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# Cluster Analysis

The routines implemented are CLUSTER which has 8 options for hierarchical clustering and PARTITION which carries out non-hierarchical clustering. We will look at the hierarchical options available first.

The automatic classification of the n row-objects of an n by mtable generally produces output in one of two forms: the assignments to clusters found for the n objects; or a series of clusterings of the n objects, from the initial situation when each object may be considered a singleton cluster to the other extreme when all objects belong to one cluster. The former is non-hierarchical clustering or partitioning.

The latter is hierarchical clustering. Brief consideration will show that a sequence of n-1 agglomerations are needed to successively merge the two closest objects and/or clusters at each stage, so that we have a set of n (singleton) clusters, n-1 clusters, ..., 2 clusters, 1 cluster. This is usually represented by a hierarchic tree or a dendrogram, and a ``slice'' through the dendrogram defines a partition of the objects. Unfortunately, no rigid guideline can be indicated for deriving such a partition from a dendrogram except that large increases in cluster criterion values (which scale the dendrogram) can indicate a partition of interest.

In carrying out the sequence of agglomerations, various criteria are feasible for defining the newly-constituted cluster:

The minimum variance criterion
(method MVAR) constructs clusters which are of minimal variance internally (i.e. compact) and maximal variance externally (i.e. isolated). It is useful for synoptic clustering, and for all clustering work where another method cannot be explicitly justified.
The minimum variance hierarchy:
All options, with the exception of MNVR, construct a set of Euclidean distances from the input set of n vectors. Thus the internal storage required is large. Option MNVR allows a minimum variance hierarchy (identical to option MVAR) to be obtained, without requiring storage of distances. Computational time is slightly higher than the latter option.
The single link method
(method SLNK) often gives a very skew or "chained" hierarchy. It is therefore not useful for summarising data, but may indicate very anomalous or outlying objects, -- these will be among the last to be agglomerated in the hierarchy.
The complete link method
(method CLNK) often does not differ unduly from the minimum variance method, but its restrictive criterion is not suitable if the data is noisy.
The average link method
(method ALNK) is a reasonable compromise between the (lax) single link method and the (rigid) complete link criterion: all of these methods may be of interest if a graph representation of the results of the clustering is desired.
The weighted average link method
(method WLNK) does not take the relative sizes of clusters into account in agglomerating them. This, and the two following methods, are included for completeness and for consistency with other software packages, but are not recommended for general use.
The median method
(method MEDN) replaces a cluster, on agglomeration, with the median value. It is not guaranteed that these criterion values will vary monotonically, and this may present difficulty with the interpretation of the dendrogram representation.
The centroid method
(method CNTR) replaces a cluster, on agglomeration, with the centroid value. As in the case of the last option, reversals or inversions in the hierarchy are possible.

The Minimal Spanning Tree, which is closely related to the single link method, has been used in such applications as interferogram analysis and in galaxy clustering studies. It is useful as a detector of outlying data points (i.e. anomalous objects).

Routine PARTITION operates in one two options. For both, a partition of minimum variance, given the number of clusters, is sought. Two iterative refinement algorithms (minimum distance or the exchange method) constitute the options available.

Next: Discriminant Analysis Up: Multivariate Analysis Methods Previous: Principal Components Analysis
Petra Nass
1999-06-15