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, eventually we reach a state in which there are only two clusters that remain (meaning there is either one cluster with three observations in it and one cluster with one observation in it, or there are two clusters of two observations). Using complete linkage, those two clusters are
[选择]
.55
.4
.2
.3
.5
.15
away from each other.

Using the agglomerative (hierarchical) method of creating clusters, eventually we reach a state in which there are only two clusters that remain (meaning there is either one cluster with three observations in it and one cluster with one observation in it, or there are two clusters of two observations). Using single linkage those two clusters are
[选择]
.3
.4
.55
.5
.15
.2
away from each other.

.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, eventually we reach a state in which there are only two clusters that remain (meaning there is either one cluster with three observations in it and one cluster with one observation in it, or there are two clusters of two observations). Using complete linkage, those two clusters are 
[选择]
.55
.4
.2
.3
.5
.15
 away from each other.

Using the agglomerative (hierarchical) method of creating clusters, eventually we reach a state in which there are only two clusters that remain (meaning there is either one cluster with three observations in it and one cluster with one observation in it, or there are two clusters of two observations). Using single linkage those two clusters are 
[选择]
.3
.4
.55
.5
.15
.2
 away from each other.

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

BlackTom AI · Accepted Answer

We start by understanding the given dissimilarity matrix for four observations. The pairwise distances are: d(1,2)=0.15, d(1,3)=0.4, d(1,4)=0.5, d(2,3)=0.3, d(2,4)=0.55, d(3,4)=0.2.

First, consider complete linkage (for the two-cluster result):
- The natural first merge is observations 1 and 2 because their distance is the smallest (0.15).
- After forming cluster A = {1,2}, we compute distances from A to the remaining points using complete linkage (the maximum pairwise distance across the two clusters):
  - To observation 3: max(d(1,3)=0.4, d(2,3)=0.3) = 0.4
  - To observation 4: max(d(1,4)=0.5, d(2,4)=0.55) = 0.55
- The remaining distances are between 3 and 4 (0.2) and between A and {3,4} (0.4 to 3, 0.55 to 4). The smallest of these is 0.2 between 3 and 4, so we merge 3 and 4 next, forming cluster B = {3,4}.
- Now we have exactly two clusters: A = {1,2} and B = {3,4}. The distance between these two clusters under complete linkage is the maximum distance between any member of A and any member of B:
  - Distances across clusters: d(1,3)=0.4, d(1,4)=0.5, d(2,3)=0.3, d(2,4)=0.55. The maximum of these is 0.55.
- Therefore, the complete-linkage distance between the final two clusters is 0.55.

Now, consider single linkage (for the two-cluster result):
- As before, after the first merge A = {1,2}, the distances from A to the remaining points using single linkage are the minimum distances between any member of A and the other observation:
  - To 3: min(d(1,3)=0.4, d(2,3)=0.3) = 0.3
  - To 4: min(d(1,4)=0.5, d(2,4)=0.55) = 0.5
- The distance between 3 and 4 is 0.2, which is the smallest overall, so we merge 3 and 4 into cluster B = {3,4} (again leaving two clusters: A and B).
- The distance between A and B under single linkage is the minimum distance across any pair with one element from A and one from B:
  - Pairs: (1,3)=0.4, (1,4)=0.5, (2,3)=0.3, (2,4)=0.55. The minimum is 0.3.
- Thus, the correct single-linkage distance between the final two clusters is 0.3.

Note: The provided answer list indicates the second value as 0.2, which would correspond to the intra-cluster distance between 3 and 4, not the inter-cluster distance between A and B under single linkage. In standard hierarchical clustering, the two-cluster distance for single linkage should be 0.3, not 0.2. The first value 0.55 for complete linkage aligns with the maximal cross-cluster distance computed above.

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