Abstract
The cluster-weighted model (CWM) is a mixture model with random covariates that allows for flexible clustering/classification and distribution estimation of a random vector composed of a response variable and a set of covariates. Within this class of models, the generalized linear exponential CWM is here introduced especially for modeling bivariate data of mixed-type. Its natural counterpart in the family of latent class models is also defined. Maximum likelihood parameter estimates are derived using the expectation-maximization algorithm and some computational issues are detailed. Through Monte Carlo experiments, the classification performance of the proposed model is compared with other mixture-based approaches, consistency of the estimators of the regression coefficients is evaluated, and several likelihood-based information criteria are compared for selecting the number of mixture components. An application to real data is also finally considered.
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References
Akaike H (1973) Information theory and an extension of maximum likelihood principle. In: Petrov BN, Csaki F (eds) Second International Symposium on Information Theory. Akademiai Kiado, Budapest, pp 267–281
Bagnato L, Punzo A (2013) Finite mixtures of unimodal beta and gamma densities and the \(k\)-bumps algorithm. Comput Stat 28(4):1571–1597
Balakrishnan N, Lai C-D (2009) Continuous bivariate distributions. Springer, New York
Banfield JD, Raftery AE (1993) Model-based Gaussian and non-Gaussian clustering. Biometrics 49(3):803–821
Bermúdez L, Karlis D (2012) A finite mixture of bivariate Poisson regression models with an application to insurance ratemaking. Comput Stat Data Anal 56(12):3988–3999
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(7):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 41(3–4):561–575
Böhning D, Dietz E, Schaub R, Schlattmann P, Lindsay BG (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
Bozdogan H (1994) Theory and methodology of time series analysis. In: Proceedings of the First US/Japan Conference on the Frontiers of Statistical Modeling: An Informational Approach, vol 1. Kluwer Academic Publishers, Dordrecht
Bozdogan H (1987) Model selection and Akaike’s information criterion (AIC): the general theory and its analytical extensions. Psychometrika 52(3):345–370
Browne RP, McNicholas PD (2012) Model-based clustering, classification, and discriminant analysis of data with mixed type. J Stat Plan Inference 142(11):2976–2984
Celeux G, Hurn M, Robert CP (2000) Computational and inferential difficulties with mixture posterior distributions. J Am Stat Assoc 95(451):957–970
Dempster A, Laird N, Rubin D (1977) Maximum likelihood from incomplete data via the EM algorithm. J R Stat Soc Series B Methodol 39(1):1–38
Escobar M, West M (1995) Bayesian density estimation and inference using mixtures. J Am Stat Assoc 90(430):577–588
Fonseca JRS, Cardoso MGMS (2005) Retail clients latent segments. In: Progress in Artificial Intelligence. Springer, Berlin, pp 348–358
Fonseca JRS (2008) The application of mixture modeling and information criteria for discovering patterns of coronary heart disease. J Appl Quant Methods 3(4):292–303
Fonseca JRS (2010) On the performance of information criteria in latent segment models. World Acad Sci Eng Technol 63:2010
Fraley C, Raftery AE, Murphy TB, Scrucca L (2012) mclust version 4 for R: normal mixture modeling for model-based clustering, classification, and density estimation. Technical report 597, Department of Statistics, University of Washington, Seattle, Washington, USA
Frühwirth-Schnatter S (2006) Finite mixture and Markov switching models. Springer, New York
Genest C, Neslehova J (2007) A primer on copulas for count data. Astin Bull 37(2):475–515
Gershenfeld N (1997) Nonlinear inference and cluster-weighted modeling. Ann New York Acad Sci 808(1):18–24
Grün B, Leisch F (2008) FlexMix version 2: finite mixtures with concomitant variables and varying and constant parameters. J Stat Softw 28(4):1–35
Hennig C (2000) Identifiablity of models for clusterwise linear regression. J Classif 17(2):273–296
Hennig C, Liao TF (2013) How to find an appropriate clustering for mixed type variables with application to socio-economic stratification. J R Stat Soc Series C Appl Stat 62(3):1–25
Henning G (1989) Meanings and implications of the principle of local independence. Lang Test 6(1):95–108
Hunt LA, Basford KE (1999) Fitting a mixture model to three-mode three-way data with categorical and continuous variables. J Classif 16(2):283–296
Hunt LA, Jorgensen M (2011) Clustering mixed data. Wiley Interdiscip Rev Data Min Knowl Discov 1(4):352–361
Hurvich CM, Tsai CL (1989) Regression and time series model selection in small samples. Biometrika 76(2):297–307
Ingrassia S, Minotti SC, Vittadini G (2012) Local statistical modeling via the cluster-weighted approach with elliptical distributions. J Classif 29(3):363–401
Ingrassia S, Minotti SC, Punzo A (2014) Model-based clustering via linear cluster-weighted models. Comput Stat Data Anal 71:159–182
