K-Means Algorithm

Assume we have the below dataset that shows 7 points in a 2-dimensional space, with x coordinates shown in the x column, and y coordinates shown in the y column.

Assuming the number of clusters is set to 2 (i.e., k=2), and distances among points are measured by Euclidean distance. Based on observation, a good choice of the initial centroids (cluster centers) are point 1 (1.0, 1.0) and point 4 (5.0, 7.0) (these two points are relatively further away from each other). Your job is to run the k-means algorithm and answer the following questions.

Part A: After the initial centroids are assigned (shown above), the first step in a clustering process is to check the rest of the points and assign them to one of the two clusters. Here we assume that the centroid remain unchanged during the first round (iteration). Determine cluster memberships (which points belong to Cluster 1 and which belong to Cluster 2) for each of these 7 points after the first iteration. Show intermediate results how you obtain the solution.

Part B: Based on the results obtained from part a, recalculate centroids (the mean vectors) for the two clusters. Then recalculate the distance from each point to its centroid. Show your results. Are there any points need to change their cluster memberships? If so, what are these points?

Part C: Continue the process shown in part b for another iteration. Do you observe any changes in cluster memberships? If so, what are these points? If not, will there be further changes if we continue the process with more iterations?

Point	x	y
1	1.0	1.0
2	1.5	2.0
3	3.0	4.0
4	5.0	7.0
5	3.5	5.0
6	4.5	5.0
7	3.5	4.5

Please draw a picture and show the changes of centroids on a 2-dimensional system at each step so that we can all learn! Thumbs up for clear work! Thanks!