{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T04:13:17Z","timestamp":1776658397924,"version":"3.51.2"},"reference-count":71,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2022,10,11]],"date-time":"2022-10-11T00:00:00Z","timestamp":1665446400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2022,10,11]],"date-time":"2022-10-11T00:00:00Z","timestamp":1665446400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100008815","name":"Libera Universit\u00e0 di Bolzano","doi-asserted-by":"crossref","id":[{"id":"10.13039\/501100008815","id-type":"DOI","asserted-by":"crossref"}]}],"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>Finite Gaussian mixture models provide a powerful and widely employed probabilistic approach for clustering multivariate continuous data. However, the practical usefulness of these models is jeopardized in high-dimensional spaces, where they tend to be over-parameterized. As a consequence, different solutions have been proposed, often relying on matrix decompositions or variable selection strategies. Recently, a methodological link between Gaussian graphical models and finite mixtures has been established, paving the way for penalized model-based clustering in the presence of large precision matrices. Notwithstanding, current methodologies implicitly assume similar levels of sparsity across the classes, not accounting for different degrees of association between the variables across groups. We overcome this limitation by deriving group-wise penalty factors, which automatically enforce under or over-connectivity in the estimated graphs. The approach is entirely data-driven and does not require additional hyper-parameter specification. Analyses on synthetic and real data showcase the validity of our proposal.<\/jats:p>","DOI":"10.1007\/s00357-022-09421-z","type":"journal-article","created":{"date-parts":[[2022,10,11]],"date-time":"2022-10-11T12:02:43Z","timestamp":1665489763000},"page":"648-674","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Group-Wise Shrinkage Estimation in Penalized Model-Based Clustering"],"prefix":"10.1007","volume":"39","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2929-3850","authenticated-orcid":false,"given":"Alessandro","family":"Casa","sequence":"first","affiliation":[]},{"given":"Andrea","family":"Cappozzo","sequence":"additional","affiliation":[]},{"given":"Michael","family":"Fop","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2022,10,11]]},"reference":[{"key":"9421_CR1","first-page":"485","volume":"9","author":"O Banerjee","year":"2008","unstructured":"Banerjee, O., Ghaoui, L.E., & d\u2019Aspremont, A. (2008). Model selection through sparse maximum likelihood estimation for multivariate gaussian or binary data. Journal of Machine Learning Research, 9, 485\u2013516.","journal-title":"Journal of Machine Learning Research"},{"issue":"3","key":"9421_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(3), 803\u2013821.","journal-title":"Biometrics"},{"issue":"512","key":"9421_CR3","doi-asserted-by":"publisher","first-page":"1479","DOI":"10.1080\/01621459.2014.960967","volume":"110","author":"A Bhattacharya","year":"2015","unstructured":"Bhattacharya, A., Pati, D., Pillai, N.S., & Dunson, D.B. (2015). Dirichlet\u2013Laplace priors for optimal shrinkage. Journal of the American Statistical Association, 110(512), 1479\u20131490.","journal-title":"Journal of the American Statistical Association"},{"issue":"6","key":"9421_CR4","doi-asserted-by":"publisher","first-page":"2577","DOI":"10.1214\/08-AOS600","volume":"36","author":"PJ Bickel","year":"2008","unstructured":"Bickel, P.J., & Levina, E. (2008). Covariance