Supervised learning: classification

Introduction

In this practical, we will learn to use three different classification methods: K-nearest neighbours, logistic regression, and linear discriminant analysis.

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.

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

Make sure to load MASS before tidyverse otherwise the function MASS::select() will overwrite dplyr::select()

Default dataset

The default dataset contains credit card loan data for 10 000 people. The goal is to classify credit card cases as yes or no based on whether they will default on their loan.

1. 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?

Default |>  
  arrange(default) |> # so the yellow dots are plotted after the blue ones
  ggplot(aes(x = balance, y = income, colour = default)) +
  geom_point(size = 1.3) +
  theme_minimal() +
  scale_colour_viridis_d() # optional custom colour scale

# People with high remaining balance are more likely to default. 
# There seems to be a low-income group and a high-income group

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

Default |> 
  arrange(default) |> # so the yellow dots are plotted after the blue ones
  ggplot(aes(x = balance, y = income, colour = default)) +
  geom_point(size = 1.3) +
  theme_minimal() +
  scale_colour_viridis_d() +
  facet_grid(cols = vars(student))

# The low-income group is students!

3. 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 test set default_test (20%)

default_df <- 
  Default |> 
  mutate(student = ifelse(student == "Yes", 1, 0)) |> 
  mutate(split = sample(rep(c("train", "test"), times = c(8000, 2000))))

default_train <- 
  default_df |> 
  filter(split == "train") |> 
  select(-split)

default_test <- 
  default_df |> 
  filter(split == "test") |> 
  select(-split)

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.

4. Create class predictions for the test set using the knn() function.

Use student, balance, and income (but no functions of those variables) in the default_train dataset. Set k to 5. Store the predictions in a variable called knn_5_pred.

knn_5_pred <- knn(
  train = default_train |> select(-default),
  test  = default_test  |> select(-default),
  cl    = as_factor(default_train$default),
  k     = 5
)

5. Create two scatter plots with income and balance as in the first plot you made. 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_test dataset before starting your ggplot() call of the second plot. What do you see?

# first plot is the same as before
default_test |> 
  arrange(default) |> 
  ggplot(aes(x = balance, y = income, colour = default)) +
  geom_point(size = 1.3) + 
  scale_colour_viridis_d() +
  theme_minimal() +
  labs(title = "True class")

# second plot maps pred to colour
bind_cols(default_test, pred = knn_5_pred) |> 
  arrange(default) |> 
  ggplot(aes(x = balance, y = income, colour = pred)) +
  geom_point(size = 1.3) + 
  scale_colour_viridis_d() +
  theme_minimal() +
  labs(title = "Predicted class (5nn)")

# there are quite some misclassifications, esp. many "No" predictions
# with "Yes" true class.

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

knn_2_pred <- knn(
  train = default_train |> select(-default),
  test  = default_test  |> select(-default),
  cl    = as_factor(default_train$default),
  k     = 2
)

# second plot maps pred to colour
bind_cols(default_test, pred = knn_2_pred) |> 
  arrange(default) |> 
  ggplot(aes(x = balance, y = income, colour = pred)) +
  geom_point(size = 1.3) + 
  scale_colour_viridis_d() +
  theme_minimal() +
  labs(title = "Predicted class (2nn)")

# compared to the 5-nn model, more people get classified as "Yes"
# Still, the method is not perfect

Confusion matrix

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 factor or character vectors:

table(
  observed  = default_test$default, 
  predicted = knn_2_pred
)

        predicted
observed   No  Yes
     No  1899   31
     Yes   55   15

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

# All the observations would fall in the yes-yes or no-no categories; 
# the off-diagonal elements would be 0 like so:

table(
  observed  = default_test$default, 
  predicted = default_test$default
)

        predicted
observed   No  Yes
     No  1930    0
     Yes    0   70

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

table(
  observed  = default_test$default, 
  predicted = knn_5_pred
)

        predicted
observed   No  Yes
     No  1922    8
     Yes   61    9

# the 2nn model has more true positives (yes-yes) but also more false
# positives (truly no but predicted yes). Overall the 5nn method has 
# slightly better accuracy (proportion of correct classifications).

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.

9. Use glm() with argument family = binomial to fit a logistic regression model lr_mod to the default_train data.

lr_mod <- glm(default ~ ., family = binomial, data = default_train)

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.

