate Out of Bag error estimate for your random forset and compare it to the test error.

3. Research how class imbalance is addressed in random forests. Compensate for class imbalance in your random forest and repeat 2c. Compare the results with those of 2c.

3. Model Trees

In the case of a univariate tree, only one input dimension is used at a tree split. In a multivariate tree, or model tree, at a decision node all input dimensions can be used and thus it is more general. In univariate classi cation trees, majority polling is used at each node to determine the split of that node as the decision rule. In model trees, a (linear) model that relies on all of the variables is used

• In reality, wehn we have a model and we want to ll in missing values, we do not have access to training data, so we only use the statistics of test data to ll in the missing values. For simplicity, in this exercise, you rst ll in the missing values and then split your data to training and test sets.

⁵They are called data imputation techniques.

⁶You are welcome to test more than one method.

Homework 4 DSCI 552, Instructor: Mohammad Reza Rajati

to determine the split of that node (i.e. instead of using X_j > s as the decision

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rule, one has _{j j} X_j > s. as the decision rule). Alternatively, in a regression tree, instead of using average in the region associated with each node, a linear regression model is used to determine the value associated with that node.

One of the methods that can be used at each node is Logistic Regression. One can use scikit learn to call Weka⁷ to train Logistic Model Trees for classi cation. Train Logistic Model Trees for the APS data set without compensation for class imbalance. Use one of 5 fold, 10 fold, and leave-one-out cross validation methods to estimate the error of your trained model and compare it with the test error. Report the Confusion Matrix, ROC, and AUC for training and test sets.

1. 6. Use SMOTE (Synthetic Minority Over-sampling Technique) to pre-process your data to compensate for class imbalance.⁸ Train a Logistic Model Tree using the pre-processed data and repeat 2e. Do not forget that there is a right and a wrong way of cross validation here. Compare the uncompensated case with SMOTE.

3. ISLR 6.8.3

3. ISLR, 6.8.5

3. ISLR 8.4.5

3. ISLR 9.7.3

3. Extra Practice: ISLR 5.4.2, 6.8.4, 8.4.4, 9.7.2

Appendix

Weka for Mac users:

1. Download JDK 9 from http://www.oracle.com/technetwork/java/javase/downloads/ index.html

2. Add environment variables in Terminal using : vi_~/.bash_profile

1. 1. export JAVA_HOME=/Library/Java/JavaVirtualMachines/jdk-9.0.4.jdk/Contents/ Home

1. 2. export PATH=$JAVA_HOME/bin:$PATH

3. Restart Terminal

4. Get brew (package installer for Mac, if you don’t have it) and install python (not necessary)

⁷http://fracpete.github.io/python-weka-wrapper/install.html. may help.

• If you did not start doing this homework on time, downsample the common class to 6,000 so that you have 12,000 data points after applying SMOTE. Remember that the purpose of this homework is to apply SMOTE to the whole training set, not the downsampled dataset.

Homework 4 DSCI 552, Instructor: Mohammad Reza Rajati

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