Figure

$3$

$($

c

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$$The

$$

$-$

means clustering algorithm was then used to cluster the data. From Figure

$4$

$,$

$$how many clusters would you suggest are present in the dataset? Give a reason for your answer.

$[$

$5$

$$marks

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$$Figure

$4$

$($

d

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$$The adjusted Rand index can be used to compare two clustering solutions of the cities. Explain why the adjusted Rand index can be used even if the two clustering solutions contain different numbers of clusters.

$[$

$5$

$$marks

$]$

$($

e

$)$

$$A model

$-$

based clustering approach was then taken to cluster the cities. The optimal model was deemed to be an EVV model with

$2$

$$components$.$ i

$.$

$$Briefly outline the Bayesian information criterion used to select the optimal model and explain its rationale.

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$1$

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$$marks

$]$

$$Cont$.$

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ii

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$$Sketch a simple plot to explain what the choice EVV means in terms of the clusters produced. Explain your motivation for what you sketch. marks

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$$iii$.$ Write down the parameterisation of the covariance matrix that corresponds to the EVV model. Explain any notation you use. marks

$]$

$($

f

$)$

$$The researcher also applied multidimensional scaling

$($

MDS

$)$

$$to these data. i

$.$

$$Explain why the researcher might have applied multidimensional scaling to these data and outline decisions that the researcher would need to make in the process. marks

$]$

$$ii

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$$How might the researcher choose the number of dimensions when applying classical multidimensional scaling? marks The researcher who collected the air pollution data in question

$2$

$$was also interested in understanding if there are groups of US cities that are similar in terms of air pollution and cluster analysis was used to investigate the data outlined in Table

$2$

$.$

$($

a

$)$

$$The researcher initially employed hierarchical clustering to cluster the cities. The researcher decided to employ the Euclidean distance as a dissimilarity measure and used complete linkage. Explain why the researcher may have made these decisions. marks