Abstract
Model-based clustering is a popular technique relying on the notion of finite mixture models that proved to be efficient in modeling heterogeneity in data. The underlying idea is to model each data group by a particular mixture component. This relationship between mixed distributions and clusters forms an attractive interpretation of groups: each cluster is assumed to be a sample from the corresponding distribution. In practice, however, there are many issues that have to be accounted for by the researcher. The area of model-based clustering is very dynamic and rapidly developing, with many questions yet to be answered. In this paper, we review and discuss the latest developments in model-based clustering including semi-supervised clustering, non-parametric mixture modeling, choice of initialization strategies, merging mixture components for clustering, handling spurious solutions, and assessing variability of obtained partitions. We also demonstrate the utility of model-based clustering by considering several challenging applications to real-life problems.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Akaike H (1973) Information theory and an extension of the maximum likelihood principle. In: Second international symposium on information theory, pp 267–281
Anh NK, Tam NT, Van Linh N (2013) Document clustering using Dirichlet process mixture model of von Mises-Fisher distributions. In: Proceedings of the fourth symposium on information and communication technology, New York, pp 131–138
Attias H (1999) Inferring parameters and structure of latent variable models by variational Bayes. In: Proceedings of the fifteenth conference on uncertainty in artificial intelligence
Azzalini A, Bowman AW (1990) A look at some data on the old faithful geyser. J R Stat Soc C 39:357–365
Azzalini A, Menardi G (2013) Package pdfCluster: cluster analysis via nonparametric density estimation. http://cran.r-project.org/web/packages/pdfCluster
Azzalini A, Torelli N (2007) Clustering via nonparametric density estimation. Stat Comput 17:71–80
Banerjee A, Dhillon IS, Ghosh J, Sra S (2005) Clustering on the unit hypersphere using von Mises-Fisher distributions. J Mach Learn Res 6:1345–1382
Banfield JD, Raftery AE (1993) Model-based Gaussian and non-Gaussian clustering. Biometrics 49:803–821
Bar-Hillel A, Hertz T, Shental N, Weinshall D (2003) Learning distance functions using equivalence relations. In: Proceedings of the twentieth international conference on machine learning, pp 11–18
Basso R, Lachos V, Cabral C, Ghosh P (2010) Robust mixture modeling based on scale mixtures of skew-normal distributions. Comput Stat Data Anal 54(12):2926–2941
Basu S, Banerjee A, Mooney R (2002) Semi-supervised clustering by seeding. In: Proceedings of the 19th international conference on machine learning, pp 19–26
Basu S, Bilenko M, Mooney RJ (2004) A probabilistic framework for semi-supervised clustering. In: Proceedings of the tenth ACM SIGKDD international conference on knowledge discovery and data mining, pp 59–68
Baudry JP, Raftery A, Celeux G, Lo K, Gottardo R (2010) Combining mixture components for clustering. J Comput Graph Stat 19(2):332–353
Benaglia T, Chauveau D, Hunter DR (2009) An EM-like algorithm for semi- and nonparametric estimation in multivariate mixtures. J Comput Graph Stat 18(2):505–526
Benaglia T, Chauveau D, Hunter DR, S YD (2009) mixtools: an R package for analyzing mixture models. J Stat Softw 32(6):1–29
Benaglia T, Chauveau D, Hunter DR (2011) Bandwidth selection in an EM-like algorithm for nonparametric multivariate mixtures. In: Hunter D, Richards DSP, Rosenberger J (eds) Nonparametric statistics and mixture models, A Festschrift in honor of Thomas P Hettmansperger. World Scientific, Singapore, pp 15–27
