Part 1: to be completed at home before the lab

In this lab at home, two different classification methods will be covered: K-nearest neighbours and logistic regression. You can download the student zip including all needed files for practical 5 here.

Note: the completed homework has to be handed in on Black Board and will be graded (pass/fail, counting towards your grade for individual assignment). The deadline is two hours before the start of your lab. Hand-in should be a PDF or HTML file. If you know how to knit pdf files, you can hand in the knitted pdf file. However, if you have not done this before, you are advised to knit to a html file as specified below, and within the html browser, ‘print’ your file as a pdf file.

One of the packages we are going to use is class. For this, you will probably need to install.packages("class") before running the library() functions. In addition, you will again need the caret package to create a training and a validation split for the used dataset (note: to keep this at home lab compact, we will only use a training and validation split, and omit the test dataset to evaluate model fit). You can download the student zip including all needed files for practical 5 here.

library(MASS)
library(class)
library(caret)
library(ISLR)
library(tidyverse)

This practical will be mainly based around the default dataset which contains credit card loan data for 10 000 people. With the goal being to classify credit card cases as yes or no based on whether they will default on their loan.

Create a scatterplot of the Default dataset, where balance is mapped to the x position, income is mapped to the y position, and default is mapped to the colour. Can you see any interesting patterns already?

Add facet_grid(cols = vars(student)) to the plot. What do you see?

For use in the KNN algorithm, transform “student” into a dummy variable using ifelse() (0 = not a student, 1 = student). Then, randomly split the Default dataset into a training set default_train (80%) and a validation set default_valid (20%) using the createDataPartition() function of the caret package.

If you haven’t used the function ifelse() before, please feel free to review it in Chapter 5 Control Flow (particular section 5.2.2) in Hadley Wickham’s Book Advanced R, this provides a concise overview of choice functions (if()) and vectorised if (ifelse()).

K-Nearest Neighbours

Now that we have explored the dataset, we can start on the task of classification. We can imagine a credit card company wanting to predict whether a customer will default on the loan so they can take steps to prevent this from happening.

The first method we will be using is k-nearest neighbours (KNN). It classifies datapoints based on a majority vote of the k points closest to it. In R, the class package contains a knn() function to perform knn.

Create class predictions on the default_valid data using the test parameter from the knn() function. Use student, balance, and income (but no basis functions of those variables) in the default_train dataset. Set k to 5. Store the predictions in a variable called knn_5_pred.

Remember: make sure to review the knn() function through the help panel on the GUI or through typing “?knn” into the console. For further guidance on the knn() function, please see Section 4.7.6 in An introduction to Statistical Learning

Create two scatter plots with income and balance as in the first plot you made but now using only the default_valid data. One with the true class (default) mapped to the colour aesthetic, and one with the predicted class (knn_5_pred) mapped to the colour aesthetic. Hint: Add the predicted class knn_5_pred to the default_valid dataset before starting your ggplot() call of the second plot. What do you see?

Repeat the same steps, but now with a knn_2_pred vector generated from a 2-nearest neighbours algorithm. Are there any differences?

During this we have manually tested two different values for K, this although useful in exploring your data. To know the optimal value for K, you should use cross validation.

Part 2: to be completed during the lab

Assessing classification

The confusion matrix is an insightful summary of the plots we have made and the correct and incorrect classifications therein. A confusion matrix can be made in R with the table() function by entering two factors:

conf_2NN <- table(predicted = knn_2_pred, true = default_valid$default)
conf_2NN

To learn more these, please see Section 4.4.3 in An Introduction to Statistical Learning, where it discusses Confusion Matrices in the context of another classification method Linear Discriminant Analysis (LDA).

What would this confusion matrix look like if the classification were perfect?

Make a confusion matrix for the 5-nn model and compare it to that of the 2-nn model. What do you conclude?

Comparing performance becomes easier when obtaining more specific measures. Calculate the specificity, sensitivity, accuracy and the precision of the 2nn and 5nn model, and compare them. Which model would you choose? Keep the goal of our prediction in mind when answering this question.

Logistic regression

KNN directly predicts the class of a new observation using a majority vote of the existing observations closest to it. In contrast to this, logistic regression predicts the log-odds of belonging to category 1. These log-odds can then be transformed to probabilities by performing an inverse logit transform:

$p = \frac{1}{1 + e^{-\alpha}}$

where $\alpha$; indicates log-odds for being in class 1 and $p$ is the probability.

Therefore, logistic regression is a probabilistic classifier as opposed to a direct classifier such as KNN: indirectly, it outputs a probability which can then be used in conjunction with a cutoff (usually 0.5) to classify new observations.

Logistic regression in R happens with the glm() function, which stands for generalized linear model. Here we have to indicate that the residuals are modeled not as a Gaussian (normal distribution), but as a binomial distribution.

Use glm() with argument family = binomial to fit a logistic regression model lr_mod to the default_train data. Use student, income and balance as predictors.

Now we have generated a model, we can use the predict() method to output the estimated probabilities for each point in the training dataset. By default predict outputs the log-odds, but we can transform it back using the inverse logit function of before or setting the argument type = "response" within the predict function.

Visualise the predicted probabilities versus observed class for the training dataset in lr_mod. You can choose for yourself which type of visualisation you would like to make. Write down your interpretations along with your plot.

Another advantage of logistic regression is that we get coefficients we can interpret.

Look at the coefficients of the lr_mod model and interpret the coefficient for balance. What would the probability of default be for a person who is not a student, has an income of 40000, and a balance of 3000 dollars at the end of each month? Is this what you expect based on the plots we’ve made before?

Let’s visualise the effect balance has on the predicted default probability.

Create a data frame called balance_df with 3 columns and 500 rows: student always 0, balance ranging from 0 to 3000, and income always the mean income in the default_train dataset.

Use this dataset as the newdata in a predict() call using lr_mod to output the predicted probabilities for different values of balance. Then create a plot with the balance_df$balance variable mapped to x and the predicted probabilities mapped to y. Is this in line with what you expect?

Use lr_mod to predict the probability of defaulting for the observations in the validation dataset. Use these to create a confusion matrix just as the one for the KNN models by using a cutoff predicted probability of 0.5. Does logistic regression perform better?

Calculate the specificity, sensitivity, accuracy and the precision for the logistic regression using the above confusion matrix. Again, compare the logistic regression to KNN.

Final exercise

Now let’s do another - slightly less guided - round of KNN and/or logistic regression on a new dataset in order to predict the outcome for a specific case. We will use the Titanic dataset also discussed in the lecture. The data can be found in the /data folder of your project. Before creating a model, explore the data, for example by using summary().

Create a model (using knn or logistic regression) to predict whether a 14 year old boy from the 3rd class would have survived the Titanic disaster.

Would the passenger have survived if they were a 14 year old girl in 2nd class?