k-medoids

转 http://en.wikipedia.org/wiki/K-medoids

The k-medoids algorithm is aclustering algorithm related to thek-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) and works with an arbitrary matrix of distances between datapoints instead of

. This method was proposed in 1987 for the work with

norm and other distances.

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 thesilhouette.

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

Amedoid 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:

Initialize: randomly select k of the n data points as the medoids

Associate each data point to the closest medoid. ("closest" here is defined using any validdistance metric, most commonlyEuclidean distance,Manhattan distance orMinkowski distance)

For each medoid mFor each non-medoid data point oSwap m and o and compute the total cost of the configuration

Select the configuration with the lowest cost.

repeat steps 2 to 4 until there is no change in the medoid.

Contents

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Demonstration of PAM[edit]

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

Consider a data set of ten objects as follows:

Figure 1.1 – distribution of the data

X126

X234

X338

X447

X562

X664

X773

X874

X985

X1076

Step 1[edit]

Figure 1.2 – clusters after step 1

Initialize k centers.

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

So here c1 and c2 are selected as medoids.

Calculate distance so as to associate each data object to its nearest medoid. Cost is calculated usingManhattan distance (Minkowski distance metric with r = 1). Costs to the nearest medoid are shown bold in the table.

ic1Data objects (Xi)Cost (distance)

134263

334384

434474

534625

634643

734735

934856

1034766

ic2Data objects (Xi)Cost (distance)

174267

374388

474476

574623

674641

774731

974852

1074762

Then 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[edit]

Figure 1.3 – clusters after step 2

Select one of the nonmedoids O′

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

ic1Data objects (Xi)Cost (distance)

134263

334384

434474

534625

634643

734744

934856

1034764

iO′Data objects (Xi)Cost (distance)

173268

373389

473477

573622

673642

773741

973853

1073763

Figure 2. K-medoids versus k-means. Figs 2.1a-2.1f present a typical example of the k-means convergence to a local minimum. This result of k-means clustering contradicts the obvious cluster structure of data set. In this example, k-medoids algorithm (Figs 2.2a-2.2h) with the same initial position of medoids (Fig. 2.2a) converges to the obvious cluster structure. The small circles are data points, the four ray stars are centroids (means), the nine ray stars are medoids.

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. So we try other nonmedoids and found that our first choice was the best. So the configuration does not change 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.

In some standard situations, k-medoids demonstrate better performance than k-means. An example is presented in Fig. 2. The most time-consuming part of the k-medoids algorithm is the calculation of the distances between objects. If a quadratic preprocessing and storage is applicable, the distances matrix can be precomputed to achieve consequent speed-up. See for example, where the authors also introduce a heuristic to choose the initial k medoids. A comparative study of K-means and k-medoids algorithms was performed for normal and for uniform distributions of data points. It was demonstrated that in the asymptotic of large data sets the k-medoids algorithm takes less time.