1932

Abstract

The important role of finite mixture models in the statistical analysis of data is underscored by the ever-increasing rate at which articles on mixture applications appear in the statistical and general scientific literature. The aim of this article is to provide an up-to-date account of the theory and methodological developments underlying the applications of finite mixture models. Because of their flexibility, mixture models are being increasingly exploited as a convenient, semiparametric way in which to model unknown distributional shapes. This is in addition to their obvious applications where there is group-structure in the data or where the aim is to explore the data for such structure, as in a cluster analysis. It has now been three decades since the publication of the monograph by McLachlan & Basford (1988) with an emphasis on the potential usefulness of mixture models for inference and clustering. Since then, mixture models have attracted the interest of many researchers and have found many new and interesting fields of application. Thus, the literature on mixture models has expanded enormously, and as a consequence, the bibliography here can only provide selected coverage.

Loading

Article metrics loading...

/content/journals/10.1146/annurev-statistics-031017-100325
2019-03-07
2024-06-14
Loading full text...

Full text loading...

/deliver/fulltext/statistics/6/1/annurev-statistics-031017-100325.html?itemId=/content/journals/10.1146/annurev-statistics-031017-100325&mimeType=html&fmt=ahah

Literature Cited

  1. Aitkin M, Rubin DB. 1985. Estimation and hypothesis testing in finite mixture models. J. R. Stat. Soc. B 47:67–75
    [Google Scholar]
  2. Anderson E. 1935. The irises of the Gaspé Peninsula. Bull. Am. Iris Soc. 59:2–5
    [Google Scholar]
  3. Arellano-Valle RB, Azzalini A. 2006. On the unification of families of skew–normal distributions. Scand. J. Stat. 33:561–74
    [Google Scholar]
  4. Arellano-Valle RB, Genton M. 2005. On fundamental skew distributions. J. Multivar. Anal. 96:93–116
    [Google Scholar]
  5. Azzalini A, Capitanio A. 2003. Distributions generated by perturbation of symmetry with emphasis on a multivariate skew t-distribution. J. R. Stat. Soc. B 65:367–89
    [Google Scholar]
  6. Azzalini A, Dalla Valle A. 1996. The multivariate skew-normal distribution. Biometrika 83:715–26
    [Google Scholar]
  7. Baek J, McLachlan GJ. 2011. Mixtures of common t-factor analyzers for clustering high-dimensional microarray data. Bioinformatics 27:1269–76
    [Google Scholar]
  8. Baek J, McLachlan GJ, Flack L. 2010. Mixtures of factor analyzers with common factor loadings: applications to the clustering and visualization of high-dimensional data. IEEE Trans. Pattern Anal. Mach. Intell. 32:1298–309
    [Google Scholar]
  9. Banfield JD, Raftery AE. 1993. Model-based Gaussian and non-Gaussian clustering. Biometrics 49:803–21
    [Google Scholar]
  10. Barndorff-Nielsen O. 1977. Exponentially decreasing distributions for the logarithm of particle size. Proc. R. Soc. Lond. A 353:401–19
    [Google Scholar]
  11. Baudry JP, Raftery AE, Celeux G, Lo K, Gottardo R. 2010. Combining mixture components for clustering. J. Comput. Graph. Stat. 19:332–53
    [Google Scholar]
  12. Baum LE, Petrie T. 1966. Statistical inference for probabilistic functions of finite Markov chains. Ann. Math. Stat. 37:1554–63
    [Google Scholar]
  13. Benaglia T, Chauveau D, Hunter D, Young D. 2009. Mixtools: an R package for analyzing mixture models. J. Stat. Softw. 32:1–29
