Consider a 1-dimensional dataset {1, 2, 3, 4, 5, 8, 9, 10, 11, 12, 24, 28, 32, 36, 40} with the three natural clusters {1, 2, 3, 4, 5}, {8, 9, 10, 11, 12} and {24, 28, 32, 36, 40}. Apply the k-means algorithm with k = 3 to this dataset and show the resulting clusters and cluster centroids for every iteration. Use the algorithm ClusteringByVarianceMinimization, not the incremental algorithm that has originally been published as k-means. We are assuming that ties are broken by assigning an object to the smallest of the closest centroids.

(a) Start with initial centroids of 1, 11, and 28. Does the algorithm detect the natural clusters?

(b) Start with initial centroids of 1, 2, and 3. Does the algorithm detect the natural clusters? What does this tell you?