k-medoids

The k-medoids algorithm is a clustering algorithm related to the k-means algorithm and the medoidshift algorithm. Both the k-means and k-medoids algorithms are partitional (breaking the dataset up into groups) and both attempt to minimize the distance between points labeled to be in a cluster and a point designated as the center of that cluster. In contrast to the k-means algorithm, k-medoids chooses datapoints as centers (medoids or exemplars).

k-medoid is a classical partitioning technique of clustering that clusters the data set of n objects into k clusters known a priori. A useful tool for determining k is the silhouette.

It is more robust to noise and outliers as compared to k-means because it minimizes a sum of pairwise dissimilarities instead of a sum of squared Euclidean distances.

A medoid can be defined as the object of a cluster, whose average dissimilarity to all the objects in the cluster is minimal i.e. it is a most centrally located point in the cluster.

The most common realisation of k-medoid clustering is the Partitioning Around Medoids (PAM) algorithm and is as follows:^[1]

1. Initialize: randomly select k of the n data points as the medoids
2. Associate each data point to the closest medoid. ("closest" here is defined using any valid distance metric, most commonly Euclidean distance, Manhattan distance or Minkowski distance)
3. For each medoid m
   1. For each non-medoid data point o
      1. Swap m and o and compute the total cost of the configuration
4. Select the configuration with the lowest cost.
5. repeat steps 2 to 5 until there is no change in the medoid.

Contents 1 Demonstration of PAM 1.1 Step 1 1.2 Step 2 2 See also 3 External links 4 References

Demonstration of PAM

Cluster the following data set of ten objects into two clusters i.e. k = 2.

Consider a data set of ten objects as follows:

X1	2	6
X2	3	4
X3	3	8
X4	4	7
X5	6	2
X6	6	4
X7	7	3
X8	7	4
X9	8	5
X10	7	6

Step 1

Initialise k centre

Let us assume c₁ = (3,4) and c₂ = (7,4)

So here c₁ and c₂ are selected as medoid.

Calculating distance so as to associate each data object to its nearest medoid. Cost is calculated using Minkowski distance metric with r = 1.

c1		Data objects (Xi)		Cost (distance)
3	4	2	6	3
3	4	3	8	4
3	4	4	7	4
3	4	6	2	5
3	4	6	4	3
3	4	7	3	5
3	4	8	5	6
3	4	7	6	6

c2		Data objects (Xi)		Cost (distance)
7	4	2	6	7
7	4	3	8	8
7	4	4	7	6
7	4	6	2	3
7	4	6	4	1
7	4	7	3	1
7	4	8	5	2
7	4	7	6	2

Then so the clusters become:

Cluster₁ = {(3,4)(2,6)(3,8)(4,7)}

Cluster₂ = {(7,4)(6,2)(6,4)(7,3)(8,5)(7,6)}

Since the points (2,6) (3,8) and (4,7) are closer to c₁ hence they form one cluster whilst remaining points form another cluster.

So the total cost involved is 20.

Where cost between any two points is found using formula

where x is any data object, c is the medoid, and d is the dimension of the object which in this case is 2.

Total cost is the summation of the cost of data object from its medoid in its cluster so here:

Step 2

Selection of nonmedoid O′ randomly

Let us assume O′ = (7,3)

So now the medoids are c₁(3,4) and O′(7,3)

If c1 and O′ are new medoids, calculate the total cost involved

By using the formula in the step 1

c1		Data objects (Xi)		Cost (distance)
3	4	2	6	3
3	4	3	8	4
3	4	4	7	4
3	4	6	2	5
3	4	6	4	3
3	4	7	4	4
3	4	8	5	6
3	4	7	6	6

O′		Data objects (Xi)		Cost (distance)
7	3	2	6	8
7	3	3	8	9
7	3	4	7	7
7	3	6	2	2
7	3	6	4	2
7	3	7	4	1
7	3	8	5	3
7	3	7	6	3

So cost of swapping medoid from c₂ to O′ is

So moving to O′ would be a bad idea, so the previous choice was good and algorithm terminates here (i.e. there is no change in the medoids).

It may happen some data points may shift from one cluster to another cluster depending upon their closeness to medoid.

See also

• cluster analysis
• k-means
• k-medians
• medoid
• silhouette

External links

K-means and K-medoids (Applet), University of Leicester

References