    [Google Scholar]
  14. Besag J. 1986. On the statistical analysis of dirty pictures (with discussion). J. R. Stat. Soc. B 48:259–302
    [Google Scholar]
  15. Bickel PJ, Chernoff H. 1993. Asymptotic distribution of the likelihood ratio statistic in a prototypical non regular problem. Statistics and Probability: A Raghu Raj Bahadur Festschrift JK Ghosh, SK Mitra, BP Rao83–96 New Delhi: Wiley Eastern
    [Google Scholar]
  16. Biernacki C, Celeux G, Govaert G. 2000. Assessing a mixture model for clustering with the integrated completed likelihood. IEEE Trans. Pattern Anal. Mach. Intell. 22:719–25
    [Google Scholar]
  17. Böhning D 1999. Computer-Assisted Analysis of Mixtures and Applications: Meta-Analysis, Disease Mapping and Others New York: Chapman & Hall/CRC
    [Google Scholar]
  18. Browne RP, McNicholas PD. 2015. A mixture of generalized hyperbolic distributions. Can. J. Stat. 43:176–98
    [Google Scholar]
  19. Cabral CRB, Lachos VH, Prates MO. 2012. Multivariate mixture modeling using skew-normal independent distributions. Comput. Stat. Data Anal. 56:126–42
    [Google Scholar]
  20. Chen J. 2017. Consistency of the MLE under mixture models. Stat. Sci. 7:1–26
    [Google Scholar]
  21. Chen H, Chen J. 2001. The likelihood ratio test for homogeneity in finite mixture models. Can. J. Stat. 29:201–15
    [Google Scholar]
  22. Chen H, Chen J, Kalbfleisch JD. 2001. A modified likelihood ratio test for homogeneity in finite mixture models. J. R. Stat. Soc. B 63:19–29
    [Google Scholar]
  23. Chen H, Chen J, Kalbfleisch JD. 2004. Testing for a finite mixture model with two components. J. R. Stat. Soc. B 66:95–115
    [Google Scholar]
  24. Chen J, Li P. 2009. Hypothesis test for normal mixture models: the EM approach. Ann. Stat. 37:2523–42
    [Google Scholar]
  25. Chen J, Li P, Fu Y. 2012. Inference on the order of a normal mixture. J. Am. Stat. Assoc. 107:1096–105
    [Google Scholar]
  26. Coleman D, Dong X, Hardin J, Rocke DM, Woodruff DL. 1999. Some computational issues in cluster analysis with no a priori metric. Comput. Stat. Data Anal. 31:1–11
    [Google Scholar]
  27. Coretto P, Hennig C. 2017. Robust improper maximum likelihood: tuning, computation, and a comparison with other methods for robust Gaussian clustering. J. Am. Stat. Assoc. 111:1648–59
    [Google Scholar]
  28. Dacunha-Castelle D, Gassiat E. 1997. The estimation of the order of a mixture model. Bernoulli 3:279–99
    [Google Scholar]
  29. Day NE. 1969. Estimating the components of a mixture of normal distributions. Biometrika 56:463–74
    [Google Scholar]
  30. Dempster AP, Laird NM, Rubin DB. 1977. Maximum likelihood from incomplete data via the EM algorithm. J. R. Stat. Soc. B 39:1–38
    [Google Scholar]
  31. Diebolt J, Robert CP. 1994. Estimation of finite mixture distributions through Bayesian sampling. J. R. Stat. Soc. B 56:363–75
    [Google Scholar]
  32. Drton M, Plummer M. 2017. A Bayesian information criterion for singular models (with discussion). J. R. Stat. Soc. B 79:323–80
    [Google Scholar]
  33. Escobar MD, West M. 1995. Bayesian density estimation and inference using mixtures. J. Am. Stat. Assoc. 90:577–88
    [Google Scholar]
  34. Everitt B, Hand D 1981. Finite Mixture Distributions New York: Chapman & Hall
    [Google Scholar]
  35. Fisher RA. 1936. The use of multiple measurements in taxonomic problems. Ann. Eugen. 7:179–88
    [Google Scholar]
  36. Fraley C, Raftery AE. 2002. Model-based clustering, discriminant analysis, and density estimation. J. Am. Stat. Assoc. 97:611–31
