  There are many cluster analysis methods that can produce quite different
clusterings on the same dataset. Cluster validation is about the evaluation of
the quality of a clustering; "relative cluster validation" is about using such
criteria to compare clusterings. This can be used to select one of a set of
clusterings from different methods, or from the same method ran with different
parameters such as different numbers of clusters.
  There are many cluster validation indexes in the literature. Most of them
attempt to measure the overall quality of a clustering by a single number, but
this can be inappropriate. There are various different characteristics of a
clustering that can be relevant in practice, depending on the aim of
clustering, such as low within-cluster distances and high between-cluster
separation.
  In this paper, a number of validation criteria will be introduced that refer
to different desirable characteristics of a clustering, and that characterise a
clustering in a multidimensional way. In specific applications the user may be
interested in some of these criteria rather than others. A focus of the paper
is on methodology to standardise the different characteristics so that users
can aggregate them in a suitable way specifying weights for the various
criteria that are relevant in the clustering application at hand.
