kmeans, an Octave code which handles the K-Means problem, which organizes a set of N points in M dimensions into K clusters;
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.
Z(J) = Sum ( all X(I) in cluster J ) X(I) /
Sum ( all X(I) in cluster J ) 1.
The energy of cluster J is
E(J) = Sum ( all 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 the "H-Means algorithm". It alternates between two procedures:
A more sophisticated algorithm, known as the "K-Means algorithm", 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:
A natural extension of the K-Means problem allows us to include some more information, namely, a set of weights associated with the data points. These might represent a measure of importance, a frequency count, or some other information. The intent is that a point with a weight of 5.0 is twice as "important" as a point with a weight of 2.5, for instance. This gives rise to the "weighted" K-Means problem.
In the weighted K-Means problem, we are given a set of N points X(I) in M-dimensions, and a corresponding set of nonnegative weights W(I). The goal is to arrange the points into K clusters, with each cluster having a representative point Z(J), usually chosen as the weighted centroid of the points in the cluster:
Z(J) = Sum ( all X(I) in cluster J ) W(I) * X(I) /
Sum ( all X(I) in cluster J ) W(I).
The weighted energy of cluster J is
E(J) = Sum ( all X(I) in cluster J ) W(I) * || X(I) - Z(J) ||^2
The computer code and data files described and made available on this web page are distributed under the MIT license
kmeans is available in a C version and a C++ version and a Fortran90 version and a MATLAB version and an Octave version.
asa058, an Octave code which implements the K-means algorithm of Sparks.
asa136, an Octave code which implements the Hartigan and Wong clustering algorithm.
cities, an Octave code which handles various problems associated with a set of "cities" on a map.
cities, a dataset directory which contains sets of data defining groups of cities.
image_quantization, an Octave code which demonstrates how the KMEANS algorithm can be used to reduce the number of colors or shades of gray in an image.
kmeans_fast, an Octave code which contains several different algorithms for the K-Means problem, which organizes a set of N points in M dimensions into K clusters, by Charles Elkan.
lorenz_ode_cluster_test, an Octave code which takes a set of N points on a trajectory of solutions to the Lorenz equations, and applies the K-means algorithm to organize the data into K clusters.
sammon_data, an Octave code which generates six sets of M-dimensional data for cluster analysis.
spaeth, a dataset directory which contains a set of test data.
spaeth2, a dataset directory which contains a set of test data.