- Year not given · 2+8 marks
Cluster analysis; K-means on height/width
Short answer
Cluster analysis discovers natural groups, summarizes data, and detects outliers. With K=2 and initial centroids (1,2), K-means converges to Cluster 1: instances 1,3,4,6, centroid (1.05, 1.625); Cluster 2: instances 2,5, centroid (2.35, 3.00).
Why do we perform cluster analysis in Data Mining? With K-means clustering algorithm, find out the clusters based on the following instances. (Choose K at least 2)
| Instances | Height | Width |
|---|---|---|
| 1 | 0.5 | 2.2 |
| 2 | 1.5 | 3.5 |
| 3 | 0.7 | 1.5 |
| 4 | 2.0 | 1.0 |
| 5 | 3.2 | 2.5 [?] |
| 6 | 1.0 | 1.8 |
Why cluster analysis?
Cluster analysis groups unlabelled data into clusters of similar objects. In data mining it is used to discover natural groupings and structure (e.g. customer segmentation), to summarise and compress large data by cluster representatives, to detect outliers, and as a preprocessing step for other tasks such as classification and indexing. It requires no class labels, so it is useful for exploring new data.
K-means on the data
Assumptions: the value for instance 5 is read as Width = 2.5; K = 2; the initial centroids are instances 1 and 2: C1 = (0.5, 2.2), C2 = (1.5, 3.5).
| Instance | Height | Width |
|---|---|---|
| 1 | 0.5 | 2.2 |
| 2 | 1.5 | 3.5 |
| 3 | 0.7 | 1.5 |
| 4 | 2.0 | 1.0 |
| 5 | 3.2 | 2.5 |
| 6 | 1.0 | 1.8 |
Iteration 1
| Instance | d to C1 (0.5, 2.2) | d to C2 (1.5, 3.5) | Cluster |
|---|---|---|---|
| 1 | 0.00 | 1.64 | C1 |
| 2 | 1.64 | 0.00 | C2 |
| 3 | 0.73 | 2.15 | C1 |
| 4 | 1.92 | 2.55 | C1 |
| 5 | 2.72 | 1.97 | C2 |
| 6 | 0.64 | 1.77 | C1 |
C1 = {1, 3, 4, 6}, C2 = {2, 5}. New centroids: C1 = ((0.5 + 0.7 + 2.0 + 1.0)/4, (2.2 + 1.5 + 1.0 + 1.8)/4) = (1.05, 1.625); C2 = ((1.5 + 3.2)/2, (3.5 + 2.5)/2) = (2.35, 3.00).
Iteration 2
| Instance | d to C1 (1.05, 1.625) | d to C2 (2.35, 3.0) | Cluster |
|---|---|---|---|
| 1 | 0.80 | 2.02 | C1 |
| 2 | 1.93 | 0.99 | C2 |
| 3 | 0.37 | 2.23 | C1 |
| 4 | 1.14 | 2.03 | C1 |
| 5 | 2.32 | 0.99 | C2 |
| 6 | 0.18 | 1.81 | C1 |
No change, so the algorithm has converged.
Answer: Cluster 1 = {instances 1, 3, 4, 6}, centroid (1.05, 1.625); Cluster 2 = {instances 2, 5}, centroid (2.35, 3.00).
✓ Answer checked by Claude Opus 5.5 (high effort)
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