Benjamini Y, Hochberg Y (1995) Controlling the false discovery rate: a practical and powerful approach to multiple testing. J R Stat Soc 57:289–300
Berlinet AF, Roland C (2012) Acceleration of the em algorithm: P-em versus epsilon algorithm. Comput Stat Data Anal 56(12):4122–4137
Biernacki C, Celeux G, Gold EM (2000) Assessing a mixture model for clustering with the integrated completed likelihood. IEEE Trans Pattern Anal Mach Intell 22:719–725
Biernacki C, Celeux G, Govaert G (2003) Choosing starting values for the EM algorithm for getting the highest likelihood in multivariate Gaussian mixture models. Comput Stat Data Anal 413:561–575
Böhning D, Dietz E, Schaub R, Schlattmann P, Lindsay B (1994) The distribution of the likelihood ratio for mixtures of densities from the one-parameter exponential family. Ann Inst Stat Math 46(2):373–388
Bouveyron C, Brunet C (2014) Model-based clustering of high-dimensional data: a review. Comput Stat Data Anal 71:52–78
Bouveyron C, Girard S, Schmid C (2007) High-dimensional data clustering. Comput Stat Data Anal 52(1):502–519. http://lear.inrialpes.fr/pubs/2007/BGS07a
Bridge M (2012) Locating the origins of wood resources: a review of dendroprovenancing. J Archaeol Sci 39(8):2828–2834
Butts CT, Handcock MS, Hunter DR (2014) Network: classes for relational data. Irvine. R package version 1.9.0, http://statnet.org/
Cadez I, Heckerman D, Meek C, Smyth P, White S (2003) Model-based clustering and visualization of navigation patterns on a web site. Data Min Knowl Discov 7:399–424
Campbell NA, Mahon RJ (1974) A multivariate study of variation in two species of rock crab of Genus Leptograsus. Aust J Zool 22:417–25
Celebi ME, Kingravi H, Vela PA (2013) A comparative study of efficient initialization methods for the K-means clustering algorithm. Expert Syst Appl 40(1):200–210
Celeux G, Govaert (1995) Gaussian parsimonious clustering models. Comput Stat Data Anal 28:781–93
Celeux C, Martin-Magniette ML, Maugis C, Raftery A (2011) Letter to the editor. J Am Stat Assoc 106:383
Chandra S (1977) On the mixtures of probability distributions. Scand J Stat 4:105–112
Chen WC, Maitra R (2011) Model-based clustering of regression time series data via APECM – an AECM algorithm sung to an even faster beat. Stat Anal Data Min 4:567–578
Chen J, Tan X, Zhang R (2008) Consistency of penalized MLE for normal mixtures in mean and variance. Stat Sin 18:443–465
Ciuperca G, Ridolfi A, Idier J (2003) Penalized maximum likelihood estimator for normal mixtures. Scand J Stat 30(1):45–59
Corduneanu A, Bishop CM (2001) Variational Bayesian model selection for mixture distributions. In: Proceedings eighth international conference on artificial intelligence and statistics, pp 27–34
Dean N, Raftery A, Scrucca L (2013) Package clustvarsel: variable selection for model-based clustering. http://cran.r-project.org/web/packages/clustvarsel
Demiriz A, Bennett K, Embrechts MJ (1999) Semi-supervised clustering using genetic algorithms. In: Artificial neural networks in engineering (ANNIE-99). ASME Press, New York, pp 809–814
Dempster AP, Laird NM, Rubin DB (1977) Maximum likelihood for incomplete data via the EM algorithm (with discussion). J R Stat Soc Ser B 39:1–38
Dertinger JJ, Walker AV (2013) Ionic liquid matrix-enhanced secondary ion mass spectrometry: the role of proton transfer. J Am Soc Mass Spectrom 24:348–355
Dhillon IS, Modha DS (2001) Concept decompositions for large sparse text data using clustering. Mach Learn 42:143–175
Diebolt J, Robert C (1994) Estimation of finite mixture distributions by Bayesian sampling. J R Stat Soc Ser B 56:363–375
Digalakis VV, Rtischev D, Neumeyer LG (1995) Speaker adaptation using constrained estimation of Gaussian mixtures. IEEE Trans Speech Audio Process 3(5):357–366
Dortet-Bernadet J, Wicker N (2008) Model-based clustering on the unit sphere with an illustration using gene expression profiles. Biostatistics 9(1):66–80
