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The CLUSTER Procedure |
Ultrametrics |
A dissimilarity measure is called an ultrametric if it satisfies the following conditions:
for all
for all
,
for all
,
for all
,
, and
Any hierarchical clustering method induces a dissimilarity measure on the observations—say, . Let
be the cluster with the fewest members that contains both
and
. Assume
was formed by joining
and
. Then define
.
If the fusion of and
reduces the number of clusters from
to
, then define
. Johnson (1967) shows that if
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then is an ultrametric. A method that always satisfies this condition is said to be a monotonic or ultrametric clustering method. All methods implemented in PROC CLUSTER except CENTROID, EML, and MEDIAN are ultrametric (Milligan; 1979; Batagelj; 1981).
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