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Consider a set of one-dimensional points: 6, 12, 18, 24, 30, 42, 48.
(1) Use 15, 40 as the initial centroids, and apply K-means to create two clusters.
(2) Calculate the Sum of Squared Errors (SSE) for the clustering result. (1 point) (3) Repeat (1), but use 10, 20 as the initial centroids.

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Answer:

1) With initial centroids 10-40, final clusters are

first cluster (6,12,18,24,30)

second cluster (42,48)

And the Sum of Squared Errors (SSE) for the clustering result is 378

2) With initial centroids 10-20, final clusters are

first cluster (6,12,18,24)  

second cluster (30,42,48)

And the Sum of Squared Errors (SSE) for the clustering result is 348

Step-by-step explanation:

K-means works as follows

  • the points will be clustered according to their distance to the centroids.
  • Then centroids are updated as the cluster means.
  • This process continues until clusters doesn't change anymore

1)  initial centroids 10-40

first cluster (6,12,18,24)  mean:15

second cluster (30,42,48) mean:40

new centroids 15-40

first cluster (6,12,18,24,30) mean:18

second cluster (42,48) mean:45

final centroids 18-45

first cluster (6,12,18,24,30) mean:18

second cluster (42,48) mean:45

Sum of Squared errors = [tex](18-6)^{2}[/tex]+[tex](18-12)^{2}[/tex]+[tex](18-18)^{2}[/tex]+[tex](18-24)^{2}[/tex]+[tex](18-30)^{2}[/tex]+[tex](45-42)^{2}[/tex]+[tex](45-48)^{2}[/tex]=378

2)initial centroids 10-20

first cluster (6,12)  mean:9

second cluster (18,24,30,42,48)  mean:32.4

new centroids 9-32.4

first cluster (6,12,18) mean:12

second cluster (24,30,42,48) mean:36

new centroids 12-36

first cluster (6,12,18,24) mean:15

second cluster (30,42,48) mean:40

final centroids 15-40

first cluster (6,12,18,24) mean:15  

second cluster (30,42,48) mean:40

Sum of Squared errors = [tex](15-6)^{2}[/tex]+[tex](15-12)^{2}[/tex]+[tex](15-18)^{2}[/tex]+[tex](15-24)^{2}[/tex]+[tex](40-30)^{2}[/tex]+[tex](40-42)^{2}[/tex]+[tex](40-48)^{2}[/tex]=348

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