NAG FL Interface
g03eff (cluster_​kmeans)

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1 Purpose

g03eff performs K-means cluster analysis.

2 Specification

Fortran Interface
Subroutine g03eff ( weight, n, m, x, ldx, isx, nvar, k, cmeans, ldc, wt, inc, nic, css, csw, maxit, iwk, wk, ifail)
Integer, Intent (In) :: n, m, ldx, isx(m), nvar, k, ldc, maxit
Integer, Intent (Inout) :: ifail
Integer, Intent (Out) :: inc(n), nic(k), iwk(n+3*k)
Real (Kind=nag_wp), Intent (In) :: x(ldx,m), wt(*)
Real (Kind=nag_wp), Intent (Inout) :: cmeans(ldc,nvar)
Real (Kind=nag_wp), Intent (Out) :: css(k), csw(k), wk(n+2*k)
Character (1), Intent (In) :: weight
C Header Interface
#include <nag.h>
void  g03eff_ (const char *weight, const Integer *n, const Integer *m, const double x[], const Integer *ldx, const Integer isx[], const Integer *nvar, const Integer *k, double cmeans[], const Integer *ldc, const double wt[], Integer inc[], Integer nic[], double css[], double csw[], const Integer *maxit, Integer iwk[], double wk[], Integer *ifail, const Charlen length_weight)
The routine may be called by the names g03eff or nagf_mv_cluster_kmeans.

3 Description

Given n objects with p variables measured on each object, xij, for i=1,2,,n and j=1,2,,p, g03eff allocates each object to one of K groups or clusters to minimize the within-cluster sum of squares:
k=1KiSkj=1p (xij-x¯kj) 2,  
where Sk is the set of objects in the kth cluster and x¯kj is the mean for the variable j over cluster k. This is often known as K-means clustering.
In addition to the data matrix, a K×p matrix giving the initial cluster centres for the K clusters is required. The objects are then initially allocated to the cluster with the nearest cluster mean. Given the initial allocation, the procedure is to iteratively search for the K-partition with locally optimal within-cluster sum of squares by moving points from one cluster to another.
Optionally, weights for each object, wi, can be used so that the clustering is based on within-cluster weighted sums of squares:
k=1KiSkj=1pwi (xij-x~kj) 2,  
where x~kj is the weighted mean for variable j over cluster k.
The routine is based on the algorithm of Hartigan and Wong (1979).

4 References

Everitt B S (1974) Cluster Analysis Heinemann
Hartigan J A and Wong M A (1979) Algorithm AS 136: A K-means clustering algorithm Appl. Statist. 28 100–108
Kendall M G and Stuart A (1976) The Advanced Theory of Statistics (Volume 3) (3rd Edition) Griffin
Krzanowski W J (1990) Principles of Multivariate Analysis Oxford University Press