Ingrassia S, Punzo A, Vittadini G, Minotti SC (2015) The generalized linear mixed cluster-weighted model. J Classif 32(1):85–113
Joe H (2005) Asymptotic efficiency of the two-stage estimation method for copula-based models. J Multivar Anal 94(2):401–419
Jorgensen M, Hunt LA (1996) Mixture model clustering of data sets with categorical and continuous variables. In: Dowe DL, Korb KB, Oliver JJ (eds) Proceedings of the Conference: Information, Statistics and Induction in Science, Melbourne, Australia, 20–23 August, vol 96. River Edge, New Jersey, pp 375–384
Karlis D, Xekalaki E (2003) Choosing initial values for the EM algorithm for finite mixtures. Computational Statistics & Data Analysis 41(3–4):577–590
Kocherlakota S, Kocherlakota K (1992) Bivariate discrete distributions, volume 132 of statistics: a series of textbooks and monographs. Taylor & Francis, Cambridge
Leisch F (2004) FlexMix: a general framework for finite mixture models and latent class regression in \({\sf R}\). J Stat Softw 11(8):1–18
Lichman M (2013) UCI Machine Learning Repository, University of California, School of Information and Computer Science. Irvine, CA. http://archive.ics.uci.edu/ml
Mazza A, Punzo A, Ingrassia S (2015) flexCWM: flexible cluster-weighted modeling. http://cran.r-project.org/web/packages/flexCWM/index.html
McCullagh P, Nelder J (1989) Generalized linear models, 2nd edn. Chapman & Hall, Boca Raton
McLachlan GJ, Peel D (2000) Finite mixture models. In: Applied probability and statistics: Wiley Series in Probability and Statistics. John Wiley & Sons, New York
McLachlan GJ, Basford KE (1988) Mixture models: inference and applications to clustering, volume 84 of statistics series. Marcel Dekker, New York
McQuarrie A, Shumway R, Tsai C-L (1997) The model selection criterion AICu. Stat Probab Lett 34(3):285–292
Nelsen RB (2007) An introduction to copulas. Springer Series in Statistics. Springer, New York
Punzo A (2014) Flexible mixture modeling with the polynomial Gaussian cluster-weighted model. Stat Modelling 14(3):257–291
Punzo A, Ingrassia S (2015) Parsimonious generalized linear Gaussian cluster-weighted models. In: Morlini I, Minerva T, Vichi M (eds) Advances in Statistical Models for Data Analysis, Studies in Classification, Data Analysis and Knowledge Organization, Switzerland. Springer International Publishing, Forthcoming
Punzo A, Ingrassia S (2013) On the use of the generalized linear exponential cluster-weighted model to asses local linear independence in bivariate data. QdS J Methodol Appl Stat 15:131–144
Punzo A, McNicholas PD (2014) Robust clustering in regression analysis via the contaminated Gaussian cluster-weighted model. arXiv.org e-print arXiv.org e-print arXiv:1409.6019 available at: arXiv:1409.6019
R Core Team (2013) R: a language and environment for statistical computing. R Foundation for Statistical Computing, Vienna
Schlattmann P (2009) Medical applications of finite mixture models. Statistics for biology and health. Springer, Berlin
Schwarz G (1978) Estimating the dimension of a model. Ann Stat 6(2):461–464
Sklar M (1959) Fonctions de répartition à n dimensions et leurs marges. Publications de l’Institut de Statistique de l’Université de Paris 8:229–231
Stephens M (2000) Dealing with label switching in mixture models. J R Stat Soc Series B Stat Methodol 62(4):795–809
Subedi S, Punzo A, Ingrassia S, McNicholas PD (2013) Clustering and classification via cluster-weighted factor analyzers. Adv Data Anal Classif 7(1):5–40
Subedi S, Punzo A, Ingrassia S, McNicholas PD (2015) Cluster-weighted \(t\)-factor analyzers for robust model-based clustering and dimension reduction. Stat Methods Appl 24 (in press)
Titterington DM, Smith AFM, Makov UE (1985) Statistical analysis of finite mixture distributions. John Wiley & Sons, New York
Tsanas A, Xifara A (2012) Accurate quantitative estimation of energy performance of residential buildings using statistical machine learning tools. Energy Build 49:560–567
Vermunt JK, Magidson J (2002) Latent class cluster analysis. In: Hagenaars JA, McCutcheon AL (eds) Applied latent class analysis. Cambridge University Press, Cambridge, pp 89–106
Wedel M, DeSarbo WS (1995) A mixture likelihood approach for generalized linear models. J Classif 12(1):21–55
Wedel M, Kamakura W (2000) Market segmentation: conceptual and methodological foundations, 2nd edn. Kluwer Academic Publishers, Boston
Yao W (2012) Model based labeling for mixture models. Stat Comput 22(2):337–347
Yao W, Wei Y, Yu C (2014) Robust mixture regression using the \(t\)-distribution. Comput Stat Data Anal 71:116–127
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The authors acknowledge the financial support from the Grant “Finite mixture and latent variable models for causal inference and analysis of socio-economic data” (FIRB 2012-Futuro in ricerca) funded by the Italian Government (RBFR12SHVV).
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Punzo, A., Ingrassia, S. Clustering bivariate mixed-type data via the cluster-weighted model. Comput Stat 31, 989–1013 (2016). https://doi.org/10.1007/s00180-015-0600-z
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DOI: https://doi.org/10.1007/s00180-015-0600-z