NAG CL Interface
g03efc (cluster_​kmeans)

1 Purpose

g03efc performs K -means cluster analysis.

2 Specification

#include <nag.h>
void  g03efc (Integer n, Integer m, const double x[], Integer tdx, const Integer isx[], Integer nvar, Integer k, double cmeans[], Integer tdc, const double wt[], Integer inc[], Integer nic[], double css[], double csw[], Integer maxit, NagError *fail)
The function may be called by the names: g03efc, nag_mv_cluster_kmeans or nag_mv_kmeans_cluster_analysis.

3 Description

Given n objects with p variables measured on each object, x ij for i = 1 , 2 , , n and j = 1 , 2 , , p , g03efc allocates each object to one of K groups or clusters to minimize the within-cluster sum of squares:
k=1 K i S k j=1 p x ij - x ¯ kj 2 ,  
where S k is the set of objects in the k th 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 by 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, w i , can be used so that the clustering is based on within-cluster weighted sums of squares:
k=1 K i S k j=1 p w i x ij - x ~ kj 2 ,  
where x ~ kj is the weighted mean for variable j over cluster k .
The function 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: n Integer Input
On entry: the number of observations, n .
Constraint: n2 .
2: m Integer Input
On entry: the number of variables in the array x.
Constraint: mnvar .
3: x[n×tdx] const double Input
On entry: x[i-1×tdx+j-1] must contain the value of j th variable for the i th object, for i=1,2,,n and j=1,2,,m.
4: tdx Integer Input
On entry: the stride separating matrix column elements in the array x.
Constraint: tdxm .
5: isx[m] const Integer Input
On entry: isx[j-1] indicates whether or not the j th variable is to be included in the analysis.
If isx[j-1] > 0 , then the j th variable contained in the j th column of x is included, for j=1,2,,m.
Constraint: isx[j-1] > 0 for nvar values of j .
6: nvar Integer Input
On entry: the number of variables included in the sum of squares calculations, p .
Constraint: 1 nvar m .
7: k Integer Input
On entry: the number of clusters, K .
Constraint: k2 .
8: cmeans[k×tdc] double Input/Output
On entry: cmeans[i-1×tdc+j-1] must contain the value of the j th variable for the i th initial cluster centre, for i=1,2,,K and j=1,2,,p.
On exit: cmeans[i-1×tdc+j-1] contains the value of the j th variable for the i th computed cluster centre, for i=1,2,,K and j=1,2,,p.
9: tdc Integer Input
On entry: the stride separating matrix column elements in the array cmeans.
Constraint: tdcnvar .
10: wt[n] const double Input
On entry: the elements of wt must contain the weights to be used in the analysis. The effective number of observations is the sum of the weights. If wt[i-1] = 0.0 then the i th observation is not included in the analysis.
If weights are not provided then wt must be set to NULL and the effective number of observations is n.
Constraints:
  • wt[i-1] 0.0 , for i=1,2,,n;
  • wt[i-1] > 0.0 for at least two values of i .
11: inc[n] Integer Output
On exit: inc[i-1] contains the cluster to which the i th object has been allocated, for i=1,2,,n.
12: nic[k] Integer Output
On exit: nic[i-1] contains the number of objects in the i th cluster, for i=1,2,,K.
13: css[k] double Output
On exit: css[i-1] contains the within-cluster (weighted) sum of squares of the i th cluster, for i=1,2,,K.
14: csw[k] double Output
On exit: csw[i-1] contains the within-cluster sum of weights of the i th cluster, for i=1,2,,K. If wt = NULL the sum of weights is the number of objects in the cluster.
15: maxit Integer Input
On entry: the maximum number of iterations allowed in the analysis.
Suggested value: maxit=10 .
Constraint: maxit>0 .
16: fail NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_2_INT_ARG_LT
On entry, m=value while nvar=value . These arguments must satisfy mnvar .
On entry, tdc=value while nvar=value . These arguments must satisfy tdcnvar .
On entry, tdx=value while m=value . These arguments must satisfy tdxm .
NE_ALLOC_FAIL
Dynamic memory allocation failed.
NE_CLUSTER_EMPTY
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.
NE_INT_ARG_LE
On entry, maxit=value.
Constraint: maxit>0.
NE_INT_ARG_LT
On entry, k=value.
Constraint: k2.
On entry, n=value.
Constraint: n2.
On entry, nvar=value.
Constraint: nvar1.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
NE_NEG_WEIGHT_ELEMENT
On entry, wt[value] = value.
Constraint: When referenced, all elements of wt must be non-negative.
NE_TOO_MANY
Too many iterations (value). Convergence has not been achieved within the maximum number of iterations given by maxit. Try increasing maxit and, if possible, use the returned values in cmeans as the initial cluster centres.
NE_VAR_INCL_INDICATED
The number of variables, nvar in the analysis =value , while number of variables included in the analysis via array isx=value .
Constraint: these two numbers must be the same.
NE_WT_ZERO
At least two elements of wt must be greater than zero.

7 Accuracy

g03efc 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

g03efc 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 (Kendall and Stuart (1976)). The data is read in, the K -means clustering performed and the results printed.

10.1 Program Text

Program Text (g03efce.c)

10.2 Program Data

Program Data (g03efce.d)

10.3 Program Results

Program Results (g03efce.r)