regularization by thresholding. The Annals of Statistics, 36(6), 2577\u20132604.","journal-title":"The Annals of Statistics"},{"issue":"4","key":"9421_CR5","doi-asserted-by":"publisher","first-page":"807","DOI":"10.1093\/biomet\/asr054","volume":"98","author":"J Bien","year":"2011","unstructured":"Bien, J., & Tibshirani, R.J. (2011). Sparse estimation of a covariance matrix. Biometrika, 98(4), 807\u2013820.","journal-title":"Biometrika"},{"issue":"6","key":"9421_CR6","doi-asserted-by":"publisher","first-page":"953","DOI":"10.1007\/s11222-013-9413-5","volume":"24","author":"C Biernacki","year":"2014","unstructured":"Biernacki, C., & Lourme, A. (2014). Stable and visualizable gaussian parsimonious clustering models. Statistics and Computing, 24(6), 953\u2013969.","journal-title":"Statistics and Computing"},{"issue":"1","key":"9421_CR7","doi-asserted-by":"publisher","first-page":"301","DOI":"10.1007\/s11222-011-9249-9","volume":"22","author":"C Bouveyron","year":"2012","unstructured":"Bouveyron, C., & Brunet, C. (2012). Simultaneous model-based clustering and visualization in the fisher discriminative subspace. Statistics and Computing, 22(1), 301\u2013324.","journal-title":"Statistics and Computing"},{"key":"9421_CR8","doi-asserted-by":"publisher","first-page":"52","DOI":"10.1016\/j.csda.2012.12.008","volume":"71","author":"C Bouveyron","year":"2014","unstructured":"Bouveyron, C., & Brunet-Saumard, C. (2014). Model-based clustering of high-dimensional data: A review. Computational Statistics & Data Analysis, 71, 52\u201378.","journal-title":"Computational Statistics & Data Analysis"},{"key":"9421_CR9","doi-asserted-by":"publisher","DOI":"10.1017\/9781108644181","volume-title":"Model-based clustering and classification for data science: with applications in R","author":"C Bouveyron","year":"2019","unstructured":"Bouveyron, C., Celeux, G., Murphy, T.B., & Raftery, A.E. (2019). Model-based clustering and classification for data science: with applications in R. Cambridge: Cambridge University Press."},{"issue":"1","key":"9421_CR10","doi-asserted-by":"publisher","first-page":"502","DOI":"10.1016\/j.csda.2007.02.009","volume":"52","author":"C Bouveyron","year":"2007","unstructured":"Bouveyron, C., Girard, S., & Schmid, C. (2007). High-dimensional data clustering. Computational Statistics & Data Analysis, 52(1), 502\u2013519.","journal-title":"Computational Statistics & Data Analysis"},{"key":"9421_CR11","unstructured":"Carter, J.S., Rossell, D., & Smith, J.Q. (2021). Partial correlation graphical lasso. arXiv:2104.10099."},{"issue":"3","key":"9421_CR12","doi-asserted-by":"publisher","first-page":"599","DOI":"10.1007\/s11634-020-00423-6","volume":"15","author":"A Casa","year":"2021","unstructured":"Casa, A., Scrucca, L., & Menardi, G. (2021). Better than the best? Answers via model ensemble in density-based clustering. Advances in Data Analysis and Classification, 15(3), 599\u2013623.","journal-title":"Advances in Data Analysis and Classification"},{"issue":"5","key":"9421_CR13","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(5), 781\u2013793.","journal-title":"Pattern Recognition"},{"issue":"1","key":"9421_CR14","doi-asserted-by":"publisher","first-page":"199","DOI":"10.1093\/biomet\/asm007","volume":"94","author":"S Chaudhuri","year":"2007","unstructured":"Chaudhuri, S., Drton, M., & Richardson, T.S. (2007). Estimation of a covariance matrix with zeros. Biometrika, 94(1), 199\u2013216.","journal-title":"Biometrika"},{"issue":"2","key":"9421_CR15","doi-asserted-by":"publisher","first-page":"373","DOI":"10.1111\/rssb.12033","volume":"76","author":"P