    [Google Scholar]
  37. Franczak BC, Browne RP, McNicholas PD. 2014. Mixtures of shifted asymmetric Laplace distributions. IEEE Trans. Pattern Anal. Mach. Intell. 36:1149–57
    [Google Scholar]
  38. Frühwirth-Schnatter S 2006. Finite Mixture and Markov Switching Models New York: Springer
    [Google Scholar]
  39. Furman WD, Lindsay BG. 1994a. Measuring the effectiveness of moment estimators as starting values in maximizing mixture likelihoods. Comput. Stat. Data Anal. 17:493–507
    [Google Scholar]
  40. Furman WD, Lindsay BG. 1994b. Testing for the number of components in a mixture of normal distributions using moment estimators. Comput. Stat. Data Anal. 17:473–92
    [Google Scholar]
  41. Galton F 1869. Hereditary Genius: An Inquiry into Its Laws and Consequences London: Macmillan
    [Google Scholar]
  42. Ganesalingam S, McLachlan GJ. 1978. The efficiency of a linear discriminant function based on unclassified initial samples. Biometrika 65:658–65
    [Google Scholar]
  43. García-Escudero LA, Gordaliza A, Greselin F, Ingrassia I, Mayo-Iscar A. 2016. The joint role of trimming and constraints in robust estimation for mixtures of Gaussian factor analyzers. Comput. Stat. Data Anal. 99:131–47
    [Google Scholar]
  44. García-Escudero LA, Gordaliza A, Greselin F, Ingrassia I, Mayo-Iscar A. 2018. Eigenvalues and constraints in mixture modeling: geometric and computational issues. Adv. Data Anal. Classif. 12:20333
    [Google Scholar]
  45. García-Escudero LA, Gordaliza A, Matrán C, Mayo-Iscar A. 2008. A general trimming approach to robust cluster analysis. Ann. Stat. 36:1324–45
    [Google Scholar]
  46. Garel B. 2005. Asymptotic theory of the likelihood ratio test for the identification of a mixture. J. Stat. Plan. Inference 131:271–96
    [Google Scholar]
  47. Ghahramani Z, Hinton G. 1997. The EM algorithm for factor analyzers Tech. Rep. CRG-TR-96-1 Dep. Comput. Sci., Univ. Toronto:
    [Google Scholar]
  48. Ghosh JK, Sen PK. 1985. On the asymptotic performance of the log likelihood ratio statistic for the mixture model and related results. Proceedings of the Berkeley Conference in Honor of Jerzy Neyman and Jack Kiefer 2 L LeCam, R Olshen789–806 Monterey, CA: Wadsworth
    [Google Scholar]
  49. Grün B, Leisch F. 2008. FlexMix Version 2: finite mixtures with concomitant variables and varying and constant parameters. J. Stat. Softw. 28:1–35
    [Google Scholar]
  50. Hall P, Zhou XH. 2003. Nonparametric estimation of component distributions in a multivariate mixture. Ann. Stat. 31:201–24
    [Google Scholar]
  51. Hartigan JA 1975. Clustering Algorithms New York: Wiley
    [Google Scholar]
  52. Hartigan JA. 1985a. A failure of likelihood asymptotics for normal mixtures. Proceedings of the Berkeley Conference in Honor of Jerzy Neyman and Jack Kiefer 2 L LeCam, R Olshen807–10 Monterey, CA: Wadsworth
    [Google Scholar]
  53. Hartigan JA. 1985b. Statistical theory in clustering. J. Classif. 2:63–76
    [Google Scholar]
  54. Hennig C. 2010. Methods for merging Gaussian mixture components. Adv. Data. Anal. Classif. 4:3–34
    [Google Scholar]
  55. Hinton GE, Dayan P, Revow M. 1997. Modeling the manifolds of images of handwritten digits. IEEE Trans. Neural. Netw. 8:65–74
    [Google Scholar]
  56. Holmes GK. 1892. Measures of distribution. J. Am. Stat. Assoc. 3:141–57
    [Google Scholar]
  57. Hunter DR, Lange K. 2004. A tutorial on MM algorithms. Am. Stat. 58:30–37
    [Google Scholar]
  58. Jacobs RA, Jordan MI, Nowlan SJ, Hinton GE. 1991. Adaptive mixtures of local experts. Neur. Comput. 3:79–87
