Question

In an effort to inform political leaders and economists discussing the deregulation of electric and gas utilities, data on eight numerical variables from utility companies have been grouped using hierarchical clustering based on Euclidean distance as the similarity measure and complete linkage as the clustering method.

(a)	Based on the following dendrogram, what is the most appropriate number of clusters to organize these utility companies?
	clusters are appropriate based on complete linkage.
(b)	Using the following data on the Observations 10, 13, 4, and 20, confirm that the complete linkage distance between the cluster containing {10, 13} and the cluster containing {4, 20} is 2.577 units as displayed in the dendrogram.
	If required, round your answers to three decimal places. Do not round intermediate calculations.
	Distance from Observation 10 and Observation 4: Distance from Observation 10 and Observation 20: Distance from Observation 13 and Observation 4: Distance from Observation 13 and Observation 20:

Answer 1

(a) From the given dendrogram, it can be seen that the observations \{7,12,15,17,8,16,11\} are

very closely spreaded, and hence should be grouped together, so the number of clusters should be 2 as in making 4 clusters, the left side clusters works well but the extreme right most cluster seems absurd leaving only 3 points in a cluster. Hence, 2 clusters are appropriate based on complete linkage.

(b) The Euclidean distance between observation 10 and 4 is calculated as follows:

\(\begin{aligned} d_{10,4} &=\sqrt{(0.032+0.510)^{2}+(0.741-0.207)^{2}+(0.7+0.004)^{2}+(-0.892+0.219)^{2} +(-0.173+0.943)^{2}+(-0.693+0.702)^{2}+(1.62-1.328)^{2}+(-0.863+0.724)^{2}} \\ &=1.4916 \\ d_{10,4} & \approx 1.492 \end{aligned}\)

The Euclidean distance between observation 10 and 20 is calculated as follows:

$$ \begin{aligned} d_{10,20} &=\sqrt{(0.032-0.466)^{2}+(0.741-0.474)^{2}+(0.7+0.490)^{2}+(-0.892-0.655)^{2} +(-0.173-0.083)^{2}+(-0.693+0.458)^{2}+(1.62-1.733)^{2}+(-0.863+0.721)^{2}} \\ &=2.0549 \\ d_{10,20} & \approx 20.55 \end{aligned} $$

The Euclidean distance between observation 13 and 4 is calculated as follows:

\(d_{13,4}=\sqrt{(0.195+0.510)^{2}+(0.875-0.207)^{2}+(0.748+0.004)^{2}+(-0.735+0.219)^{2} +(1.013+0.943)^{2}+(-0.489+0.702)^{2}+(2.275-1.328)^{2}+(-1.035+0.724)^{2}}\)

$$ \begin{aligned} &=2.5768 \\ d_{13,4} & \approx 2.577 \end{aligned} $$

The Euclidean distance between observation 13 and 20 is calculated as follows:

\(d_{13,20}=\sqrt{(0.195-0.466)^{2}+(0.875-0.474)^{2}+(0.748+0.490)^{2}+(-0.735-0.655)^{2} +(1.013-0.083)^{2}+(-0.489+0.458)^{2}+(2.275-1.733)^{2}+(-1.035+0.721)^{2}}\)

$$ \begin{aligned} &=2.2264 \\ d_{13.20} & \approx 2.226 \end{aligned} $$

Hence, the correct answers are as follows:

Distance from observation 10 and 4 is 1.492

Distance between observation 10 and 20 is 2.055 Distance between observation 13 and 4 is 2.577

Distance between observation 13 and 20 is 2.226