Danaher","year":"2014","unstructured":"Danaher, P., Wang, P., & Witten, D.M. (2014). The joint graphical lasso for inverse covariance estimation across multiple classes. Journal of the Royal Statistical Society: Series B (Methodological), 76(2), 373.","journal-title":"Journal of the Royal Statistical Society: Series B (Methodological)"},{"issue":"1","key":"9421_CR16","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1111\/j.2517-6161.1977.tb01600.x","volume":"39","author":"AP Dempster","year":"1977","unstructured":"Dempster, A.P., Laird, N.M., & Rubin, D.B. (1977). Maximum likelihood from incomplete data via the EM algorithm. Journal of the Royal Statistical Society: Series B (Methodological), 39(1), 1\u201322.","journal-title":"Journal of the Royal Statistical Society: Series B (Methodological)"},{"issue":"3","key":"9421_CR17","doi-asserted-by":"publisher","first-page":"1102","DOI":"10.1214\/09-AOAS249","volume":"3","author":"IL Dryden","year":"2009","unstructured":"Dryden, I.L., Koloydenko, A., & Zhou, D. (2009). Non-Euclidean statistics for covariance matrices, with applications to diffusion tensor imaging. The Annals of Applied Statistics, 3(3), 1102\u20131123.","journal-title":"The Annals of Applied Statistics"},{"issue":"1","key":"9421_CR18","first-page":"17","volume":"5","author":"P Erd\u0151s","year":"1960","unstructured":"Erd\u0151s, P., & R\u00e9nyi, A. (1960). On the evolution of random graphs. Publications of the Mathematical Institute of the Hungarian Academy of Sciences, 5(1), 17\u201360.","journal-title":"Publications of the Mathematical Institute of the Hungarian Academy of Sciences"},{"issue":"2","key":"9421_CR19","doi-asserted-by":"publisher","first-page":"521","DOI":"10.1214\/08-AOAS215","volume":"3","author":"J Fan","year":"2009","unstructured":"Fan, J., Feng, Y., & Wu, Y. (2009). Network exploration via the adaptive lasso and scad penalties. The Annals of Applied Statistics, 3(2), 521.","journal-title":"The Annals of Applied Statistics"},{"key":"9421_CR20","unstructured":"Fop, M. (2020). covglasso: Sparse covariance matrix estimation. R package version 2.0. https:\/\/CRAN.R-project.org\/package=covglasso"},{"key":"9421_CR21","doi-asserted-by":"publisher","first-page":"18","DOI":"10.1214\/18-SS119","volume":"12","author":"M Fop","year":"2018","unstructured":"Fop, M., & Murphy, T.B. (2018). Variable selection methods for model-based clustering. Statistics Surveys, 12, 18\u201365.","journal-title":"Statistics Surveys"},{"issue":"4","key":"9421_CR22","doi-asserted-by":"publisher","first-page":"791","DOI":"10.1007\/s11222-018-9838-y","volume":"29","author":"M Fop","year":"2019","unstructured":"Fop, M., Murphy, T.B., & Scrucca, L. (2019). Model-based clustering with sparse covariance matrices. Statistics and Computing, 29(4), 791\u2013819.","journal-title":"Statistics and Computing"},{"key":"9421_CR23","unstructured":"Forina, M., Armanino, C., Lanteri, S., & Tiscornia, E. (1983). Classification of olive oils from their fatty acid composition. In Food research and data analysis: proceedings from the IUFoST Symposium September 20-23 1982, Oslo, Norway\/edited by H. Martens and H. Russwurm, Jr. London: Applied Science Publishers."