Comprehensive cluster validity Index based on structural simplicity

Nonhierarchical clustering depending on unsupervised algorithms may not retrieve the optimal partition of datasets. Determining if clusters fit ``natural partitions`` can be achieved using cluster validity indices (CVIs). Most existing CVIs consider criteria such as cohesion, separation, and their equivalents. However, these binary relations may provide neither the optimal measure of partition suitability nor reference values corresponding to the worst partition. Moreover, previous CVI studies have been mostly focused on fitting correct partitions according to researchers' a priori assumptions. In contrast, we investigated desirable properties of CVIs, namely, scale shift transform invariance, optimal clustering, and unbiased clustering with representing the worst partition. Then, we conducted experiments to evaluate whether existing CVIs fulfill these properties. As none of these CVIs fulfilled the desired properties, we propose the simplicity index, which measures the simplicity of tree structures in clusters. The simplicity index is the unique index invariant to the ``correct rate`` and provides both a reference indicating the most complex partition and the best value indicating the simplest one.