The cross validation concepts described above can be

The cross-validation concepts described above can be applied including
the validation data set and testing data set in analogy.
4.2 Cluster Evaluation
There are numerous methods to choose from to perform cluster analysis and
the same now applies to cluster evaluation. As with the choice of algorithm,
one needs to understand the specifics of the data, to which it is applied, and
the objectives to be able to use the parameters of evaluation correctly to
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Fig. 21 K-fold cross-validation procedure

16 Mathematical Background of Machine Learning
299
determine which cluster analysis is more appropriate for the case. There is
usually not a single or universal measure, but rather a bundle of them.
The general requirements of evaluation include determining the perfor-
mance of the applied algorithm and deciding on the number size, shape, etc.
of clusters or sometimes even sub-clusters. Cluster validity indices (CVI) can
be used to assess the issue.
There are the two types of CVIs: internal and external. External CVIs can
be used only when the true number of clusters is known (hence the name)
since it then compares the result of a procedure partition to it. Internal CVIs
are to be applied to all other cases. The other distinction is their method
applicability, which can either be a crisp (hard) or a fuzzy (soft) clustering. 11
However, the former might be converted into the latter if needed.
Of over half a hundred different indices, we describe some of those
we had an option to choose from 12 when evaluating the clusters created
throughout this chapter.
The following external CVIs were selected in our cluster evaluation
process:
Crisp partitions :
• Rand index (to be maximized).
• Adjusted rand index (to be maximized).
• Jaccard index (to be maximized).
• Folkes–Mallows (to be maximized).
• Variation of information (to be minimized).
Fuzzy partitions :
• Soft rand index (to be maximized).
• Soft adjusted rand index (to be maximized).
• Soft variation of information (to be minimized).
• Soft normalized mutual information based on max entropy (to be
maximized).
Since the external cluster validity indices gauge the propriety of data distri-
bution into the simulated clusters in accordance with the distribution in the
true ones, one needs to deal with a confusion matrix, which is based on the
11
Hard or crisp clustering involves strict and excluding placement of a data point in relation to a par-
ticular cluster, meaning that an observation cannot belong to two or more clusters at the same time.
This would be possible under using fuzzy or soft partitioning and one point would belong to clusters to
differing extents.
12
The list of CVIs is taken from Sarda-Espinosa ( 2019 ).