On cluster validation

Clustering is an unsupervised learning process, hence it is difficult to find the optimal cluster number. Cluster validation is a process in which a cluster validity index (CVI) is constructed to evaluate the quality of clustering results and determine the optimal cluster number. Firstly, the mathematical description of clustering and the classification of CVIs are introduced. Then, 12 CVIs only considering the geometry information of the data set, 6 CVIs only considering the degree of membership, and 9 CVIs considering both the geometry information of the data set and the degree of membership are reviewed respectively based on different components in the indices. And the status quo of each type of CVIs is analyzed. Afterwards, the studies of other types of CVIs, such as external and stability-based indices, are briefly summarized. Finally, we point out the main challenges and research directions in the area of cluster validation.