# ASA136 The K-Means Algorithm

ASA136 is a MATLAB library which divides N points in M dimensions into K clusters so that the within-clusters sum of squares is minimized, by Hartigan and Wong.

ASA136 is Applied Statistics Algorithm 136.

In the K-Means problem, a set of N points X(I) in M-dimensions is given. The goal is to arrange these points into K clusters, with each cluster having a representative point Z(J), usually chosen as the centroid of the points in the cluster. The energy of each cluster is

```        E(J) = Sum ( all points X(I) in cluster J ) || X(I) - Z(J) ||^2
```

For a given set of clusters, the total energy is then simply the sum of the cluster energies E(J). The goal is to choose the clusters in such a way that the total energy is minimized. Usually, a point X(I) goes into the cluster with the closest representative point Z(J). So to define the clusters, it's enough simply to specify the locations of the cluster representatives.

This is actually a fairly hard problem. Most algorithms do reasonably well, but cannot guarantee that the best solution has been found. It is very common for algorithms to get stuck at a solution which is merely a "local minimum". For such a local minimum, every slight rearrangement of the solution makes the energy go up; however a major rearrangement would result in a big drop in energy.

A simple algorithm for the problem is known as "H-Means". It alternates between two procedures:

• Using the given cluster centers, assign each point to the cluster with the nearest center;
• Using the given cluster assignments, replace each cluster center by the centroid or average of the points in the cluster.
These steps are repeated until no points are moved, or some other termination criterion is reached.

A more sophisticated algorithm, known as "K-Means", takes advantage of the fact that it is possible to quickly determine the decrease in energy caused by moving a point from its current cluster to another. It repeats the following procedure:

• For each point, move it to another cluster if that would lower the energy. If you move a point, immediately update the cluster centers of the two affected clusters.
This procedure is repeated until no points are moved, or some other termination criterion is reached.

### Languages:

ASA136 is available in a C version and a C++ version and a FORTRAN77 version and a FORTRAN90 version and a MATLAB version.

### Related Data and Programs:

ASA058, a MATLAB library which carries out the K-means algorithm for clustering data.

ASA113, a MATLAB library which implements the Banfield and Bassill clustering algorithm using transfers and swaps.

CITIES, a MATLAB library which handles various problems associated with a set of "cities" on a map.

CITIES, a dataset directory which contains a number of city distance datasets.

KMEANS, a MATLAB library which contains several implementations of the H-Means and K-Means clustering algorithms.

LAU_NP, a FORTRAN90 library which contains heuristic algorithms for the K-center and K-median problems.

SAMMON_DATA, a MATLAB program which generates six sets of M-dimensional data for cluster analysis.

SPAETH, a FORTRAN90 library which clusters data according to various principles.

SPAETH, a dataset directory which contains test data for clustering.

SPAETH2, a FORTRAN90 library which can cluster data according to various principles.

SPAETH2, a dataset directory which contains test data for clustering.

### Author:

Original FORTRAN77 version by John Hartigan, Manchek Wong; MATLAB version by John Burkardt.

### Reference:

1. John Hartigan, Manchek Wong,
Algorithm AS 136: A K-Means Clustering Algorithm,
Applied Statistics,
Volume 28, Number 1, 1979, pages 100-108.
2. Wendy Martinez, Angel Martinez,
Computational Statistics Handbook with MATLAB,
pages 373-376,
Chapman and Hall / CRC, 2002.
3. David Sparks,
Algorithm AS 58: Euclidean Cluster Analysis,
Applied Statistics,
Volume 22, Number 1, 1973, pages 126-130.

### Source Code:

• kmns.m carries out the K-means algorithm.
• optra.m carries out the optimal transfer stage.
• qtran.m carries out the quick transfer stage.
• timestamp.m prints out the current YMDHMS date as a timestamp.

### Examples and Tests:

You can go up one level to the MATLAB source codes.

Last revised on 15 February 2008.