Incremental Cluster Validity Indices for Hard Partitions: Extensions and Comparative Study

V alidation is one of the most important aspects of clustering, but most approaches have been batch methods. Recently, interest has grown in providing incremental alternatives. This paper extends the incremental cluster validity index (iCVI) family to include incremental versions of Calinski-Harabasz (iCH), I index and Pakhira-Bandyopadhyay-Maulik (iI and iPBM), Silhouette (iSIL), Negentropy Increment (iNI), Representative Cross Information Potential (irCIP) and Representative Cross Entropy (irH), and Conn Index (iConn Index). Additionally, the effect of under-and over-partitioning on the behavior of these six iCVIs, the Partition Separation (PS) index, as well as two other recently developed iCVIs (incremental Xie-Beni (iXB) and incremental Davies-Bouldin (iDB)) was examined through a comparative study. Experimental results using fuzzy adaptive resonance theory (ART)-based clustering methods showed that while evidence of most under-partitioning cases could be inferred from the behaviors of all these iCVIs, over-partitioning was found to be a more challenging scenario indicated only by the iConn Index. The expansion of incremental validity indices provides significant novel opportunities for assessing and interpreting the results of unsupervised learning. L. E. Brito da Silva is with the Applied Computational Intelligence Laboratory, Department of Electrical and Computer Engineering, Missouri University of Science and Technology, Rolla, MO 65409 USA, and also with the CAPES Foundation, Ministry of Education of Brazil, Bras ılia, DF 70040-020, Brazil (email: leonardoenzo@ieee.org). N. M. Melton is with the Applied Computational Intelligence Laboratory, Department of Electrical and Computer Engineering, Missouri University of Science and Technology, Rolla, MO 65409 USA (email: niklasmelton@ieee.org). D. C. Wunsch II is with the Applied Computational Intelligence Laboratory, Department of Electrical and Computer Engineering, Missouri University of Science and Technology, Rolla, MO 65409 USA (email: wunsch@ieee.org). I NTRODUCTION Cluster validation [1] is a critical topic in cluster analysis.