5 Arguments

1: weight Character(1) Input
On entry: indicates if weights are to be used.
No weights are used.
Weights are used and must be supplied in wt.
Constraint: weight='U' or 'W'.
2: n Integer Input
On entry: n, the number of objects.
Constraint: n>1.
3: m Integer Input
On entry: the total number of variables in array x.
Constraint: mnvar.
4: x(ldx,m) Real (Kind=nag_wp) array Input
On entry: x(i,j) must contain the value of the jth variable for the ith object, for i=1,2,,n and j=1,2,,m.
5: ldx Integer Input
On entry: the first dimension of the array x as declared in the (sub)program from which g03eff is called.
Constraint: ldxn.
6: isx(m) Integer array Input
On entry: isx(j) indicates whether or not the jth variable is to be included in the analysis. If isx(j)>0, the variable contained in the jth column of x is included, for j=1,2,,m.
Constraint: isx(j)>0 for nvar values of j.
7: nvar Integer Input
On entry: p, the number of variables included in the sums of squares calculations.
Constraint: 1nvarm.
8: k Integer Input
On entry: K, the number of clusters.
Constraint: k2.
9: cmeans(ldc,nvar) Real (Kind=nag_wp) array Input/Output
On entry: cmeans(i,j) must contain the value of the jth variable for the ith initial cluster centre, for i=1,2,,K and j=1,2,,p.
On exit: cmeans(i,j) contains the value of the jth variable for the ith computed cluster centre, for i=1,2,,K and j=1,2,,p.
10: ldc Integer Input
On entry: the first dimension of the array cmeans as declared in the (sub)program from which g03eff is called.
Constraint: ldck.
11: wt(*) Real (Kind=nag_wp) array Input
Note: the dimension of the array wt must be at least n if weight='W', and at least 1 otherwise.
On entry: if weight='W', the first n elements of wt must contain the weights to be used.
If wt(i)=0.0, the ith observation is not included in the analysis. The effective number of observation is the sum of the weights.
If weight='U', wt is not referenced and the effective number of observations is n.
Constraint: if weight='W', wt(i)0.0 and wt(i)>0.0 for at least two values of i, for i=1,2,,n.
12: inc(n) Integer array Output
On exit: inc(i) contains the cluster to which the ith object has been allocated, for i=1,2,,n.
13: nic(k) Integer array Output
On exit: nic(i) contains the number of objects in the ith cluster, for i=1,2,,K.
14: css(k) Real (Kind=nag_wp) array Output
On exit: css(i) contains the within-cluster (weighted) sum of squares of the ith cluster, for i=1,2,,K.
15: csw(k) Real (Kind=nag_wp) array Output
On exit: csw(i) contains the within-cluster sum of weights of the ith cluster, for i=1,2,,K. If weight='U', the sum of weights is the number of objects in the cluster.
16: maxit Integer Input
On entry: the maximum number of iterations allowed in the analysis.
Suggested value: maxit=10.
Constraint: maxit>0.
17: iwk(n+3×k) Integer array Workspace
18: wk(n+2×k) Real (Kind=nag_wp) array Workspace
19: ifail Integer Input/Output
On entry: ifail must be set to 0, −1 or 1 to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of 0 causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of −1 means that an error message is printed while a value of 1 means that it is not.
If halting is not appropriate, the value −1 or 1 is recommended. If message printing is undesirable, then the value 1 is recommended. Otherwise, the value 0 is recommended. When the value -1 or 1 is used it is essential to test the value of ifail on exit.
On exit: ifail=0 unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry ifail=0 or −1, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
On entry, k=value.
Constraint: k2.
On entry, ldc=value and k=value.
Constraint: ldck.
On entry, ldx=value and n=value.
Constraint: ldxn.
On entry, m=value and nvar=value.
Constraint: mnvar.
On entry, maxit=value.
Constraint: maxit>0.
On entry, n=value.
Constraint: n>1.
On entry, nvar=value.
Constraint: nvar1.
On entry, weight=value.
Constraint: weight='U' or 'W'.
On entry, i=value and wt(i)<0.0.
Constraint: wt(i)0.0.
On entry, wt has less than two positive values.
On entry, nvar=value and value values of isx>0.
Constraint: exactly nvar elements of isx>0.
At least one cluster is empty after the initial assignment. Try a different set of initial cluster centres in cmeans and also consider decreasing the value of k. The empty clusters may be found by examining the values in nic.
Convergence has not been achieved within the maximum number of iterations maxit=value. Try increasing maxit and, if possible, use the returned values in cmeans as the initial cluster centres.
An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

g03eff produces clusters that are locally optimal; the within-cluster sum of squares may not be decreased by transferring a point from one cluster to another, but different partitions may have the same or smaller within-cluster sum of squares.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
g03eff is not threaded in any implementation.

9 Further Comments

The time per iteration is approximately proportional to npK.

10 Example

The data consists of observations of five variables on twenty soils (see Hartigan and Wong (1979)). The data is read in, the K-means clustering performed and the results printed.

10.1 Program Text

Program Text (g03effe.f90)

10.2 Program Data

Program Data (g03effe.d)

10.3 Program Results

Program Results (g03effe.r)