{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,9]],"date-time":"2026-07-09T04:16:56Z","timestamp":1783570616131,"version":"3.55.0"},"reference-count":37,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2022,9,23]],"date-time":"2022-09-23T00:00:00Z","timestamp":1663891200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2022,9,23]],"date-time":"2022-09-23T00:00:00Z","timestamp":1663891200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Classif"],"published-print":{"date-parts":[[2022,11]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>In the cluster analysis literature, there are several partitioning (non-hierarchical) methods for clustering multivariate objects based on model estimation. Distinct to these methods is the use of a system of <jats:italic>n<\/jats:italic> nested statistical models and the optimization of a loss function to best-fit a clustering model to observed data. Many hierarchical clustering methods are not model-based where hierarchy is obtained using a divisive or agglomerative greedy procedure. This paper aims to fill this gap by proposing a novel hierarchical cluster analysis methodology called Hierarchical Means Clustering. HMC produces a set of nested partitions with a centroid-based model estimated via least-squares by minimizing the total within-cluster deviance of the <jats:italic>n<\/jats:italic> partitions in the hierarchy. Hierarchical Means Clustering produces a hierarchy formed by <jats:italic>n<\/jats:italic>-1 nested partitions from 2 to <jats:italic>n<\/jats:italic> clusters with minimal total cluster deviance. Six real data examples are featured, and key links to <jats:italic>k<\/jats:italic>-means, Ward\u2019s method, Bisecting <jats:italic>k<\/jats:italic>-means and model-based hierarchical agglomerative clustering methods are discussed.<\/jats:p>","DOI":"10.1007\/s00357-022-09419-7","type":"journal-article","created":{"date-parts":[[2022,9,23]],"date-time":"2022-09-23T01:02:43Z","timestamp":1663894963000},"page":"553-577","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":41,"title":["Hierarchical Means Clustering"],"prefix":"10.1007","volume":"39","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3876-444X","authenticated-orcid":false,"given":"Maurizio","family":"Vichi","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1816-3521","authenticated-orcid":false,"given":"Carlo","family":"Cavicchia","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6683-8971","authenticated-orcid":false,"given":"Patrick J. F.","family":"Groenen","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2022,9,23]]},"reference":[{"key":"9419_CR1","unstructured":"Arthur, D., & Vassilvitskii, S. (2007). K-Means$$++$$: The Advantages of Careful Seeding. In: Proceedings of the Eighteenth Annual ACM-SIAM Symposium on Discrete Algorithms. SODA \u201907. New Orleans, Louisiana: Society for Industrial and Applied Mathematics, pp. 1027-1035. ISBN: 9780898716245"},{"key":"9419_CR2","doi-asserted-by":"publisher","first-page":"803","DOI":"10.2307\/2532201","volume":"49","author":"JD Banfield","year":"1993","unstructured":"Banfield, J. D., & Raftery, A. E. (1993). Model-based gaussian and non-gaussian clustering. Biometrics, 49, 803\u2013821.","journal-title":"Biometrics"},{"key":"9419_CR3","unstructured":"Baxter, M. J. (1994). Exploratory multivariate analysis in archaeology. Edinburgh: Edinburgh University Press. . ISBN: 0748604235"},{"key":"9419_CR4","doi-asserted-by":"publisher","first-page":"781","DOI":"10.1016\/0031-3203(94)00125-6","volume":"28","author":"G Celeux","year":"1995","unstructured":"Celeux, G., & Govaert, G. (1995). Gaussian parsimonious clustering models. Pattern Recognition, 28, 781\u2013793.","journal-title":"Pattern Recognition"},{"key":"9419_CR5","doi-asserted-by":"crossref","unstructured":"Coomans, D. et al. (1983). Comparison of multivariate discrimination techniques for clinical data-application to the thyroid functional state. In: Methods of information in medicine 22.2, pp. 93-101.","DOI":"10.1055\/s-0038-1635425"},{"key":"9419_CR6","doi-asserted-by":"crossref","unstructured":"Cormack, R. M. (1971). A Review of Classification. In: Journal of the Royal Statistical Society. Series A (General) 134.3, pp. 321-367.","DOI":"10.2307\/2344237"},{"key":"9419_CR7","unstructured":"Cormen, T. H., Leiserson, C. E., & Rivest R. L. (1990). Introduction to Algorithms. 