void	nag_mv_hierar_cluster_analysis (Nag_ClusterMethod method, Integer n, double d[], Integer ilc[], Integer iuc[], double cd[], Integer iord[], double dord[], NagError *fail)

3 Description

Given a distance or dissimilarity matrix for

n

objects (see nag_mv_distance_mat (g03eac)), cluster analysis aims to group the

n

objects into a number of more or less homogeneous groups or clusters. With agglomerative clustering methods, a hierarchical tree is produced by starting with

n

clusters, each with a single object and then at each of

n - 1

stages, merging two clusters to form a larger cluster, until all objects are in a single cluster. This process may be represented by a dendrogram (see nag_mv_dendrogram (g03ehc)).

At each stage, the clusters that are nearest are merged, methods differ as to how the distance between the new cluster and other clusters are computed. For three clusters

i

j

and

k

let

n_{i}

n_{j}

and

n_{k}

be the number of objects in each cluster and let

d_{i j}

d_{i k}

and

d_{j k}

be the distances between the clusters. Let clusters

j

and

k

be merged to give cluster

j k

, then the distance from cluster

i

to cluster

j k

d_{i . j k}

can be computed in the following ways:

1.	Single link or nearest neighbour: $d_{i . j k} = \min (d_{i j}, d_{i k})$ .
2.	Complete link or furthest neighbour: $d_{i . j k} = \max (d_{i j}, d_{i k})$ .
3.	Group average: $d_{i . j k} = \frac{n_{j}}{n_{j} + n_{k}} d_{i j} + \frac{n_{k}}{n_{j} + n_{k}} d_{i k}$ .
4.	Centroid: $d_{i . j k} = \frac{n_{j}}{n_{j} + n_{k}} d_{i j} + \frac{n_{k}}{n_{j} + n_{k}} d_{i k} - \frac{n_{j} n_{k}}{{(n_{j} + n_{k})}^{2}} d_{j k}$ .
5.	Median: $d_{i . j k} = \frac{1}{2} d_{i j} + \frac{1}{2} d_{i k} - \frac{1}{4} d_{j k}$ .
6.	Minimum variance: $d_{i . j k} = \{(n_{i} + n_{j}) d_{i j} + (n_{i} + n_{k}) d_{i k} - n_{i} d_{j k}\} / (n_{i} + n_{j} + n_{k})$ .

For further details see Everitt (1974) or Krzanowski (1990).

If the clusters are numbered

1, 2, \dots, n

then, for convenience, if clusters

j

and

k

j < k

, merge then the new cluster will be referred to as cluster

j

. Information on the clustering history is given by the values of

j

k

and

d_{j k}

for each of the

n - 1

clustering steps. In order to produce a dendrogram, the ordering of the objects such that the clusters that merge are adjacent is required. This ordering is computed so that the first element is 1. The associated distances with this ordering are also computed.

4 References

Everitt B S (1974) Cluster Analysis Heinemann

Krzanowski W J (1990) Principles of Multivariate Analysis Oxford University Press

5 Arguments

1: $method$ – Nag_ClusterMethodInput

On entry: indicates which clustering.

$method = Nag_SingleLink$: Single link.
$method = Nag_CompleteLink$: Complete link.
$method = Nag_GroupAverage$: Group average.
$method = Nag_Centroid$: Centroid.
$method = Nag_Median$: Median.
$method = Nag_MinVariance$: Minimum variance.

Constraint:

method = Nag_SingleLink

Nag_CompleteLink

Nag_GroupAverage

Nag_Centroid

Nag_Median

Nag_MinVariance

2: $n$ – IntegerInput

On entry: the number of objects,

n

Constraint:

n \geq 2

3: $d [n \times (n - 1) / 2]$ – doubleInput/Output

On entry: the strictly lower triangle of the distance matrix.

D

must be stored packed by rows, i.e.,

d [(i - 1) (i - 2) / 2 + j - 1]

i > j

must contain

d_{i j}

On exit: is overwritten.

Constraint:

d [i - 1] \geq 0.0

, for

i = 1, 2, \dots, n (n - 1) / 2

4: $ilc [n - 1]$ – IntegerOutput

On exit:

ilc [l - 1]

contains the number,

j

, of the cluster merged with cluster

k

(see iuc),

j < k

, at step

l

, for

l = 1, 2, \dots, n - 1

5: $iuc [n - 1]$ – IntegerOutput

On exit:

iuc [l - 1]

contains the number,

k

, of the cluster merged with cluster

j

j < k

, at step

l

, for

l = 1, 2, \dots, n - 1

6: $cd [n - 1]$ – doubleOutput

On exit:

cd [l - 1]

contains the distance

d_{j k}

, between clusters

j

and

k

j < k

, merged at step

l

, for

l = 1, 2, \dots, n - 1

7: $iord [n]$ – IntegerOutput

On exit: the objects in dendrogram order.

8: $dord [n]$ – doubleOutput

On exit: the clustering distances corresponding to the order in iord.

dord [l - 1]

contains the distance at which cluster

iord [l - 1]

and

iord [l]

merge, for

l = 1, 2, \dots, n - 1

dord [n - 1]

contains the maximum distance.

9: $fail$ – NagError *Input/Output

The NAG error argument (see Section 2.7 in How to Use the NAG Library and its Documentation).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
NE_BAD_PARAM: On entry, argument method had an illegal value.
NE_DENDROGRAM: A true dendrogram cannot be formed because the distances at which clusters have merged are not increasing for all steps, i.e., $cd [i - 1] < cd [i - 2]$ for some $i = 2, 3, \dots, n - 1$ . This can occur for the $method = Nag_Centroid$ and $method = Nag_Median$ methods.
NE_INT_ARG_LT: On entry, $n = 〈value〉$ .
Constraint: $n \geq 2$ .
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 contact NAG for assistance.
NE_REALARR: On entry, $d [〈value〉] = 〈value〉$ .
Constraint: $d [i - 1] \geq 0.0$ , for $i = 1, 2, \dots, n \times (n - 1) / 2$ .

7 Accuracy

For methods other than

method = Nag_SingleLink

Nag_CompleteLink

, slight rounding errors may occur in the calculations of the updated distances. These would not normally significantly affect the results, however there may be an effect if distances are (almost) equal.

If at a stage, two distances

d_{i j}

and

d_{k l}

i < k

i = k

and

j < l

, are equal then clusters

k

and

l

will be merged rather than clusters

i

and

j

. For single link clustering this choice will only affect the order of the objects in the dendrogram. However, for other methods the choice of

k l

rather than

i j

may affect the shape of the dendrogram. If either of the distances

d_{i j}

d_{k l}

are affected by rounding errors then their equality, and hence the dendrogram, may be affected.

8 Parallelism and Performance

nag_mv_hierar_cluster_analysis (g03ecc) is not threaded in any implementation.

9 Further Comments

The dendrogram may be formed using nag_mv_dendrogram (g03ehc). Groupings based on the clusters formed at a given distance can be computed using nag_mv_cluster_indicator (g03ejc).

10 Example

Data consisting of three variables on five objects are read in. Euclidean squared distances based on two variables are computed using nag_mv_distance_mat (g03eac), the objects are clustered using nag_mv_hierar_cluster_analysis (g03ecc) and the dendrogram computed using nag_mv_dendrogram (g03ehc). The dendrogram is then printed.

NAG Library Function Documentnag_mv_hierar_cluster_analysis (g03ecc)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG Library Function Document

nag_mv_hierar_cluster_analysis (g03ecc)