},{"issue":"458","key":"9421_CR24","doi-asserted-by":"publisher","first-page":"611","DOI":"10.1198\/016214502760047131","volume":"97","author":"C Fraley","year":"2002","unstructured":"Fraley, C., & Raftery, A.E. (2002). Model-based clustering, discriminant analysis, and density estimation. Journal of the American Statistical Association, 97(458), 611\u2013631.","journal-title":"Journal of the American Statistical Association"},{"issue":"3","key":"9421_CR25","doi-asserted-by":"publisher","first-page":"432","DOI":"10.1093\/biostatistics\/kxm045","volume":"9","author":"J Friedman","year":"2008","unstructured":"Friedman, J., Hastie, T., & Tibshirani, R. (2008). Sparse inverse covariance estimation with the graphical lasso. Biostatistics, 9(3), 432\u2013441.","journal-title":"Biostatistics"},{"issue":"4","key":"9421_CR26","doi-asserted-by":"publisher","first-page":"909","DOI":"10.1080\/10618600.2014.937811","volume":"24","author":"X Gao","year":"2015","unstructured":"Gao, X., & Massam, H. (2015). Estimation of symmetry-constrained gaussian graphical models: application to clustered dense networks. Journal of Computational and Graphical Statistics, 24(4), 909\u2013929.","journal-title":"Journal of Computational and Graphical Statistics"},{"issue":"536","key":"9421_CR27","doi-asserted-by":"publisher","first-page":"2087","DOI":"10.1080\/01621459.2021.1938081","volume":"116","author":"A Gelman","year":"2021","unstructured":"Gelman, A., & Vehtari, A. (2021). What are the most important statistical ideas of the past 50 years? Journal of the American Statistical Association, 116 (536), 2087\u20132097.","journal-title":"Journal of the American Statistical Association"},{"issue":"1","key":"9421_CR28","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1093\/biomet\/asq060","volume":"98","author":"J Guo","year":"2011","unstructured":"Guo, J., Levina, E., Michailidis, G., & Zhu, J. (2011). Joint estimation of multiple graphical models. Biometrika, 98(1), 1\u201315.","journal-title":"Biometrika"},{"key":"9421_CR29","doi-asserted-by":"publisher","DOI":"10.1201\/b18401","volume-title":"Statistical learning with sparsity: the lasso and generalizations","author":"T Hastie","year":"2015","unstructured":"Hastie, T., Tibshirani, R., & Wainwright, M. (2015). Statistical learning with sparsity: the lasso and generalizations. Boca Raton: CRC Press."},{"issue":"5","key":"9421_CR30","doi-asserted-by":"publisher","first-page":"1005","DOI":"10.1111\/j.1467-9868.2008.00666.x","volume":"70","author":"S H\u00f8jsgaard","year":"2008","unstructured":"H\u00f8jsgaard, S., & Lauritzen, S.L. (2008). Graphical gaussian models with edge and vertex symmetries. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 70(5), 1005\u20131027.","journal-title":"Journal of the Royal Statistical Society: Series B (Statistical Methodology)"},{"key":"9421_CR31","doi-asserted-by":"publisher","first-page":"193","DOI":"10.1007\/BF01908075","volume":"2","author":"L Hubert","year":"1985","unstructured":"Hubert, L., & Arabie, P. (1985). Comparing partitions. Journal of Classification, 2, 193\u2013218.","journal-title":"Journal of Classification"},{"key":"9421_CR32","doi-asserted-by":"publisher","first-page":"205","DOI":"10.3389\/fpsyt.2016.00205","volume":"7","author":"JV Hull","year":"2017","unstructured":"Hull, J.V., Dokovna, L.B., Jacokes, Z.J., Torgerson, C.M., Irimia, A., & Van Horn, J.D. (2017). Resting-state functional connectivity in autism spectrum disorders: A review. Frontiers in Psychiatry, 7, 205.","journal-title":"Frontiers in Psychiatry"},{"key":"9421_CR33","unstructured":"Kuhn, M. (2021). caret: Classification and Regression Training. R package version 6.0-86. https:\/\/CRAN.R-project.org\/package=caret"},{"issue":"1","key":"9421_CR34","doi-asserted-by":"publisher","first-page":"41","DOI":"10.1214\/16-AOAS990","volume":"11","author":"GG Leday","year":"2017","unstructured":"Leday, G.G., de Gunst, M.C., Kpogbezan, G.B., van der Vaart, A.W., van Wieringen, W.N., & van de