    [Google Scholar]
  59. Jeffreys SH. 1932. An alternative to the rejection of observations. Proc. R. Soc. Lond. A 137:78–87
    [Google Scholar]
  60. Kadane JB. 1974. The role of identification in Bayesian theory. Studies in Bayesian Econometrics and Statistics S Fienberg, A Zellner175–91 New York: Elsevier
    [Google Scholar]
  61. Karlis D, Santourian A. 2009. Model-based clustering with non-elliptically contoured distributions. Stat. Comput. 19:73–83
    [Google Scholar]
  62. Keribin C. 2000. Consistent estimation of the order of mixture models. Sankhya A 62:49–66
    [Google Scholar]
  63. Lange K 2013. Optimization New York: Springer
    [Google Scholar]
  64. Lavine M, West M. 1992. A Bayesian method of classification and discrimination. Can. J. Stat. 20:451–61
    [Google Scholar]
  65. Lebret R, Iovleff S, Langrognet F, Biernacki C, Celeux G, Govaert G. 2015. Rmixmod: The R package of the model-based unsupervised, supervised, and semi-supervised classification Mixmod library. J. Stat. Softw. 67:1–29
    [Google Scholar]
  66. Lee SX, Leemaqz KL, McLachlan GJ. 2018. A block EM algorithm for multivariate skew normal and skew t-mixture models. IEEE T. Neur. Net. Learn. 29:5581–91
    [Google Scholar]
  67. Lee SX, McLachlan GJ. 2013. On mixtures of skew normal and skew t-distributions. Adv. Data. Anal. Classif. 7:241–66
    [Google Scholar]
  68. Lee SX, McLachlan GJ. 2014. Finite mixtures of multivariate skew t-distributions: some recent and new results. Stat. Comput. 24:181–202
    [Google Scholar]
  69. Lee SX, McLachlan GJ. 2016. Finite mixtures of canonical fundamental skew t-distributions. Stat. Comput. 26:573–89
    [Google Scholar]
  70. Lee SX, McLachlan GJ. 2018. EMMIXcskew: an R package for the fitting of a mixture of canonical fundamental skew t-distributions. J. Stat. Softw. 83:1–32
    [Google Scholar]
  71. Leroux B. 1992. Consistent estimation of a mixing distribution. Ann. Stat. 20:1350–60
    [Google Scholar]
  72. Li JQ, Barron AR. 1999. Mixture density estimation. Advances in Neural Information Processing Systems 12 (NIPS 1999) SA Solla, TK Leen, K Müller279–85 Cambridge, MA: MIT Press
    [Google Scholar]
  73. Li P, Chen J. 2010. Testing the order of a finite mixture. J. Am. Stat. Assoc. 105:1084–92
    [Google Scholar]
  74. Li P, Chen J, Marriott P. 2009. Non-finite Fisher information and homogeneity: an EM approach. Biometrika 96:411–26
    [Google Scholar]
  75. Lin TI, McLachlan GJ, Lee SX. 2016. Extending mixtures of factor models using the restricted multivariate skew-normal distribution. J. Multivar. Anal. 143:398–413
    [Google Scholar]
  76. Lin TI, Wu PH, McLachlan GJ, Lee SX. 2015. A robust factor analysis model using the restricted skew t-distribution. TEST 24:510–31
    [Google Scholar]
  77. Lindsay BG 1995. Mixture Models: Theory, Geometry and Applications Hayward, CA: Inst. Math. Stat.
    [Google Scholar]
  78. Liu X, Shao Y. 2004. Asymptotics for the likelihood ratio test in a two-component normal mixture model. J. Stat. Plan. Inference 123:61–81
    [Google Scholar]
  79. Lo K, Gottardo R. 2012. Flexible mixture modeling via the multivariate t distribution with the Box-Cox transformation: an alternative to the skew-t distribution. Stat. Comput. 22:33–52
    [Google Scholar]
  80. Maugis C, Celeux G, Martin-Magniette ML. 2009. Variable selection for clustering with Gaussian mixture models. Biometrics 65:701–9
    [Google Scholar]
