{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,18]],"date-time":"2026-07-18T06:37:21Z","timestamp":1784356641190,"version":"3.55.0"},"reference-count":19,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2021,6,16]],"date-time":"2021-06-16T00:00:00Z","timestamp":1623801600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Grouping the objects based on their similarities is an important common task in machine learning applications. Many clustering methods have been developed, among them k-means based clustering methods have been broadly used and several extensions have been developed to improve the original k-means clustering method such as k-means ++ and kernel k-means. K-means is a linear clustering method; that is, it divides the objects into linearly separable groups, while kernel k-means is a non-linear technique. Kernel k-means projects the elements to a higher dimensional feature space using a kernel function, and then groups them. Different kernel functions may not perform similarly in clustering of a data set and, in turn, choosing the right kernel for an application could be challenging. In our previous work, we introduced a weighted majority voting method for clustering based on normalized mutual information (NMI). NMI is a supervised method where the true labels for a training set are required to calculate NMI. In this study, we extend our previous work of aggregating the clustering results to develop an unsupervised weighting function where a training set is not available. The proposed weighting function here is based on Silhouette index, as an unsupervised criterion. As a result, a training set is not required to calculate Silhouette index. This makes our new method more sensible in terms of clustering concept.<\/jats:p>","DOI":"10.3390\/e23060759","type":"journal-article","created":{"date-parts":[[2021,6,16]],"date-time":"2021-06-16T10:27:07Z","timestamp":1623839227000},"page":"759","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":320,"title":["Silhouette Analysis for Performance Evaluation in Machine Learning with Applications to Clustering"],"prefix":"10.3390","volume":"23","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1454-2962","authenticated-orcid":false,"given":"Meshal","family":"Shutaywi","sequence":"first","affiliation":[{"name":"Department of Mathematics, King Abdulaziz University, Rabigh 21911, Saudi Arabia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9397-1807","authenticated-orcid":false,"given":"Nezamoddin N.","family":"Kachouie","sequence":"additional","affiliation":[{"name":"Department of Mathematical Sciences, Florida Institute of Technology, Melbourne, FL 32901, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2021,6,16]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Gentleman, R., Carey, V., Huber, W., Irizarry, R., and Dudoit, S. 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