Efron B, Tibshirani R, d Storey J, Tusher V (2001) Empirical Bayes analysis of a microarray experiment. J Am Stat Assoc 96:1151–1160
Escobar MD, West M (1995) Bayesian density estimation and inference using mixtures. J Am Stat Assoc 90:577–588
Esper J, Cook E, Schweingruber F (2002) Low-frequency signals in long tree-ring chronologies for reconstructing past temperature variability. Science 295(5563):2250–2253
Feng Z, McCulloch C (1996) Using bootstrap likelihood ratio in finite mixture models. J R Stat Soc B 58:609–617
Forgy E (1965) Cluster analysis of multivariate data: efficiency vs. interpretability of classifications. Biometrics 21:768–780
Fraley C (1998) Algorithms for model-based Gaussian hierarchical clustering. SIAM J Sci Comput 20:270–281
Fraley C, Raftery AE (2002) Model-based clustering, discriminant analysis, and density estimation. J Am Stat Assoc 97:611–631
Fraley C, Raftery AE (2006) MCLUST version 3 for R: normal mixture modeling and model-based clustering. Technical Report 504, Department of Statistics, University of Washington, Seattle
Frühwirth-Schnatter S (2001) Markov Chain Monte Carlo estimation of classical and dynamic switching and mixture models. J Am Stat Assoc 96:194–209
Frühwirth-Schnatter S, Pyne S (2010) Bayesian inference for finite mixtures of univariate and multivariate skew-normal and skew-t distributions. Biostatistics 11:317–336
Gallegos MT, Ritter G (2009) Trimmed ML estimation of contaminated mixtures. Sankhya Ser A 71:164–220
Garcia-Escudero L, Gordaliza A, Mayo-Iscar A (2013) A constrained robust proposal for mixture modeling avoiding spurious solutions. Adv Data Anal Classif 1–17. doi:10.1007/s11634-013-0153-3
Gopal S, Yang Y (2014) von Mises-Fisher clustering models. J Mach Learn Res 32:154–162
Gormley IC, Murphy TB (2010) A mixture of experts latent position cluster model for social network data. Stat Methodol 7:385–405
Guo J, Levina E, Michailidis G, Zhu J (2010) Pairwise variable selection for high-dimensional model-based clustering. Biometrics 66:793–804
Hall P, Ormerod JT, Wand MP (2011) Theory of Gaussian variational approximation for a Poisson mixed model. Stat Sin 21:369–389
Hammer R, Hertz T, Hochstein S, Weinshall D (2007) Classification with positive and negative equivalence constraints: theory, computation and human experiments. In: Proceedings of the 2nd international conference on advances in brain, vision and artificial intelligence, Springer-Verlag Berlin, pp 264–276
Handcock MS, Raftery AE, Tantrum JM (2007) Model-based clustering for social networks. J R Stat Soc Ser A 170:301–354
Haneca K, Wazny T, Van Acker J, Beeckman H (2005) Provenancing Baltic timber from art historical objects: success and limitations. J Archaeol Sci 32(2):261–271
Hartigan JA (1981) Consistency of single linkage for high-density clusters. J Am Stat Assoc 76:388–394
Hathaway RJ (1985) A constrained formulation of maximum-likelihood estimation for normal mixture distributions. Stat Probab Lett 4:53–56
Hennig C (2004) Breakdown points for maximum likelihood-estimators of location-scale mixtures. Ann Stat 32:1313–1340
Hennig C (2010) Methods for merging Gaussian mixture components. Adv Data Anal Classif 4:3–34
Hennig C, Coretto P (2008) The noise component in model-based cluster analysis. In: Preisach C, Burkhardt H, Schmidt-Thieme L, Decker R (eds) Data analysis, machine learning and applications, studies in classification, data analysis, and knowledge organization. Springer, Berlin, Heidelberg, pp 127–138
Hoff PD, Raftery AE, Handcock MS (2002) Latent space approaches to social network analysis. J Am Stat Assoc 97:460:1090–1098
Holzmann H, Munk A, Gneiting T (2006) Identifiability of finite mixtures of elliptical distributions. Scand J Stat 33:753–763