  81. McLachlan GJ. 1975. Iterative reclassification procedure for constructing an asymptotically optimal rule of allocation in discriminant analysis. J. Am. Stat. Assoc. 70:365–69
    [Google Scholar]
  82. McLachlan GJ. 1982. The classification and mixture maximum likelihood approaches to cluster analysis. Handbook of Statistics 2 PR Krishnaiah, L Kanal199–208 Amsterdam: North-Holland
    [Google Scholar]
  83. McLachlan GJ. 1987. On bootstrapping the likelihood ratio test statistic for the number of components in a normal mixture. Appl. Stat. 36:318–24
    [Google Scholar]
  84. McLachlan GJ 1992. Discriminant Analysis and Statistical Pattern Recognition New York: Wiley
    [Google Scholar]
  85. McLachlan GJ. 2016. Mixture distributions—further developments. Wiley StatsRef: Statistics Reference Online N Balakrishnan, P Brandimarte, B Everitt, G Molenberghs, F Ruggeri, W Piegorsch Chichester, UK: Wiley https://doi.org/10.1002/9781118445112.stat00947.pub2
    [Crossref] [Google Scholar]
  86. McLachlan GJ, Basford K 1988. Mixture Models: Inference and Applications to Clustering New York: Marcel Dekker
    [Google Scholar]
  87. McLachlan GJ, Bean RW, Ben-Tovim Jones L. 2007. Extension of the mixture of factor analyzers model to incorporate the multivariate t-distribution. Comput. Stat. Data Anal. 51:5327–38
    [Google Scholar]
  88. McLachlan GJ, Bean R, Peel D. 2002. A mixture model–based approach to the clustering of microarray expression data. Bioinformatics 18:413–22
    [Google Scholar]
  89. McLachlan GJ, Lee SX. 2016. Comment on “On nomenclature for, and the relative merits of, two formulations of skew distributions” by A Azzalini, R Browne, M Genton, and P McNicholas. Stat. Probab. Lett. 116:1–5
    [Google Scholar]
  90. McLachlan GJ, Khan N. 2004. On a resampling approach for tests on the number of clusters with mixture model-based clustering of tissue samples. J. Multivar. Anal. 90:90–105
    [Google Scholar]
  91. McLachlan GJ, Krishnan T 2008. The EM Algorithm and Extensions Hoboken, NJ: Wiley. 2nd ed.
    [Google Scholar]
  92. McLachlan GJ, Peel D. 1998. Robust cluster analysis via mixtures of multivariate t-distributions. Advances in Pattern Recognition A Amin, D Dori, P Pudil, H Freeman Berlin: Springer
    [Google Scholar]
  93. McLachlan GJ, Peel D 2000a. Finite Mixture Models New York: Wiley
    [Google Scholar]
  94. McLachlan GJ, Peel D. 2000b. Mixtures of factor analyzers. Proceedings of the Seventeenth International Conference on Machine Learningpp 599–606 Burlington, MA: Morgan Kaufmann
    [Google Scholar]
  95. McLachlan GJ, Peel D, Basford KE, Adams P. 1999. The EMMIX algorithm for the fitting of normal and t-components. J. Stat. Softw. 4:1–14
    [Google Scholar]
  96. McLachlan GJ, Rathnayake SI. 2011. Testing for group structure in high-dimensional data. J. Biopharm. Stat. 21:1113–25
    [Google Scholar]
  97. McLachlan GJ, Rathnayake SI. 2014. On the number of components in a Gaussian mixture model. WIREs Data Min. Knowl. Discov. 4:341–55
    [Google Scholar]
  98. McNicholas PD 2017. Mixture Model-Based Classification Boca Raton, FL: CRC
    [Google Scholar]
  99. McNicholas PD, Murphy TB. 2008. Parsimonious Gaussian mixture models. Stat. Comput. 18:285–96
    [Google Scholar]
  100. Meng XL, Rubin DB. 1993. Maximum likelihood estimation via the ECM algorithm: a general framework. Biometrika 80:267–78
    [Google Scholar]
  101. 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. B 59:511–67
    [Google Scholar]
  102. Mengersen K, Robert C, Titterington D, eds. 2011. Mixtures: Estimation and Applications New York: Wiley
    [Google Scholar]
