Understanding K-means, K-means++ and, K-medoids Clustering Algorithms

K-Means Clustering:

K-Means algorithm is a centroid based clustering technique. This technique cluster the dataset to k different cluster having an almost equal number of points. Each cluster is k-means clustering algorithm is represented by a centroid point.

What is a centroid point?

The centroid point is the point that represents its cluster. Centroid point is the average of all the points in the set and will change in each step and will be computed by:

For the above equation,
C_i: i'th Centroid
S_i: All points belonging to set_i with centroid as C_i
x_j: j'th point from the set
||S_i||: number of points in set_i

The idea of the K-Means algorithm is to find k-centroid points and every point in the dataset will belong either of k-sets having minimum Euclidean distance.

From the image above (Image 3), the distance of point x_i from all three centroids are d1, d2, d3, x_i point is nearest to centroid_3 with distance d3, so the point x_i will belong to the cluster of centroid_3 and this process will continue for all the points in the dataset.

Cost Function of K-Means:

The idea of the K-Means algorithm is to find k centroid points (C_1, C_1, . . . C_k) by minimizing the sum over each cluster of the sum of the square of the distance between the point and its centroid.

This cost is NP-hard and has exponential time complexity. So we use the idea of approximation using Lloyd’s Algorithm.

Lloyd’s Algorithm:

Lloyd’s algorithm is an approximation iterative algorithm used to cluster points. The steps of the algorithm are as follows:

1. Initialization
2. Assignment
3. Update Centroid
4. Repeat Step 2 and 3 until convergence.

Iterative implementation of the K-Means algorithm:

Steps #1: Initialization:

The initial k-centroids are randomly picked from the dataset of points (lines 27–28).

Steps #2: Assignment:

For each point in the dataset, find the euclidean distance between the point and all centroids (line 33). The point will be assigned to the cluster with the nearest centroid.

Steps #3: Updation of Centroid:

Update the value of the centroid with the new mean value (lines 39–40).

Steps #4: Repeat:

Repeat steps 2 and 3 unless convergence is achieved. If convergence is achieved then break the loop(line 43). Convergence refers to the condition where the previous value of centroids is equal to the updated value.

Results:

Plot for the initial dataset (Image 4)

Clustering result plot for k=2 (Image 5)

Clustering result plot for k=3 (Image 6)