Huang JT, Hasegawa-Johnson M (2009) On semi-supervised learning of Gaussian mixture models for phonetic classification. In: NAACL HLT workshop on semi-supervised learning
Inbarani HH, Thangavel K (2009) Mining and analysis of clickstream patterns. In: Abraham A, Hassanien AE, Leon F de Carvalho A, Snášel V (eds) Foundations of computational, intelligence, vol 6. Studies in computational intelligence, vol 206. Springer, Berlin, Heidelberg, pp 3–27
Jasra A, Holmes CC, Stephens DA (2005) Markov chain Monte Carlo methods and the label switching problem in Bayesian mixture modeling. Stat Sci 20:50–67
Jiao S, Zhang S (2008) The t-mixture model approach for detecting differentially expressed genes in microarrays. Funct Integr Genomics 8:181–186
Jolliffe IT, Jones B, Morgan BJT (1995) Identifying influential observations in hierarchical cluster analysis. J Appl Stat 22(1):61–80
Kalman RE (1960) A new approach to linear filtering and prediction problems. J Basic Eng 82:35–45
Kent J (1983) Identifiability of finite mixtures for directional data. Ann Stat 11(3):984–988
Kiefer NM (1978) Discrete parameter variation: efficient estimation of a switching regression model. Econometrica 46:427–434
Kim D, Seo B (2014) Assessment of the number of components in Gaussian mixture models in the presence of multiple local maximizers. J Multivar Anal 125:100–120
Klein D, Kamvar SD, Manning C (2002) From instance-level constraints to space-level constraints: making the most of prior knowledge in data clustering. In: Proceedings of the nineteenth international conference on machine learning (ICML-2002), pp 307–314
Krivitsky PN, Handcock MS (2008) Fitting position latent cluster models for social networks with latentnet. J Stat Softw 24(5). http://statnetproject.org
Krivitsky PN, Handcock MS (2009) latentnet: Latent position and cluster models for statistical networks. R package version 2.2-2. http://statnetproject.org
Lauritzen SL (1996) Graphical models. Clarendon Press, Oxford
Law MHC, Topchy A, Jain AK (2005) Model-based clustering with probabilistic constraints. In: 2005 SIAM international conference on data mining, pp 641–645
Lee H, Li J (2012) Variable selection for clustering by separability based on ridgelines. J Comput Graph Stat 21:315–337
Lee S, McLachlan G (2013) On mixtures of skew normal and skew t-distributions. Adv Data Anal Classif 7:241–266
Li J, Zha H (2006) Two-way Poisson mixture models for simultaneous document classification and word clustering. Comput Stat Data Anal 50(1):163–180
Li J, Ray S, Lindsay B (2007) A nonparametric statistical approach to clustering via mode identification. J Mach Learn Res 8:1687–1723
Lin TI (2009) Maximum likelihood estimation for multivariate skew normal mixture models. J Multivar Anal 100:257–265
Lin TI, Lee JC, Yen SY (2007) Finite mixture modelling using the skew normal distribution. Stat Sin 17:909–927
Liu B (2011) Web data mining: exploring hyperlinks, contents, and usage data, 2nd edn. Springer, New York
Liu C, Rubin DB (1994) The ECME algorithm: a simple extension of EM and ECM with faster monotone convergence. Biometrika 81:633–648
Liu C, Rubin DB, Wu YN (1998) Parameter expansion to accelerate em: the PX-EM algorithm. Biometrika 85:755–770
Lotsi A, Wit E (2013) High dimensional sparse Gaussian graphical mixture model. arXiv:13083381v3
Lu Z, Leen TK (2007) Penalized probabilistic clustering. Neural Comput 19:1528–1567
MacEachern SN, Muller P (1998) Estimating mixtures of Dirichlet process models. J Comput Graph Stat 7:223–238
Maitra R (2009) Initializing partition-optimization algorithms. IEEE/ACM Trans Comput Biol Bioinform 6:144–157. http://doi.ieeecomputersociety.org/10.1109/TCBB.2007.70244
Maitra R, Melnykov V (2010) Simulating data to study performance of finite mixture modeling and clustering algorithms. J Comput Graph Stat 19(2):354–376. doi:10.1198/ jcgs.2009.08054
