This is a dissimilarity matrix that shows how far away each observation is from each other observation. As a reminder: this says that observations 1 and 2 are .15 away from each other.

Using the agglomerative (hierarchical) method of creating clusters, if we use complete linkage, what is the first cluster of two observations to form?
[选择]
Observations 1 and 4
Observations 2 and 3
Observations 1 and 3
Observations 3 and 4
Observations 2 and 4
Observations 1 and 2

Using the agglomerative (hierarchical) method of creating clusters, if we use single linkage, what is the first cluster of two observations to form?
[选择]
Observations 2 and 3
Observations 1 and 4
Observations 1 and 3
Observations 3 and 4
Observations 1 and 2
Observations 2 and 4

.15 .4 .5
.15 .3 .55
.4 .3 .2
.5 .55 .2

Question

This is a dissimilarity matrix that shows how far away each observation is from each other observation. As a reminder: this says that observations 1 and 2 are .15 away from each other.

Using the agglomerative (hierarchical) method of creating clusters, if we use complete linkage, what is the first cluster of two observations to form? 
[选择]
Observations 1 and 4
Observations 2 and 3
Observations 1 and 3
Observations 3 and 4
Observations 2 and 4
Observations 1 and 2

Using the agglomerative (hierarchical) method of creating clusters, if we use single linkage, what is the first cluster of two observations to form? 
[选择]
Observations 2 and 3
Observations 1 and 4
Observations 1 and 3
Observations 3 and 4
Observations 1 and 2
Observations 2 and 4

.15	.4	.5
.15		.3	.55
.4	.3		.2
.5	.55	.2

BlackTom AI · Accepted Answer

We start by restating the scenario: we have a dissimilarity matrix showing distances between observations, and we’re performing agglomerative hierarchical clustering with two linkage options (complete and single) to determine the first cluster that forms.

First, list the pairwise distances from the matrix provided: d12 = 0.15, d13 = 0.40, d14 = 0.50, d23 = 0.30, d24 = 0.55, d34 = 0.20. These are the distances between every pair of single observations before any clustering.

Option: Observations 1 and 4
- Reasoning: The distance between 1 and 4 is 0.50, which is not the smallest among all pairwise distances. For both complete and single linkage, the first merge should be the pair with the smallest distance, so this option is not the first cluster.

Option: Observations 2 and 3
- Reasoning: The distance between 2 and 3 is 0.30. While this is a relatively small distance, it is larger than 0.15, which is the distance between observations 1 and 2. Since the initial merge selects the closest pair, this option is incorrect for the first clustering step.

Option: Observations 1 and 3
- Reasoning: The distance between 1 and 3 is 0.40, which is not the smallest distance. Therefore, this would not be the first merge under either complete or single linkage.

Option: Observations 3 and 4
- Reasoning: The distance between 3 and 4 is 0.20. Although this is small, it is still larger than the 0.15 distance between 1 and 2, so this pair would not be the first merge in either linkage scheme.

Option: Observations 2 and 4
- Reasoning: The distance between 2 and 4 is 0.55, the largest of the listed pairwise distances. This would never be chosen first under either linkage method.

Option: Observations 1 and 2
- Reasoning: The distance between 1 and 2 is 0.15, which is the smallest pairwise distance among all observed pairs. In both complete linkage and single linkage, the first clustering step merges the pair with the smallest distance, so this is the correct first cluster for both linkage methods.

Summary of reasoning across options: For initial agglomerative clustering, the key criterion is selecting the closest pair of single observations. Since 1 and 2 are at distance 0.15, they form the first cluster under both complete linkage and single linkage. All other pairs have distances greater than 0.15, so they cannot form the first cluster. This explains why all other options are incorrect and why Observations 1 and 2 is the only viable first merge in this dataset.

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