Eigenvalues and constraints in mixture modeling: geometric and computational issues

Advances in Data Analysis and Classification - Tập 12 - Trang 203-233 - 2017
Luis Angel García-Escudero1, Alfonso Gordaliza1, Francesca Greselin2, Salvatore Ingrassia3, Agustín Mayo-Iscar1
1Department of Statistics and Operations Research and IMUVA, University of Valladolid, Valladolid, Spain
2Department of Statistics and Quantitative Methods, Milano-Bicocca University, Milan, Italy
3Department of Economics and Business, University of Catania, Catania, Italy

Tóm tắt

This paper presents a review about the usage of eigenvalues restrictions for constrained parameter estimation in mixtures of elliptical distributions according to the likelihood approach. The restrictions serve a twofold purpose: to avoid convergence to degenerate solutions and to reduce the onset of non interesting (spurious) local maximizers, related to complex likelihood surfaces. The paper shows how the constraints may play a key role in the theory of Euclidean data clustering. The aim here is to provide a reasoned survey of the constraints and their applications, considering the contributions of many authors and spanning the literature of the last 30 years.

Tài liệu tham khảo

Andrews JL, McNicholas PD (2011) Extending mixtures of multivariate \(t\)-factor analyzers. Stat Comput 21:361–373 Andrews JL, McNicholas PD, Subedi S (2011) Model-based classification via mixtures of multivariate \(t\)-distributions. Comput Stat Data Anal 55:520–529 Banfield JD, Raftery AE (1993) Model-based Gaussian and non-Gaussian clustering. Biometrics 49:803–821 Baudry J-P, Celeux G (2015) EM for mixtures. Stat Comput 22(5):1021–1029 Biernacki C (2004) Initializing EM using the properties of its trajectories in gaussian mixtures. Stat Comput 14:267–279 Biernacki C, Chrétien S (2003) Degeneracy in the maximum likelihood estimation of univariate Gaussian mixtures with the EM. Stat Probab Lett 61:373–382 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 Boyles RA (1983) On the convergence of the EM algorithm. J R Stat Soc B 45:47–50 Celeux G, Govaert G (1995) Gaussian parsimonious clustering models. Pattern Recognit 28:781–793 Cerioli A, García-Escudero LA, Mayo-Iscar A, Riani M (2016) Finding the number of groups in model-based clustering via constrained likelihoods. http://uvadoc.uva.es/handle/10324/18093 Chanda KC (1954) A note on the consistency and maxima of the roots of likelihood equations. Biometrika 41:56–61 Ciuperca G, Ridolfi A, Idier J (2003) Penalized maximum likelihood estimator for normal mixtures. Scand J Stat 30:45–59 Cramér H (1946) Math Methods Stat. Princeton University Press, Princeton Day N (1969) Estimating the components of a mixture of normal distributions. Biometrika 56(3):463–474 Dempster A, Laird N, Rubin D (1977) Maximum likelihood from incomplete data via the EM algorithm. J R Stat Soc Ser B (Methodol) 39(1):1–38 Dennis JE (1981) Algorithms for non linear fitting. In: Proceedings of the NATO advanced research symposium, Cambridge, England. Cambridge University Dykstra RL (1983) An algorithm for restricted least squares regression. J Am Stat Assoc 78(384):837–842 Fang K, Anderson T (1990) Statistical inference in elliptically contoured and related distributions. Alberton, New York Fraley C, Raftery AE (2006) Mclust version 3: an R package for normal mixture modeling and model-based clustering. Technical report, DTIC Document Fraley C, Raftery A (2007) Bayesian regularization for normal mixture estimation and model-based clustering. J Classif 24(2):155–181 Fraley C, Raftery A, Murphy T, Scrucca L (2012) mclust version 4 for R: normal mixture modeling for model-based clustering, classification, and density estimation. University of Washington, Seattle Fritz H, García-Escudero LA, Mayo-Iscar A (2012) tclust: an R package for a trimming approach to cluster analysis. J Stat Softw 47(12):1–26 Fritz H, García-Escudero LA, Mayo-Iscar A (2013) A fast algorithm for robust constrained clustering. Comput Stat Data Anal 61:124–136 Gallegos MT (2002) Maximum likelihood clustering with outliers. In: Jajuga K, Sokolowski A, Bock H-H (eds) Classification, clustering, and data analysis: recent advances and applications. Springer, Berlin, pp 247–255 Gallegos M, Ritter G (2009) Trimmed ML estimation of contaminated mixture. Sankhya (Ser A) 71:164–220 García-Escudero LA, Gordaliza A, Matrán C, Mayo-Iscar A (2008) A general trimming approach to robust cluster analysis. Ann Stat 36(3):1324–1345 García-Escudero LA, Gordaliza