Wiel, M.A. (2017). Gene network reconstruction using global-local shrinkage priors. The Annals of Applied Statistics, 11 (1), 41\u201368.","journal-title":"The Annals of Applied Statistics"},{"issue":"4","key":"9421_CR35","doi-asserted-by":"publisher","first-page":"366","DOI":"10.1002\/sam.11530","volume":"14","author":"Q Li","year":"2021","unstructured":"Li, Q., Sun, X., Wang, N., & Gao, X. (2021). Penalized composite likelihood for colored graphical gaussian models. Statistical Analysis and Data Mining: The ASA Data Science Journal, 14(4), 366\u2013378.","journal-title":"Statistical Analysis and Data Mining: The ASA Data Science Journal"},{"issue":"8","key":"9421_CR36","doi-asserted-by":"publisher","first-page":"2839","DOI":"10.1016\/j.jspi.2011.03.008","volume":"141","author":"H Lian","year":"2011","unstructured":"Lian, H. (2011). Shrinkage tuning parameter selection in precision matrices estimation. Journal of Statistical Planning and Inference, 141(8), 2839\u20132848.","journal-title":"Journal of Statistical Planning and Inference"},{"issue":"2","key":"9421_CR37","doi-asserted-by":"publisher","first-page":"257","DOI":"10.1016\/j.jmva.2008.04.010","volume":"100","author":"TI Lin","year":"2009","unstructured":"Lin, T.I. (2009). Maximum likelihood estimation for multivariate skew normal mixture models. Journal of Multivariate Analysis, 100(2), 257\u2013265.","journal-title":"Journal of Multivariate Analysis"},{"issue":"3","key":"9421_CR38","doi-asserted-by":"publisher","first-page":"343","DOI":"10.1007\/s11222-009-9128-9","volume":"20","author":"TI Lin","year":"2010","unstructured":"Lin, T.I. (2010). Robust mixture modeling using multivariate skew t distributions. Statistics and Computing, 20(3), 343\u2013356.","journal-title":"Statistics and Computing"},{"issue":"9","key":"9421_CR39","doi-asserted-by":"publisher","first-page":"e1006436","DOI":"10.1371\/journal.pcbi.1006436","volume":"14","author":"Y Lyu","year":"2018","unstructured":"Lyu, Y., Xue, L., Zhang, F., Koch, H., Saba, L., Kechris, K., & Li, Q. (2018). Condition-adaptive fused graphical lasso (CFGL): An adaptive procedure for inferring condition-specific gene co-expression network. PLoS computational Biology, 14(9), e1006436.","journal-title":"PLoS computational Biology"},{"issue":"3","key":"9421_CR40","doi-asserted-by":"publisher","first-page":"701","DOI":"10.1111\/j.1541-0420.2008.01160.x","volume":"65","author":"C Maugis","year":"2009","unstructured":"Maugis, C., Celeux, G., & Martin-Magniette, M.-L. (2009a). Variable selection for clustering with Gaussian mixture models. Biometrics, 65(3), 701\u2013709.","journal-title":"Biometrics"},{"issue":"11","key":"9421_CR41","doi-asserted-by":"publisher","first-page":"3872","DOI":"10.1016\/j.csda.2009.04.013","volume":"53","author":"C Maugis","year":"2009","unstructured":"Maugis, C., Celeux, G., & Martin-Magniette, M.-L. (2009b). Variable selection in model-based clustering: A general variable role modeling. Computational Statistics & Data Analysis, 53(11), 3872\u20133882.","journal-title":"Computational Statistics & Data Analysis"},{"key":"9421_CR42","doi-asserted-by":"crossref","unstructured":"McLachlan, G.J., & Peel, D. (1998). Robust cluster analysis via mixtures of multivariate t-distributions. In Joint IAPR international workshops on statistical techniques in pattern recognition (SPR) and structural and syntactic pattern recognition (SSPR) (pp. 658\u2013666). Springer.","DOI":"10.1007\/BFb0033290"},{"issue":"3-4","key":"9421_CR43","doi-asserted-by":"publisher","first-page":"379","DOI":"10.1016\/S0167-9473(02)00183-4","volume":"41","author":"GJ McLachlan","year":"2003","unstructured":"McLachlan, G.J., Peel, D., & Bean, R. (2003). Modelling high-dimensional data by mixtures of factor analyzers. Computational Statistics & Data Analysis, 41(3-4), 379\u2013388.","journal-title":"Computational Statistics & Data Analysis"},{"issue":"3","key":"9421_CR44","doi-asserted-by":"publisher","first-page":"331","DOI":"10.1007\/s00357-016-9211-9","volume":"33","author":"PD McNicholas","year":"2016","unstructured":"McNicholas, P.D. (2016). Model-based clustering. Journal of Classification, 33(3), 331\u2013373.","journal-title":"Journal of Classification"},{"key":"9421_CR45","unstructured":"McNicholas, P.D., ElSherbiny, A., McDaid, A.F., & Murphy, T.B. (2019). pgmm: Parsimonious gaussian mixture models. R package version 1.2.4. https:\/\/CRAN.R-project.org\/package=pgmm"},{"key":"9421_CR46","doi-asserted-by":"crossref","unstructured":"McNicholas, P. D., & Murphy, T. B. (2008). Parsimonious gaussian mixture models. Statistics and Computing, 18(3), 285\u2013296.","DOI":"10.1007\/s11222-008-9056-0"},{"issue":"3","key":"9421_CR47","doi-asserted-by":"publisher","first-page":"1436","DOI":"10.1214\/009053606000000281","volume":"34","author":"N Meinshausen","year":"2006","unstructured":"Meinshausen, N., B\u00fchlmann, P., & et al. (2006). High-dimensional graphs and variable selection with the lasso. The Annals of Statistics, 34(3), 1436\u20131462.","journal-title":"The Annals of Statistics"},{"key":"9421_CR48","unstructured":"Meyer, D., Dimitriadou, E., Hornik, K., Weingessel, A., & Leisch, F. (2020). e1071: Misc Functions of the Department of Statistics, Probability Theory Group (Formerly: E1071), TU Wien. R package version 1.7-4."},{"issue":"1","key":"9421_CR49","first-page":"445","volume":"15","author":"K Mohan","year":"2014","unstructured":"Mohan, K., London, P., Fazel, M., Witten, D., & Lee, S. (2014). Node-based learning of multiple gaussian graphical models. Journal of Machine Learning Research, 15(1), 445\u2013488.","journal-title":"Journal of Machine Learning Research"},{"key":"9421_CR50","first-page":"1145","volume":"8","author":"W Pan","year":"2007","unstructured":"Pan, W., & Shen, X. (2007). Penalized model-based clustering with application to variable selection. Journal of Machine Learning Research, 8, 1145\u20131164.","journal-title":"Journal of Machine Learning Research"},{"issue":"501-538","key":"9421_CR51","first-page":"105","volume":"9","author":"NG Polson","year":"2010","unstructured":"Polson, N.G., & Scott, J.G. (2010). Shrink globally, act locally: Sparse bayesian regularization and prediction. Bayesian Statistics, 9(501-538), 105.","journal-title":"Bayesian Statistics"},{"key":"9421_CR52","doi-asserted-by":"publisher","DOI":"10.1002\/9781118573617","volume-title":"High-dimensional covariance estimation wiley series in probability and statistics","author":"M Pourahmadi","year":"2013","unstructured":"Pourahmadi, M. (2013). High-dimensional covariance estimation wiley series in probability and statistics. New York: Wiley."},{"key":"9421_CR53","volume-title":"R: A Language and Environment for Statistical Computing","author":"R Core Team","year":"2022","unstructured":"R Core Team. (2022). R: A Language and Environment for Statistical Computing. Vienna, Austria: R Foundation for Statistical Computing."},{"issue":"473","key":"9421_CR54","doi-asserted-by":"publisher","first-page":"168","DOI":"10.1198\/016214506000000113","volume":"101","author":"AE Raftery","year":"2006","unstructured":"Raftery, A.E., & Dean, N. (2006). Variable selection for model-based clustering. Journal of the American Statistical Association, 101(473), 168\u2013178.","journal-title":"Journal of the American Statistical Association"},{"key":"9421_CR55","unstructured":"Russell, N., Murphy, T.B., & Raftery, A.E. (2015). Bayesian model averaging in model-based clustering and density estimation. arXiv:1506.09035."