nag_mv_hierar_cluster_analysis (g03ecc) performs hierarchical cluster analysis.

Given a distance or dissimilarity matrix for  objects (see g03eac (nag_mv_distance_mat)), cluster analysis aims to group the  objects into a number of more or less homogeneous groups or clusters. With agglomerative clustering methods, a hierarchical tree is produced by starting with  clusters, each with a single object and then at each of  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 g03ehc (nag_mv_dendrogram)).

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 ,  and  let ,  and  be the number of objects in each cluster and let ,  and  be the distances between the clusters. Let clusters  and  be merged to give cluster , then the distance from cluster  to cluster ,  can be computed in the following ways:

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

If the clusters are numbered  then, for convenience, if clusters  and , , merge then the new cluster will be referred to as cluster . Information on the clustering history is given by the values of ,  and  for each of the  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.

"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.,  for some . This can occur for the  and  methods.

"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_REALARR"

On entry, . Constraint: , .

For methods other than   "Nag_SingleLink" or "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  and ,  or  and , are equal then clusters  and  will be merged rather than clusters  and . For single link clustering this choice will only affect the order of the objects in the dendrogram. However, for other methods the choice of  rather than  may affect the shape of the dendrogram. If either of the distances  or  are affected by rounding errors then their equality, and hence the dendrogram, may be affected.

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