10. 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.

tibble(observed  = default_train$default, 
       predicted = predict(lr_mod, type = "response")) |> 
  ggplot(aes(y = predicted, x = observed, colour = observed)) +
  geom_point(position = position_jitter(width = 0.2), alpha = .3) +
  scale_colour_manual(values = c("dark blue", "orange"), guide = "none") +
  theme_minimal() +
  labs(y = "Predicted probability to default")

# I opted for a raw data display of all the points in the test set. Here,
# we can see that the defaulting category has a higher average probability
# for a default compared to the "No" category, but there are still data 
# points in the "No" category with high predicted probability for defaulting.

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

11. 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?

coefs <- coef(lr_mod)
coefs["balance"]

    balance 
0.005672977 

# The higher the balance, the higher the log-odds of defaulting. Precisely:
# Each dollar increase in balance increases the log-odds by 0.0058.

# Let's calculate the log-odds for our person
logodds <- coefs[1] + 4e4*coefs[4] + 3e3*coefs[3]

# Let's convert this to a probability
1 / (1 + exp(-logodds))

(Intercept) 
  0.9982497 

# probability of .998 of defaulting. This is in line with the plots of before
# because this new data point would be all the way on the right.

Visualising the effect of the balance variable

In two steps, we will visualise the effect balance has on the predicted default probability.

12. 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.

balance_df <- tibble(
  student = rep(0, 500),
  balance = seq(0, 3000, length.out = 500),
  income  = rep(mean(default_train$income), 500)
)

13. 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?

balance_df$predprob <- predict(lr_mod, newdata = balance_df, type = "response")

balance_df |> 
  ggplot(aes(x = balance, y = predprob)) +
  geom_line(col = "dark blue", linewidth = 1) +
  theme_minimal()

# Just before 2000 in the first plot is where the ratio of
# defaults to non-defaults is 50-50. So this line is exactly what we expect!

14. 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?

pred_prob <- predict(lr_mod, newdata = default_test, type = "response")
pred_lr   <- factor(pred_prob > .5, labels = c("No", "Yes"))

table(
  observed  = default_test$default, 
  predicted = pred_lr
)

        predicted
observed   No  Yes
     No  1925    5
     Yes   47   23

# logistic regression performs better in every way than knn. This depends on
# your random split so your mileage may vary

Linear discriminant analysis

The last method we will use is LDA, using the lda() function from the MASS package.

15. Train an LDA classifier lda_mod on the training set.

lda_mod <- lda(default ~ ., data = default_train)

16. Look at the lda_mod object. What can you conclude about the characteristics of the people who default on their loans?

lda_mod

Call:
lda(default ~ ., data = default_train)

Prior probabilities of groups:
      No      Yes 
0.967125 0.032875 

Group means:
      student   balance   income
No  0.2888717  803.4652 33665.79
Yes 0.3726236 1749.3143 32559.39

Coefficients of linear discriminants:
                  LD1
student -1.452216e-01
balance  2.231857e-03
income   5.306987e-06

# defaulters have a larger proportion of students than non-defaulters 
# (40% vs 29%), they have a slightly lower income, and they have a 
# much higher remaining credit card balance than non-defaulters.

17. Create a confusion matrix and compare it to the previous methods.

pred_lda <- predict(lda_mod, newdata = default_test)
table(
  observed  = default_test$default,
  predicted = pred_lda$class
)

        predicted
observed   No  Yes
     No  1926    4
     Yes   56   14

# It has slightly more false negatives, a lower false positive rate, 
# and fewer true positives than logistic regression.

OPTIONAL: Final assignment

18. Create a model (using knn, logistic regression, or LDA) to predict whether a 14 year old boy from the 3rd class would have survived the Titanic disaster. You can find the data here. Would the passenger have survived if they were a girl in 2nd class?

titanic <- read_csv("data/Titanic.csv")

# I'll do a logistic regression with all interactions
lr_mod_titanic <- glm(Survived ~ PClass * Sex * Age, family = binomial, data = titanic)

predict(
  object = lr_mod_titanic, 
  newdata = tibble(
    PClass = c( "3rd",    "2nd"),
    Age    = c(    14,       14), 
    Sex    = c("male", "female")
  ), 
  type = "response"
)

        1         2 
0.2289759 0.9215896 

# So our hypothetical passenger does not have a large survival probability:
# our model would classify the boy as not surviving. The girl would likely
# survive however. This is due to the women and children getting preferred
# access to the lifeboats. Also 3rd class was way below deck.