  Data clustering involves identifying latent similarities within a dataset and
organizing them into clusters or groups. The outcomes of various clustering
algorithms differ as they are susceptible to the intrinsic characteristics of
the original dataset, including noise and dimensionality. The effectiveness of
such clustering procedures directly impacts the homogeneity of clusters,
underscoring the significance of evaluating algorithmic outcomes. Consequently,
the assessment of clustering quality presents a significant and complex
endeavor. A pivotal aspect affecting clustering validation is the cluster
validity metric, which aids in determining the optimal number of clusters. The
main goal of this study is to comprehensively review and explain the
mathematical operation of internal and external cluster validity indices, but
not all, to categorize these indices and to brainstorm suggestions for future
advancement of clustering validation research. In addition, we review and
evaluate the performance of internal and external clustering validation indices
on the most common clustering algorithms, such as the evolutionary clustering
algorithm star (ECA*). Finally, we suggest a classification framework for
examining the functionality of both internal and external clustering validation
measures regarding their ideal values, user-friendliness, responsiveness to
input data, and appropriateness across various fields. This classification aids
researchers in selecting the appropriate clustering validation measure to suit
their specific requirements.