  103. Montanari A, Viroli C. 2010. Heteroscedastic factor mixture analysis. Stat. Model. 10:441–60
    [Google Scholar]
  104. Murray P, Browne R, McNicholas P. 2014. Mixtures of skew-t factor analyzers. Comput. Stat. Data Anal. 77:326–35
    [Google Scholar]
  105. Newcomb S. 1886. A generalized theory of the combination of observations so as to obtain the best result. Am. J. Math. 8:343–66
    [Google Scholar]
  106. Ng SK, McLachlan GJ, Wang K, Ben-Tovim Jones L, Ng SW. 2006. A mixture model with random-effects components for clustering correlated gene-expression profiles. Bioinformatics 22:1745–52
    [Google Scholar]
  107. Ng SK, McLachlan GJ, Wang K, Nagymanyoki Z, Liu S, Ng SW. 2015. Inference on differential expression using cluster-specific contrasts of mixed effects. Biostatistics 16:98–112
    [Google Scholar]
  108. Nguyen HD, McLachlan GJ. 2016a. Laplace mixtures of linear experts. Comput. Stat. Data Anal. 93:177–91
    [Google Scholar]
  109. Nguyen HD, McLachlan GJ. 2016b. Maximum likelihood estimation of triangular and polygonal distributions. Comput. Stat. Data Anal. 102:23–36
    [Google Scholar]
  110. Nguyen HD, McLachlan GJ, Orban P, Bellec P, Janke AL. 2017. Maximum pseudolikelihood estimation for a model-based clustering of time-series data. Neural Comput. 29:990–1020
    [Google Scholar]
  111. Nguyen HD, McLachlan GJ, Ullmann JFP, Janke AL. 2016. Spatial clustering of time-series via mixtures of autoregressive models and Markov random fields. Stat. Neerl. 70:414–39
    [Google Scholar]
  112. Nguyen HD, Ullmann JFP, McLachlan GJ, Voleti V, Li W et al. 2018. Whole-volume clustering of time series data from zebrafish brain calcium images via mixture model-based functional data analysis. Stat. Anal. Data Min. 11:5–16
    [Google Scholar]
  113. O'Neill TJ. 1978. Normal discrimination with unclassified observations. J. Am. Stat. Assoc. 73:821–26
    [Google Scholar]
  114. Pan W, Shen X. 2007. Penalized model-based clustering with application to variable selection. J. Mach. Learn. Res. 8:1145–64
    [Google Scholar]
  115. Panel on Nonstandard Mixtures of Distributions. 1989. Statistical models and analysis in auditing. Stat. Sci. 4:2–33
    [Google Scholar]
  116. Pearson K. 1894. Contributions to the mathematical theory of evolution. Phil. Trans. R. Soc. Lond. A 185:71–110
    [Google Scholar]
  117. Peel D, McLachlan GJ. 2000. Robust mixture modelling using the t distribution. Stat. Comput. 10:339–48
    [Google Scholar]
  118. Prates MO, Cabral CRB, Lachos VH. 2013. Mixsmsn: fitting finite mixture of scale mixture of skew-normal distributions. J. Stat. Softw. 54:1–20
    [Google Scholar]
  119. Pyne S, Hu X, Wang K, Rossin E, Lin TI et al. 2009. Automated high-dimensional flow cytometric data analysis. PNAS 106:8519–24
    [Google Scholar]
  120. Quetelet A. 1846. Lettres à S.A.R. Le Duc Régnant de Saxe-Coburg et Gotha: sur la Théorie des Probabilités, Appliquée aux Sciences Morales Et Politiques Brussels: Hayez
    [Google Scholar]
  121. Quetelet A. 1852. Sur quelques propritiétés curieuses que présentent les résultats d'une serie d'observations, faites dans la vue de déterminer une constante, lorsque les chances de rencontrer des écarts en plus et en moins sont égales et indépendantés les unes des autres. Bull. Acad. R. Sci. Lett. Beaux-Arts Belg. 19:303–17
    [Google Scholar]
  122. Quinn BG, McLachlan GJ, Hjort NL. 1987. A note on the Aitkin-Rubin approach to hypothesis testing in mixture models. J. R. Stat. Soc. B 49:311–14
    [Google Scholar]
