STAT7305 Assignment 2 - Clustering
Due: Friday 26/9/2025 by 5pm ; Weighting: 17%
The R package gapminder offers a dataset summarising a small number of attributes of 142 countries, recorded every 5 years for 55 years from 1952 to 2007. The data can also be downloaded from Blackboard.
The dataset is complete, meaning that there are no missing values for any of the listed countries or years. We will focus on two different years: 1952 and 2007.
Two important variables in describing average quality of life in a country are life expectancy and gross domestic product (GDP) per capita. The GDP per capita is measured in “international dollars”, which here is “a hypothetical unit of currency that has the same purchasing power parity that the U.S. dollar had in the United States in 2005” . We also have a record of country populations, each either from censuses or from United Nations estimates.
You will aim to use this data to find clusters of countries which are similar, in the experience of the average inhabitant. Detailed questions are given below.
a) After looking at the data, potentially attempting clustering, and thinking about this, decide whether you prefer to retain all three quantitative variables for clustering or drop one. In either case, provide justification for your decision. [2 marks]
b) You will use two forms of clustering to cluster the data over the variables chosen above. A Gaussian mixture model must be one of the methods. You are free to choose another method, but make an argument as to why you think it might be useful for clustering this dataset. [1 mark]
c) Give the assumptions of each clustering method. Consider transformations of each variable to try to suit the assumptions of each clustering method. Explain which transformations you chose and why. Plot the transformed data in each case, illustrating the relationships between variables. [2 marks]
d) For the Gaussian mixture model, use MClust in R or another package offering a similar range of models and methods of model selection.
You will also need to use an effective method to select the optimal number of clusters with each form of clustering. Define the methods of selecting the number of components that you choose for each clustering method and give pseudo-code. [2 marks]
e) Write out the statistical model for both the VVV and the VVE mixture models, as used by the R package MClust, with p variables. [1 mark]
f) Do any countries in either 1952 or 2007 seem like such outliers that you would be better off removing them from the dataset? Explain why/why not. [1 mark]
g) Use each clustering method to select an optimal collection of clusters for the 2007 data.
Then fix the number of clusters (for each method) and use each clustering method to cluster the 1952 data. Plot the resulting clusters from each method at each time point (including a readable form. of country labelling). For the mixture model fit, add a set of contours for each of the weighted components of this fitted mixture distribution. Use the same set of weighted density levels for the contours of each component, except where infeasible. Also present all the estimated model parameters for each model. [2 marks]
h) With respect to your preferred mixture model fit to your chosen variables from the dataset, use a resampling approach to approximate 95% (marginal) confidence intervals for the component proportions. Explain the concept of label switching and why it could be of concern for the production of these confidence intervals. Look for evidence of label switching and explain why you think it was or wasn’t present. [2 marks]
i) Which countries seem to have changed cluster from 1952 to 2007? Give a table of countries that (on your evidence) have changed cluster, including their cluster number at each time. [1 mark]
j) Various categorisations of countries exist, particularly for their level of development. Find one such form. of categorisation, including lists of countries in each category in 1952 and 2007, and compare it to your preferred clustering of the countries in the gapminder data. How do you account for any differences? Note that you will need to research the basis for thecountry categorisation and explain this as part of your answer. [3 marks]
Notes:
• Your main response to the questions should consist of a single .pdf file, submitted by the relevant link for this assignment on Blackboard. You may use other software prepare your document, but the submitted file must be in .pdf format and contain all your answers. The assignments will be marked via Gradescope, and you will need to allocate pages to question parts.
• You should not include any raw output from software except figures and these should have a title, axis labels, a legend where appropriate, a caption, a figure number and be referenced by the figure number at least once in your report text. Any other output should be manually processed/selected before being included in e.g. text or tables.
• All your code and any supplementary files should be submitted via a separate .zip file to a second link for this assignment, also on Blackboard. No code should be included in the .pdf file. All code should be written in R or Python and be readable via a text editor.
• Name your files e.g. student_number_STAT3006_A2_report.pdf and student_number_STAT3006_A2_supp.zip to assist with marking.
• As perhttps://my.uq.edu.au/information-and-services/manage-my-program/student- integrity-and-conduct/academic-integrity-and-student-conduct ,
you must submit work that you prepared. Even where working from sources, you should endeavour to write in your own words. Equations are either correct or not, but you should use consistent notation throughout your assignment and define all of it.