Mardia KV, Jupp PE (2000) Directional statistics. Wiley, New York
Markitsis A, Lai Y (2010) The t-mixture model approach for detecting differentially expressed genes in microarrays. Bioinformatics 26:640–646
Martinez-Uso A, Pla F, Sotoca J (2010) A semi-supervised Gaussian mixture model for image segmentation. In: International conference on pattern recognition, pp 2941–2944
Masseran N, Razali A, Ibrahim K, Latif M (2013) Fitting a mixture of von Mises-distributions in order to model data on wind direction in Peninsular Malaysia. Energy Convers Manag 72:94–102
Maugis C, Celeux G, Martin-Magniette ML (2009) Variable selection for clustering with Gaussian mixture models. Biometrics 65(3):701–709
Maugis C, Celeux G, Martin-Magniette ML (2009) Variable selection in model-based clustering: a general variable role modeling. Comput Stat Data Anal 53(11):3872–3882
Maugis-Rabusseau C, Martin-Magniette ML, Pelletier S (2012) Selvarclustmv: variable selection approach in model-based clustering allowing for missing values. J Soc Fr Stat 153(2):21–36
McGrory C, Titterington D (2007) Variational approximations in Bayesian model selection for finite mixture distributions. Comput Stat Data Anal 51(11):5352–5367. doi:10.1016/j. csda.2006.07.020, http://www.sciencedirect.com/science/article/B6V8V-4KMYRPW-1/2/42 8635340ac2d823187a0c04164508c5. Advances in Mixture Models
McLachlan G (1987) On bootstrapping the likelihood ratio test statistic for the number of components in a normal mixture. Appl Stat 36:318–324
McLachlan GJ, Basford KE (1988) Mixture models: inference and applications to clustering. Marcel Dekker, New York
McLachlan G, Krishnan T (2008) The EM algorithm and extensions, 2nd edn. Wiley, New York
McLachlan G, Peel D (2000) Finite mixture models. Wiley, New York
McLachlan G, Peel G, Basford K, Adams P (1999) Fitting of mixtures of normal and \(t\)-components. J Stat Softw 4:2
McLachlan G, Been R, Jones LT (2006) A simple implementation of a normal mixture approach to differential gene expression in multiclass microarrays. Bioinformatics 22:1608–1615
McNeil DR (1977) Interactive data analysis. Wiley, New York
Melnikov V, Litvinov V, Koppe V, Bobkov V (2008) Sims study of the processes in buffer solutions of bioorganic systems. Bull Russ Acad Sci Phys 72:929–933
Melnykov V (2012) Efficient estimation in model-based clustering of Gaussian regression time series. Stat Anal Data Min 5:95–99
Melnykov V (2013) Challenges in model-based clustering. Wiley Interdiscip Rev Comput Stat 5:135–148
Melnykov V (2013) Finite mixture modelling in mass spectrometry analysis. J R Stat Soc Ser C 62:573–592
Melnykov V (2013) On the distribution of posterior probabilities in finite mixture models with application in clustering. J Multivar Anal 122:175–189
Melnykov V (2014) Merging mixture components for clustering through pairwise overlap. J Comput Graph Stat (tentatively accepted)
Melnykov V (2014) Model-based biclustering of clickstream data. Comput Stat Data Anal (under minor revision)
Melnykov V, Maitra R (2011) CARP: software for fishing out good clustering algorithms. J Mach Learn Res 12:69–73
Melnykov V, Melnykov I (2012) Initializing the EM algorithm in Gaussian mixture models with an unknown number of components. Comput Stat Data Anal 56:1381–1395
Melnykov I, Melnykov V (2014) On k-means algorithm with the use of Mahalanobis distances. Stat Probab Lett 84:88–95
Melnykov V, Michael S (2014) Finite mixture modeling of Gaussian regression time series with application to dendrochronology. J Classif (under review)
Melnykov V, Chen WC, Maitra R (2012) MixSim: an R package for simulating data to study performance of clustering algorithms. J Stat Softw 51:1–25
Meng XL, van Dyk D (1997) The EM algorithm - an old folk song sung to a fast new tune (with discussion). J R Stat Soc Ser B 59:511–567
