You are given the following four pairs of observations:

[[math]] x_1 = (-1,0), \, x_2 = (1,1), \, x_3 = (2,-1), \, x_4 = (5,10) [[/math]]

A hierarchical clustering algorithm is used with complete linkage and Euclidean distance.

Calculate the intercluster dissimilarity between [math]\{x_1,x_2\}[/math] and [math]\{x_4\}[/math].

• 2.2
• 3.2
• 9.9
• 10.8
• 11.7

Copyright 2023. The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.

Key: E

First, calculate the distance between pairs of elements in each set. There are two pairs here:

[[math]] \begin{aligned} x_1,x_4 &: \sqrt{} = \sqrt{136} = 11.66 \\ x2,x4 &: \sqrt{(1-5)^2 + (1-10)^2} = \sqrt{97} = 9.85 \end{aligned} [[/math]]

For complete linkage, the dissimilarity measure used is the maximum, which is 11.66.

Copyright 2023. The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.