},{"key":"9421_CR56","unstructured":"Scheinberg, K., Ma, S., and Goldfarb, D. (2010). Sparse inverse covariance selection via alternating linearization methods. In Proceedings of the 23rd International Conference on Neural Information Processing Systems."},{"issue":"2","key":"9421_CR57","doi-asserted-by":"publisher","first-page":"461","DOI":"10.1214\/aos\/1176344136","volume":"6","author":"G Schwarz","year":"1978","unstructured":"Schwarz, G. (1978). Estimating the dimension of a model. The Annals of Statistics, 6(2), 461\u2013464.","journal-title":"The Annals of Statistics"},{"issue":"1","key":"9421_CR58","doi-asserted-by":"publisher","first-page":"289","DOI":"10.32614\/RJ-2016-021","volume":"8","author":"L Scrucca","year":"2016","unstructured":"Scrucca, L., Fop, M., Murphy, T.B., & Raftery, A.E. (2016). mclust 5: Clustering, classification and density estimation using gaussian finite mixture models. The R Journal, 8(1), 289\u2013317.","journal-title":"The R Journal"},{"issue":"4","key":"9421_CR59","doi-asserted-by":"publisher","first-page":"447","DOI":"10.1007\/s11634-015-0220-z","volume":"9","author":"L Scrucca","year":"2015","unstructured":"Scrucca, L., & Raftery, A.E. (2015). Improved initialisation of model-based clustering using Gaussian hierarchical partitions. Advances in Data Analysis and Classification, 9(4), 447\u2013460.","journal-title":"Advances in Data Analysis and Classification"},{"issue":"1","key":"9421_CR60","doi-asserted-by":"crossref","first-page":"267","DOI":"10.1111\/j.2517-6161.1996.tb02080.x","volume":"58","author":"R Tibshirani","year":"1996","unstructured":"Tibshirani, R. (1996). Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society: Series B (Methodological), 58(1), 267\u2013288.","journal-title":"Journal of the Royal Statistical Society: Series B (Methodological)"},{"key":"9421_CR61","doi-asserted-by":"publisher","first-page":"196","DOI":"10.1016\/j.csda.2013.07.008","volume":"71","author":"I Vrbik","year":"2014","unstructured":"Vrbik, I., & McNicholas, P.D. (2014). Parsimonious skew mixture models for model-based clustering and classification. Computational Statistics & Data Analysis, 71, 196\u2013210.","journal-title":"Computational Statistics & Data Analysis"},{"issue":"4","key":"9421_CR62","doi-asserted-by":"publisher","first-page":"867","DOI":"10.1214\/12-BA729","volume":"7","author":"H Wang","year":"2012","unstructured":"Wang, H. (2012). Bayesian graphical lasso models and efficient posterior computation. Bayesian Analysis, 7(4), 867\u2013886.","journal-title":"Bayesian Analysis"},{"issue":"4","key":"9421_CR63","doi-asserted-by":"publisher","first-page":"521","DOI":"10.1007\/s11222-013-9385-5","volume":"24","author":"H Wang","year":"2014","unstructured":"Wang, H. (2014). Coordinate descent algorithm for covariance graphical lasso. Statistics and Computing, 24(4), 521\u2013529.","journal-title":"Statistics and Computing"},{"issue":"2","key":"9421_CR64","doi-asserted-by":"publisher","first-page":"197","DOI":"10.1007\/s11634-014-0182-6","volume":"9","author":"Y Wei","year":"2015","unstructured":"Wei, Y., & McNicholas, P.D. (2015). Mixture model averaging for clustering. Advances in Data Analysis and Classification, 9(2), 197\u2013217.","journal-title":"Advances in Data Analysis and Classification"},{"key":"9421_CR65","volume-title":"Graphical models in applied multivariate statistics","author":"J Whittaker","year":"1990","unstructured":"Whittaker, J. (1990). Graphical models in applied multivariate statistics. New York: Wiley."