1st. Cambridge, MA: The MIT Press."},{"key":"9419_CR8","doi-asserted-by":"crossref","unstructured":"Everitt, B. et al. (2011). Cluster analysis. 5th. Wiley. ISBN: 978-0-470-74991-3.","DOI":"10.1002\/9780470977811"},{"key":"9419_CR9","doi-asserted-by":"publisher","first-page":"270","DOI":"10.1137\/S1064827596311451","volume":"20","author":"C Fraley","year":"1998","unstructured":"Fraley, C. (1998). Algorithms for model-based Gaussian hierarchical clustering. SIAM Journal on Scientific Computing, 20, 270\u2013281.","journal-title":"SIAM Journal on Scientific Computing"},{"key":"9419_CR10","doi-asserted-by":"publisher","unstructured":"Galili, T. (2015). dendextend: an R package for visualizing, adjusting, and comparing trees of hierarchical clustering. In: Bioinformatics. https:\/\/doi.org\/10.1093\/bioinformatics\/btv428","DOI":"10.1093\/bioinformatics\/btv428"},{"key":"9419_CR11","unstructured":"Gordon, A. D. (1981). Classification. 1st ed. Chapman and Hall\/CRC."},{"key":"9419_CR12","doi-asserted-by":"crossref","unstructured":"Gordon, A. D. (1999). Classification. 2nd ed. Chapman and Hall\/CRC.","DOI":"10.1201\/9781584888536"},{"key":"9419_CR13","doi-asserted-by":"crossref","unstructured":"Hartigan, J. A. (1967). Representation of similarity matrices by trees. In: Journal of the American Statistical Association 62.320, pp. 1140-1158.","DOI":"10.1080\/01621459.1967.10500922"},{"key":"9419_CR14","unstructured":"Hartigan, J. A. (1975). Clustering algorithms. New York: Wiley."},{"key":"9419_CR15","doi-asserted-by":"crossref","unstructured":"Hubert, L., Arabie, P. (1985). Comparing partitions. In: Journal of Classification 2.1, pp. 193-218.","DOI":"10.1007\/BF01908075"},{"key":"9419_CR16","doi-asserted-by":"crossref","unstructured":"Hurley, C. B. (2004). Clustering visualizations of multidimensional data. In: Journal of Computational and Graphical Statistics 13.4, pp. 788-806.","DOI":"10.1198\/106186004X12425"},{"key":"9419_CR17","doi-asserted-by":"publisher","unstructured":"Khomtchouk, B. B. (2020). Codon usage bias levels predict taxonomic identity and genetic composition. In: bioRxiv. https:\/\/doi.org\/10.1101\/2020.10.26.356295.","DOI":"10.1101\/2020.10.26.356295."},{"key":"9419_CR18","doi-asserted-by":"crossref","unstructured":"K\u0159iv\u00e1nek, M. & Mor\u00e1vek, J. (1986). NP-hard problems in hierarchical-tree clustering. In: Acta Informatica 23.3, pp. 311-323.","DOI":"10.1007\/BF00289116"},{"issue":"4","key":"9419_CR19","first-page":"373","volume":"9","author":"GN Lance","year":"1967","unstructured":"Lance, G. N., & Williams, W. T. (1967). A general theory of classificatory sorting strategies: 1. Hierarchical Systems. In: The Computer Journal, 9(4), 373\u2013380.","journal-title":"Hierarchical Systems. In: The Computer Journal"},{"key":"9419_CR20","doi-asserted-by":"crossref","unstructured":"Lloyd, S. (1982). Least squares quantization in PCM. In: IEEE Transactions on Information Theory 28.2, pp. 129-137.","DOI":"10.1109\/TIT.1982.1056489"},{"key":"9419_CR21","unstructured":"Luketa, S. (2012). New views on the megaclassification of life. In: Protistology 7.2, pp. 218-237."},{"key":"9419_CR22","unstructured":"MacQueen, J. (1967). Some methods for classification and analysis of multivariate observations. In: Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability, Volume 1: Statistics. Berkeley, Calif.: University of California Press, pp. 281-297."