  123. R Development Team. 2012. R: A language and environment for statistical computing Vienna: R Found. Stat. Comput.
    [Google Scholar]
  124. Raftery A, Dean N. 2006. Variable selection for model-based clustering. J. Am. Stat. Assoc. 101:168–78
    [Google Scholar]
  125. Rao CR. 1948. The utilization of multiple measurements in problems of biological classification. J. R. Stat. Soc. B 10:159–203
    [Google Scholar]
  126. Redner R. 1981. Note on the consistency of the maximum likelihood estimate for nonidentifiable distributions. Ann. Stat. 9:225–28
    [Google Scholar]
  127. Richardson S, Green PJ. 1997. On Bayesian analysis of mixtures with an unknown number of components (with discussion). J. R. Stat. Soc. B 59:731–92
    [Google Scholar]
  128. Robert CP. 1996. Mixtures of distributions: inference and estimation. Markov Chain Monte Carlo in Practice WR Gilks, S Richardson, DJ Spiegelhalter441–64 London: Chapman & Hall
    [Google Scholar]
  129. Roeder K, Wasserman L. 1997. Practical Bayesian density estimation using mixtures of normals. J. Am. Stat. Assoc. 92:894–902
    [Google Scholar]
  130. Schwarz G. 1978. Estimating the dimension of a model. Ann. Stat. 6:461–64
    [Google Scholar]
  131. Scrucca L, Fop M, Murphy TB, Raftery AE. 2016. Mclust 5: clustering, classification and density estimation using Gaussian finite mixture models. R J. 8:205–33
    [Google Scholar]
  132. Spurek P. 2017. General split Gaussian cross-entropy clustering. Expert Syst. Appl. 68:58–68
    [Google Scholar]
  133. Stigler SM 1986. The History of Statistics: The Measurement of Uncertainty Before 1900 Cambridge, MA: Harvard Univ. Press
    [Google Scholar]
  134. Tanner MA, Wong WH. 1987. The calculation of posterior distributions by data augmentation (with discussion). J. Am. Stat. Assoc. 82:528–50
    [Google Scholar]
  135. Teicher H. 1960. On the mixture of distributions. Ann. Math. Stat. 31:55–73
    [Google Scholar]
  136. Tibshirani R, Walther G, Hastie T. 2001. Estimating the number of clusters in a data set via the gap statistic. J. R. Stat. Soc. B 63:411–23
    [Google Scholar]
  137. Titterington DM. 1981. Contribution to the discussion of paper by M. Aitkin, D. Anderson, and J. Hinde. J. R. Stat. Soc. A 144:459
    [Google Scholar]
  138. Titterington DM, Smith A, Makov U 1985. Statistical Analysis of Finite Mixture Distributions New York: Wiley
    [Google Scholar]
  139. Tortora C, McNicholas P, Browne R. 2016. A mixture of generalized hyperbolic factor analyzers. Adv. Data Anal. Classif. 10:423–40
    [Google Scholar]
  140. Viroli C. 2010. Dimensionally reduced model-based clustering through mixtures of factor mixture analyzers. J. Classif. 27:363–88
    [Google Scholar]
  141. Viroli C, McLachlan GJ 2017. Deep Gaussian mixture models. Stat. Comput https://doi.org/10.1007/s11222-017-9793-z
    [Crossref] [Google Scholar]
  142. Wald A. 1949. Note on the consistency of the maximum likelihood estimate. Ann. Math. Stat. 20:595–601
    [Google Scholar]
  143. Weldon WFR. 1892. Certain correlated variations in Crangon vulgaris. Proc. R. Soc. Lond. 51:1–21
    [Google Scholar]
  144. Weldon WFR. 1893. On certain correlated variations in Carcinus maenas. Proc. R. Soc. Lond. 54:318–29
    [Google Scholar]
  145. Witten D, Tibshirani R. 2010. A framework for feature selection in clustering. J. Am. Stat. Assoc. 105:713–26
    [Google Scholar]
  146. Wolfe JH. 1970. Pattern clustering by multivariate mixture analysis. Multivar. Behav. Res. 5:329–50
    [Google Scholar]
  147. Wu CFJ. 1983. On the convergence properties of the EM algorithm. Ann. Math. Stat. 11:95–103
    [Google Scholar]
  148. Wraith D, Forbes F. 2015. Location and scale mixtures of Gaussians with flexible tail behaviour: properties, inference and application to multivariate clustering. Comput. Stat. Data Anal. 90:61–73
    [Google Scholar]
  149. Xie B, Pan W, Shen X. 2010. Penalized mixtures of factor analyzers with application to clustering high-dimensional microarray data. Bioinformatics 26:501–8
    [Google Scholar]
  150. Yakowitz SJ, Spragins JD. 1968. On the identifiability of finite mixtures. Ann. Math. Stat. 39:209–14
    [Google Scholar]
  151. Zhou H, Pan W. 2009. Penalized model-based clustering with unconstrained covariance matrices. Electron. J. Stat. 3:1473–96
    [Google Scholar]
  152. Zhu X, Melnykov V. 2018. Manly transformation in finite mixture modeling. Comput. Stat. Data Anal. 121:190–208
    [Google Scholar]
/content/journals/10.1146/annurev-statistics-031017-100325
Loading
/content/journals/10.1146/annurev-statistics-031017-100325
Loading

Data & Media loading...

  • Article Type: Review Article
This is a required field
Please enter a valid email address
Approval was a Success
Invalid data
An Error Occurred
Approval was partially successful, following selected items could not be processed due to error