Clustering validity assessment: finding the optimal partitioning of a data set

By Maria Halkidi, Michalis Vazirgiannis. Proceedings 2001 IEEE International Conference on Data Mining, 2001

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NbClust: An R Package for Determining the Relevant Number of Clusters in a Data Set (2014, methodology): “2000), and SDbw index which is based on the criteria of compactness and separation between clusters (Halkidi and Vazirgiannis 2001).”
Understanding of Internal Clustering Validation Measures (2010, background): “The S Dbw index (í µí± í µí°·í µí±í µí±¤) [16] takes density into account to measure the inter-cluster separation.”
Density-Based Clustering Validation (2014, background): “The SD and S_Dbw measures [17, 14] have similar definitions, based on concepts of scattering inside clusters and separation between clusters considering the variance and distance between centroids of the clusters.”
An extensive comparative study of cluster validity indices (2013, background): “S_Dbw index (SDbwk) [38]: This is a ratio-type index that has a more complex formulation based on the Euclidean norm JxJ1⁄4 ðx xÞ, the standard deviation of a set of objects, sðXÞ 1⁄4 1=9X9 P xi AX ðxi xÞ(2) and the standard deviation of a partition, stdevðCÞ 1⁄4 1=K ffiffiffiffiffiffiffiffiffiffiff”
Dynamic clustering using particle swarm optimization with application in image segmentation (2006, background): “3.1.3 S_Dbw validity index [49]”