},{"key":"9419_CR23","doi-asserted-by":"publisher","unstructured":"M\u00fcllner, D. (2011). Modern hierarchical, agglomerative clustering algorithms. arXiv. https:\/\/doi.org\/10.48550\/ARXIV.1109.2378","DOI":"10.48550\/ARXIV.1109.2378"},{"key":"9419_CR24","doi-asserted-by":"publisher","first-page":"274","DOI":"10.1007\/s00357-014-9161-z","volume":"31","author":"F Murtagh","year":"2014","unstructured":"Murtagh, F., & Legendre, P. (2014). Ward\u2019s hierarchical agglomerative clustering method: Which algorithms implement ward\u2019s criterion? Journal of Classification, 31, 274\u2013295.","journal-title":"Journal of Classification"},{"key":"9419_CR25","unstructured":"R Core Team (2021). R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing. Vienna, Austria. https:\/\/www.R-project.org\/"},{"key":"9419_CR26","doi-asserted-by":"crossref","unstructured":"Rand, W. M. (1971). Objective criteria for the evaluation of clustering methods. In: Journal of the American Statistical Association 66.336, pp. 846-850.","DOI":"10.1080\/01621459.1971.10482356"},{"key":"9419_CR27","doi-asserted-by":"crossref","unstructured":"Rubin, J. (1967). Optimal classification into groups: An approach for solving the taxonomy problem. In: Journal of Theoretical Biology 15.1, pp. 103-144.","DOI":"10.1016\/0022-5193(67)90046-X"},{"key":"9419_CR28","doi-asserted-by":"crossref","unstructured":"Ruspini, E. H. (1970). Numerical methods for fuzzy clustering. In: Inf. Sci. 2.3, pp. 319-350. ISSN: 0020-0255.","DOI":"10.1016\/S0020-0255(70)80056-1"},{"key":"9419_CR29","doi-asserted-by":"crossref","unstructured":"Scrucca, L. et al. (2016). mclust 5: clustering, classification and density estimation using gaussian finite mixture models. In: The R Journal 8.1, pp. 289-317.","DOI":"10.32614\/RJ-2016-021"},{"key":"9419_CR30","doi-asserted-by":"publisher","first-page":"33","DOI":"10.2307\/1217208","volume":"11","author":"RR Sokal","year":"1962","unstructured":"Sokal, R. R., & Rohlf, F. J. (1962). The comparison of dendrograms by objective methods. Taxon, 11, 33\u201340.","journal-title":"Taxon"},{"key":"9419_CR31","doi-asserted-by":"crossref","unstructured":"Sriram, N. (1990). Clique optimization: A method to construct parsimonious ultrametric trees from similarity data. In: Journal of Classification 7.1, pp. 33-52.","DOI":"10.1007\/BF01889702"},{"key":"9419_CR32","unstructured":"Steinbach, M., Karypis, G., & Kumar V. (2000). A comparison of document clustering techniques. In: In KDD Workshop on Text Mining."},{"key":"9419_CR33","unstructured":"Streuli, H. (1973). Der heutige Stand der Kaffee-Chemie. In Association Scientifique International du Cafe, 6th International Colloquium on Coffee Chemistry, Bogata, Colombia (pp. 61\u201372)."},{"key":"9419_CR34","doi-asserted-by":"crossref","unstructured":"Vichi, M. (2008). Fitting semiparametric clustering models to dissimilarity data. In: Advances in Data Analysis and Classification 2.2, pp. 121-161.","DOI":"10.1007\/s11634-008-0025-4"},{"key":"9419_CR35","doi-asserted-by":"crossref","unstructured":"Ward, J. H. (1963). Hierarchical grouping to optimize an objective function. In: Journal of the American Statistical Association 58.301, pp. 236-244.","DOI":"10.1080\/01621459.1963.10500845"},{"key":"9419_CR36","doi-asserted-by":"crossref","unstructured":"Woese, C. R., Kandler, O., & Wheelis, M. L. (1990). Towards a natural system of organisms: proposal for the domains Archaea, Bacteria, and Eucarya. In: Proceedings of the National Academy of Sciences 87.12, pp. 4576-4579.","DOI":"10.1073\/pnas.87.12.4576"},{"key":"9419_CR37","unstructured":"Zangwill, W. I. (1969). Nonlinear programming: a unified approach. Englewood Cliffs: Prentice-Hall."}],"container-title":["Journal of Classification"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00357-022-09419-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00357-022-09419-7\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00357-022-09419-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,12,15]],"date-time":"2022-12-15T14:13:46Z","timestamp":1671113626000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00357-022-09419-7"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,9,23]]},"references-count":37,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2022,11]]}},"alternative-id":["9419"],"URL":"https:\/\/doi.org\/10.1007\/s00357-022-09419-7","relation":{},"ISSN":["0176-4268","1432-1343"],"issn-type":[{"value":"0176-4268","type":"print"},{"value":"1432-1343","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,9,23]]},"assertion":[{"value":"15 August 2022","order":1,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"23 September 2022","order":2,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare no competing interests.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of Interest"}}]}}