Meng XL, Rubin DB (1993) Maximum likelihood estimation via the ECM algorithm: a general framework. Biometrika 80(2):267–278
Michael S, Melnykov V (2014) Studying complexity of model-based clustering. Commun Stat Simul Comput (accepted)
Moore A (1998) Very fast EM-based mixture model clustering using multiresolution kd-trees. In: In advances in neural information processing systems 11. MIT Press, Cambridge, pp 543–549
Neal R (2000) Markov chain sampling methods for Dirichlet process mixture models. J Comput Graph Stat 9:249–265
Neal RM, Hinton GE (1993) A new view of the EM algorithm that justifies incremental and other variants. In: Learning in graphical models. Kluwer, Dordrecht, pp 355–368
Newcomb S (1886) A generalized theory of the combination of observations so as to obtain the best result. Am J Math 8:343–366
Neykov N, Filzmoser P, Dimova R, Neytchev P (2007) Robust fitting of mixtures using the trimmed likelihood estimator. Comput Stat Data Anal 17:299–308
Nigam K, McCallum AK, Thrun S, Mitchell T (2000) Text classification from labeled and unlabeled documents using EM. Mach Learn 39:103–134
Ortega JM, Rheinboldt WC (1970) Iterative solutions of nonlinear equations in several variables. Academic, Princeton
Pan W, Shen X (2007) Penalized model-based clustering with application to variable selection. J Mach Learn Res 8:1145–1164
Pan W, Shen X, Jiang A, Hebbel R (2006) Semisupervised learning via penalized mixture model with application to microarray sample classification. Bioinformatics 22(19):2388–2395
Papastamoulis P, Iliopoulos G (2010) An artificial allocations based solution to the label switching problem in Bayesian analysis of mixtures of distributions. J Comput Graph Stat 19:313–331
Pearl J (1988) Probabilistic reasoning in intelligent systems: networks of plausible inference. Morgan Kaufmann, Los Altos
Pearson K (1894) Contribution to the mathematical theory of evolution. Philos Trans R Soc 185:71–110
Peel D, McLachlan G (2000) Robust mixture modeling using the t distribution. Stat Comput 10:339–348
Peel D, Whiten W, McLachlan G (2001) Fitting mixtures of Kent distributions to aid in joint set identifications. J Am Stat Assoc 96:56–63
Raftery AE, Dean N (2006) Variable selection for model-based clustering. J Am Stat Assoc 101:168–178
Raftery AE, Niu X, Hoff PD, Yeung KY (2012) Fast inference for the latent space network model using a case-control approximate likelihood. J Comput Graph Stat 21(4):901–919
Ray S, Cheng Y (2014) Package Modalclust: hierarchical modal clustering. http://cran.r-project.org/web/packages/Modalclust
Ray S, Lindsay B (2005) The topography of multivariate normal mixtures. Ann Stat 33(5):2042–2065
Richardson S, Green PJ (1997) On Bayesian analysis of mixtures with an unknown number of components (with discussion). J R Stat Soc Ser B 59:731–792
Robin S, Bar-Hen A, Daudin JJ, Pierre L (2007) A semi-parametric approach for mixture models: application to local false discovery rate estimation. Comput Stat Data Anal 51:5483–5493
Rodriguez CE, Walker SG (2014) Label switching in Bayesian mixture models: deterministic relabeling strategies. J Comput Graph Stat 23(1):25–45
Saídaoui F (2010) Acceleration of the em algorithm via extrapolation methods: review, comparison and new methods. Comput Stat Data Anal 54(3):750–766
Salter-Townshend M, Murphy TB (2013) Variational Bayesian inference for the latent position cluster model for network data. Comput Stat Data Anal 57:661–671
Sampson SF (1969) Crisis in a cloister. Ph.D. thesis, Department of Sociology, Cornell University, Ithaca
Schwarz G (1978) Estimating the dimensions of a model. Ann Stat 6:461–464
Seo B, Kim D (2012) Root selection in normal mixture models. Comput Stat Data Anal 56:2454–2470