},{"issue":"4","key":"9421_CR66","doi-asserted-by":"publisher","first-page":"892","DOI":"10.1198\/jcgs.2011.11051a","volume":"20","author":"DM Witten","year":"2011","unstructured":"Witten, D.M., Friedman, J.H., & Simon, N. (2011). New insights and faster computations for the graphical lasso. Journal of Computational and Graphical Statistics, 20(4), 892\u2013900.","journal-title":"Journal of Computational and Graphical Statistics"},{"key":"9421_CR67","doi-asserted-by":"publisher","first-page":"168","DOI":"10.1214\/08-EJS194","volume":"2","author":"B Xie","year":"2008","unstructured":"Xie, B., Pan, W., & Shen, X. (2008). Penalized model-based clustering with cluster-specific diagonal covariance matrices and grouped variables. Electronic Journal of Statistics, 2, 168.","journal-title":"Electronic Journal of Statistics"},{"issue":"1","key":"9421_CR68","first-page":"1059","volume":"13","author":"T Zhao","year":"2012","unstructured":"Zhao, T., Liu, H., Roeder, K., Lafferty, J., & Wasserman, L. (2012). The huge package for high-dimensional undirected graph estimation in R. Journal of Machine Learning Research, 13(1), 1059\u20131062.","journal-title":"Journal of Machine Learning Research"},{"key":"9421_CR69","doi-asserted-by":"publisher","first-page":"1473","DOI":"10.1214\/09-EJS487","volume":"3","author":"H Zhou","year":"2009","unstructured":"Zhou, H., Pan, W., & Shen, X. (2009). Penalized model-based clustering with unconstrained covariance matrices. Electronic Journal of Statistics, 3, 1473\u20131496.","journal-title":"Electronic Journal of Statistics"},{"issue":"476","key":"9421_CR70","doi-asserted-by":"publisher","first-page":"1418","DOI":"10.1198\/016214506000000735","volume":"101","author":"H Zou","year":"2006","unstructured":"Zou, H. (2006). The adaptive lasso and its oracle properties. Journal of the American Statistical Association, 101(476), 1418\u20131429.","journal-title":"Journal of the American Statistical Association"},{"issue":"5","key":"9421_CR71","doi-asserted-by":"publisher","first-page":"2173","DOI":"10.1214\/009053607000000127","volume":"35","author":"H Zou","year":"2007","unstructured":"Zou, H., Hastie, T., & Tibshirani, R. (2007). On the \u201cdegrees of freedom\u201d of the lasso. The Annals of Statistics, 35(5), 2173\u20132192.","journal-title":"The Annals of Statistics"}],"container-title":["Journal of Classification"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00357-022-09421-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00357-022-09421-z\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00357-022-09421-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,10,5]],"date-time":"2024-10-05T15:37:39Z","timestamp":1728142659000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00357-022-09421-z"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,10,11]]},"references-count":71,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2022,11]]}},"alternative-id":["9421"],"URL":"https:\/\/doi.org\/10.1007\/s00357-022-09421-z","relation":{},"ISSN":["0176-4268","1432-1343"],"issn-type":[{"value":"0176-4268","type":"print"},{"value":"1432-1343","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,10,11]]},"assertion":[{"value":"12 September 2022","order":1,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"11 October 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":"<!--Emphasis Type='Bold' removed-->Conflict of Interest"}}]}}