Matrix(1..n, 1..tdx, datatype=float[8], order=C_order);

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NAG[g03efc] NAG[nag_mv_kmeans_cluster_analysis] - -means

Calling Sequence

g03efc(x, isx, cmeans, inc, nic, css, csw, 'n'=n, 'm'=m, 'tdx'=tdx, 'nvar'=nvar, 'k'=k, 'tdc'=tdc, 'wt'=wt, 'maxit'=maxit, 'fail'=fail)

nag_mv_kmeans_cluster_analysis(. . .)

Parameters

x - Matrix(1..n, 1..tdx, datatype=float[8], order=C_order);

On entry: must contain the value of th variable for the th object for ; .

isx - Vector(1..m, datatype=integer[kernelopts('wordsize')/8]);

On entry: indicates whether or not the th variable is to be included in the analysis.

If , then the th variable contained in the th column of x is included, for .

Constraint: for nvar values of .

cmeans - Matrix(1..k, 1..tdc, datatype=float[8], order=C_order);

On entry: must contain the value of the th variable for the th initial cluster centre, for ; .

On exit: contains the value of the th variable for the th computed cluster centre, for ; .

inc - Vector(1..n, datatype=integer[kernelopts('wordsize')/8]);

On exit: contains the cluster to which the th object has been allocated, for .

nic - Vector(1..k, datatype=integer[kernelopts('wordsize')/8]);

On exit: contains the number of objects in the th cluster, for

css - Vector(1..k, datatype=float[8]);

On exit: contains the within-cluster (weighted) sum of squares of the th cluster, for .

csw - Vector(1..k, datatype=float[8]);

On exit: contains the within-cluster sum of weights of the th cluster, for . If the sum of weights is the number of objects in the cluster.

'n'=n - integer; (optional)

Default value: the first dimension of the arrays x, wt, inc.

On entry: the number of observations, .

Constraint: . .

'm'=m - integer; (optional)

Default value: the first dimension of the array isx and the second dimension of the array isxthe array x.

On entry: the number of variables in the array x.

Constraint: . .

'tdx'=tdx - integer; (optional)

On entry: the second dimension of the array x as declared in the function from which nag_mv_kmeans_cluster_analysis (g03efc) is called.

Constraint: . .

'nvar'=nvar - integer; (optional)

Default value: the second dimension of the array cmeans.

On entry: the number of variables included in the sum of squares calculations, .

Constraint: . .

'k'=k - integer; (optional)

Default value: the first dimension of the arrays cmeans, nic, css, csw.

On entry: the number of clusters, .

Constraint: . .

'tdc'=tdc - integer; (optional)

On entry: the second dimension of the array cmeans as declared in the function from which nag_mv_kmeans_cluster_analysis (g03efc) is called.

Constraint: . .

'wt'=wt - Vector(1..n, datatype=float[8]); (optional)

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 then the th observation is not included in the analysis.

Constraints:

, for ;

for at least two values of .

Note: if wt is set to the null pointer NULL, i.e., (double *)0, then wt is not referenced and the effective number of observations is .

'maxit'=maxit - integer; (optional)

On entry: the maximum number of iterations allowed in the analysis.

Constraint: . .

Suggested value: (default: ) .

'fail'=fail - table; (optional)

The NAG error argument, see the documentation for NagError.

Description

	Purpose
	nag_mv_kmeans_cluster_analysis (g03efc) performs -means cluster analysis.

Description

Given objects with variables measured on each object, for ; , nag_mv_kmeans_cluster_analysis (g03efc) allocates each object to one of groups or clusters to minimize the within-cluster sum of squares:

(∑)(∑)[i in S[k]](∑)(x[ij],-,(x)[kj])^2

where is the set of objects in the th cluster and is the mean for the variable over cluster . This is often known as -means clustering.

In addition to the data matrix, a by matrix giving the initial cluster centres for the 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 -partition with locally optimal within-cluster sum of squares by moving points from one cluster to another.

Optionally, weights for each object, , can be used so that the clustering is based on within-cluster weighted sums of squares:

(∑)(∑)[i in S[k]](∑)w[i](x[ij],-,(x˜)[kj])^2

where is the weighted mean for variable over cluster .

The function is based on the algorithm of Hartigan and Wong (1979).

Error Indicators and Warnings

"NE_2_INT_ARG_LT"

On entry, while . These arguments must satisfy .

"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 must not be less than or equal to 0: .

"NE_INT_ARG_LT"

On entry, n must not be less than 2: .

"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 consult NAG for assistance.

"NE_NEG_WEIGHT_ELEMENT"

On entry, . Constraint: When referenced, all elements of wt must be non-negative.

"NE_TOO_MANY"

Too many iterations (). 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 , while number of variables included in the analysis via array . Constraint: these two numbers must be the same.

"NE_WT_ZERO"

At least two elements of wt must be greater than zero.

	Accuracy
	nag_mv_kmeans_cluster_analysis (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.

	Further Comments
	The time per iteration is approximately proportional to .

Examples

n := 20:
m := 5:
tdx := 5:
nvar := 5:
k := 3:
tdc := 5:
maxit := 10:
x := Matrix([[77.3, 13, 9.699999999999999, 1.5, 6.4], [82.5, 10, 7.5, 1.5, 6.5], [66.90000000000001, 20.6, 12.5, 2.3, 7], [47.2, 33.8, 19, 2.8, 5.8], [65.3, 20.5, 14.2, 1.9, 6.9], [83.3, 10, 6.7, 2.2, 7], [81.59999999999999, 12.7, 5.7, 2.9, 6.7], [47.8, 36.5, 15.7, 2.3, 7.2], [48.6, 37.1, 14.3, 2.1, 7.2], [61.6, 25.5, 12.9, 1.9, 7.3], [58.6, 26.5, 14.9, 2.4, 6.7], [69.3, 22.3, 8.4, 4, 7], [61.8, 30.8, 7.4, 2.7, 6.4], [67.7, 25.3, 7, 4.8, 7.3], [57.2, 31.2, 11.6, 2.4, 6.5], [67.2, 22.7, 10.1, 3.3, 6.2], [59.2, 31.2, 9.6, 2.4, 6], [80.2, 13.2, 6.6, 2, 5.8], [82.2, 11.1, 6.7, 2.2, 7.2], [69.7, 20.7, 9.6, 3.1, 5.9]], datatype=float[8], order='C_order'):
isx := Vector([1, 1, 1, 1, 1], datatype=integer[kernelopts('wordsize')/8]):
cmeans := Matrix([[82.5, 10, 7.5, 1.5, 6.5], [47.8, 36.5, 15.7, 2.3, 7.2], [67.2, 22.7, 10.1, 3.3, 6.2]], datatype=float[8], order='C_order'):
wt := Vector([], datatype=float[8]):
inc := Vector(20, datatype=integer[kernelopts('wordsize')/8]):
nic := Vector(3, datatype=integer[kernelopts('wordsize')/8]):
css := Vector(3, datatype=float[8]):
csw := Vector(3, datatype=float[8]):
NAG:-g03efc(x, isx, cmeans, inc, nic, css, csw, 'n' = n, 'm' = m, 'tdx' = tdx, 'nvar' = nvar, 'k' = k, 'tdc' = tdc, 'wt' = wt, 'maxit' = maxit):