1 Introduction

In this practical, we will apply model-based clustering on a data set of bank note measurements.

We use the following packages:

library(mclust)
library(tidyverse)
library(patchwork)

The data is built into the mclust package and can be loaded as a tibble by running the following code:

df <- as_tibble(banknote)

2 Take-home exercises

2.1 Data exploration

1. Read the help file of the banknote data set to understand what it’s all about.

2. Create a scatter plot of the left (x-axis) and right (y-axis) measurements on the data set. Map the Status column to colour. Jitter the points to avoid overplotting. Are the classes easy to distinguish based on these features?

3. From now on, we will assume that we don’t have the labels. Remove the Status column from the data set.

4. Create density plots for all columns in the data set. Which single feature is likely to be best for clustering?

2.2 Univariate model-based clustering

5. Use Mclust to perform model-based clustering with 2 clusters on the feature you chose. Assume equal variances. Name the model object fit_E_2. What are the means and variances of the clusters?

6. Use the formula from the slides and the model’s log-likelihood (fit_E_2$loglik) to compute the BIC for this model. Compare it to the BIC stored in the model object (fit_E_2$bic). Explain how many parameters (m) you used and which parameters these are.

7. Plot the model-implied density using the plot() function. Afterwards, add rug marks of the original data to the plot using the rug() function from the base graphics system.

8. Use Mclust to perform model-based clustering with 2 clusters on this feature again, but now assume unequal variances. Name the model object fit_V_2. What are the means and variances of the clusters? Plot the density again and note the differences.

9. How many parameters does this model have? Name them.

10. According to the deviance, which model fits better?

11. According to the BIC, which model is better?

3 Lab exercises

3.1 Multivariate model-based clustering

We will now use all available information in the data set to cluster the observations.

12. Use Mclust with all 6 features to perform clustering. Allow all model types (shapes), and from 1 to 9 potential clusters. What is the optimal model based on the BIC?

13. How many mean parameters does this model have?

14. Run a 2-component VVV model on this data. Create a matrix of bivariate contour (“density”) plots using the plot() function. Which features provide good component separation? Which do not?

15. Create a scatter plot just like the first scatter plot in this tutorial, but map the estimated class assignments to the colour aesthetic. Map the uncertainty (part of the fitted model list) to the size aesthetic, such that larger points indicate more uncertain class assignments. Jitter the points to avoid overplotting. What do you notice about the uncertainty?

3.2 Challenge assignment: High-dimensional Gaussian Mixture modeling

NB: this procedure is very technical and will not be tested in-depth in the exam. It is meant to give you a start in high-dimensional clustering and an example of how to explore new packages.

16. Install and load the package HDclassif. Read the introduction and section 4.2, parts “First results” and “PCA representation” from the associated paper here.

This paper is from the Journal of Statistical Software, a very high-quality open journal describing statistical software packages. If a package has a JSS paper, always start there!

17. Run high-dimensional data clustering on the Crabs dataset using demo("hddc"). Choose the EM algorithm with random initialization with the AkBkQkDk model. Explain what happens in the plot window.