Shental N, Bar-Hillel A, Hertz T, Weinshall D (2003) Computing Gaussian mixture models with EM using equivalence constraints. In: Advances in NIPS, A Bradford Book, vol 15
Steiner P, Hudec M (2007) Classification of large data sets with mixture models via sufficient em. Comput Stat Data Anal 51:5416–5428
Stuetzle W (2003) Estimating the cluster tree of a density by analyzing the minimal spanning tree of a sample. J Classif 20:25–47
Stuetzle W, Nugent R (2010) A generalized single linkage method for estimating the cluster tree of a density. J Comput Graph Stat 19:397–418
Tanabe A, Fukumizu K, Oba S, Takenouchi T, Ishii S (2007) Parameter estimation for von Mises-Fisher distributions. Comput Stat 22:145–157
Teicher H (1963) Identifiability of finite mixtures. Ann Math Stat 34:1265–1269
Vardi Y, Shepp LA, Kaufman LA (1985) A statistical model for positron emission tomography. J Am Stat Assoc 80:8–37
Vicari D, Alfó M (2014) Model based clustering of customer choice data. Comput Stat Data Anal 71:3–13
Vu DQ, Hunter DR, Schweinberger M (2013) Model-based clustering of large networks. Ann Appl Stat 7:1010–1039
Wagstaff K, Cardie C, Rogers S, Schroedl S (2001) Constrained K-means clustering with background knowledge. In: Proceedings of the eighteenth international conference on machine learning (ICML-2001), pp 577–584
Wang B, Titterington D (2006) Convergence properties of a general algorithm for calculating variational Bayesian estimates for a normal mixture model. Bayesian Anal 1(3):625–650
Wang S, Zhu J (2008) Variable selection for model-based high-dimensional clustering and its application to microarray data. Biometrics 64:440–448
Wang H, Zhang Q, Luo B, Wei S (2004) Robust mixture modelling using multivariate t-distribution with missing information. Pattern Recognit Lett 25:701–710
Wei GCG, Tanner MA (1990) A Monte Carlo implementation of the EM algorithm and the Poor Man’s data augmentation algorithms. J Am Stat Assoc 85(411):699–704
Wishart D (1969) Mode analysis: a generalization of nearest neighbor which reduces chaining effect. In: Cole AJ (ed) Numerical taxonomy. Academic, London, pp 282–311
Wolfe JH (1967) NORMIX: computational methods for estimating the parameters of multivariate normal mixture distributions. Technical bulletin USNPRA SRM 6
Xie B, Pan W, Shen X (2010) Penalized mixtures of factor analyzers with application to clustering high-dimensional microarray data. Bioinformatics 26:501–508
Xing EP, Ng AY, Jordan MI, Russell S (2003) Distance metric learning with application to clustering with side-information. In: Thrun S, Becker S, Obermayer K (eds) Advances in neural information processing systems, vol 15. MIT Press, Cambridge, pp 505–512
Yakowitz SJ, Spragins JD (1968) On the identifiability of finite mixtures. Ann Math Stat 39(1):209–214
Ypma A, Heskes T (2002) Categorization of web pages and user clustering with mixtures of hidden Markov models. In: Proceedings of the international workshop on web knowledge discovery and data mining WEBKDD’02, Edmonton, pp 31–43
Yuan M, Lin Y (2007) Model selection and estimation in the Gaussian graphical model. Biometrika 94:19–35
Zhou H, Pan W, X S (2009) Penalized model-based clustering with unconstrained covariance matrices. Electron J Stat 3:1473–1496
Zhu X (2005) Semi-supervised learning literature survey. Technical Report 1530, Computer Sciences, University of Wisconsin-Madison
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Melnykov, V., Michael, S., Melnykov, I. (2015). Recent Developments in Model-Based Clustering with Applications. In: Celebi, M. (eds) Partitional Clustering Algorithms. Springer, Cham. https://doi.org/10.1007/978-3-319-09259-1_1
Download citation
DOI: https://doi.org/10.1007/978-3-319-09259-1_1
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-09258-4
Online ISBN: 978-3-319-09259-1
eBook Packages: EngineeringEngineering (R0)