A, Mayo-Iscar A (2014) A constrained robust proposal for mixture modeling avoiding spurious solutions. Adv Data Anal Classif 8(1):27–43 García-Escudero LA, Gordaliza A, Matrán C, Mayo-Iscar A (2015) Avoiding spurious local maximizers in mixture modelling. Stat Comput 25:619–633 García-Escudero LA, Gordaliza A, Greselin F, Ingrassia S, Mayo-Iscar A (2016) The joint role of trimming and constraints in robust estimation for mixtures of Gaussian factor analyzers. Comput Stat Data Anal 99:131–147 García-Escudero LA, Gordaliza A, Greselin F, Ingrassia S, Mayo-Iscar A (2017) Eigenvalues in robust approaches to mixture modeling: a review. Technical report, in preparation Ghahramani Z, Hinton G (1997) The EM algorithm for factor analyzers. Technical report CRG-TR-96-1, University of Toronto Greselin F, Ingrassia S (2010) Constrained monotone em algorithms for mixtures of multivariate \(t\)-distributions. Stat Comput 20:9–22 Greselin F, Ingrassia S (2015) Maximum likelihood estimation in constrained parameter spaces for mixtures of factor analyzers. Stat Comput 25(2):215–226 Greselin F, Ingrassia S, Punzo A (2011) Assessing the pattern of covariance matrices via an augmentation multiple testing procedure. Stat Methods Appl 20:141–170 Hathaway RJ (1985) A constrained formulation of maximum-likelihood estimation for normal mixture distributions. Ann Stat 13(2):795–800 Hathaway RJ (1996) A constrained EM algorithm for univariate normal mixtures. J Stat Comput Simul 23:211–230 Hennig C (2004) Breakdown points for maximum likelihood estimators of location-scale mixtures. Ann Stat 32:1313–1340 Ingrassia S (1992) A comparison between the simulated annealing and the EM algorithms in normal mixture decompositions. Stat Comput 2:203–211 Ingrassia S (2004) A likelihood-based constrained algorithm for multivariate normal mixture models. Stat Methods Appl 13(4):151–166 Ingrassia S, Rocci R (2007) Constrained monotone EM algorithms for finite mixture of multivariate gaussians. Comput Stat Data Anal 51(11):5339–5351 Ingrassia S, Rocci R (2011) Degeneracy of the EM algorithm for the MLE of multivariate Gaussian mixtures and dynamic constraints. Comput Stat Data Anal 55(4):1715–1725 Kiefer J, Wolfowitz J (1956) Consistency of the maximum likelihood estimator in the presence of infinitely many incidental parameters. Ann Math Stat 27(4):887–906 Kiefer NM (1978) Discrete parameter variation: efficient estimation of a switching regression model. Econometrica 46(2):427–434 Lindsay BG (1995) Mixture models: theory, geometry and applications. NSF-CBMS regional conference series in probability and statistics, vol 5. Institute of Mathematical Statistics, Hayward, CA McLachlan G, Krishnan T (2008a) The EM algorithm and extensions, 2nd edn, vol 589. Wiley, New York McLachlan GJ, Krishnan T (2008b) The EM algorithm and its extensions, 2nd edn. Wiley, New York McLachlan GJ, Peel D (2000) Finite mixture models. Wiley, New York McNicholas PD, Murphy TB (2008) Parsimonious Gaussian mixture models. Stat Comput 18:285–296 Meng X-L (1994) On the rate of convergence of the ECM algorithm. Ann Stat 22(1):326–339 Meng X-L, van Dyk D (1997) The EM algorithm. An old folk song sung to a fast new tune. J R Stat Soc B 59(3):511–567 Nettleton D (1999) Convergence properties of the EM algorithm in constrained spaces. Canad J Stat 27(3):639–644 O’Hagan A, White A (2016) Improved model-based clustering performance using bayes initialization averaging. Technical report, arXiv:1504.06870v4 O’Hagan A, Murphy TB, Gormley C (2013) Computational aspects of fitting mixture model via the expectation-maximisation algorithm. Comput Stat Data Anal 56(12):3843–3864 Puntanen S, Styan GP, Isotalo J (2011) Matrix tricks for linear statistical models. Springer, Berlin Redner RA, Walker HF (1984) Mixture densities maximum likelihood and the EM algorithm. SIAM Rev 26(2):195–239 Ritter G (2014) Cluster analysis and variable selection. CRC Press, Boca Raton Rocci R, Gattone SA, Di Mari R (2016) A data driven equivariant approach to constrained Gaussian mixture modeling. Adv Data Anal Classif. doi:10.1007/s11634-016-0279-1 Rousseeuw PJ, Leroy AM (2005) Robust regression and outlier detection. Wiley, New York Rudin W (1976) Principles of mathematical analysis, 3rd edn. McGraw-Hill, Tokyo Scrucca L, Fop M, Murphy TB, Raftery A (2016) mclust 5: clustering, classification and density estimation using Gaussian finite mixture models. R J 8(1):289–317 Seo B, Kim D (2012) Root selection in normal mixture models. Comput Stat Data Anal 56:2454–2470 Subedi S, Punzo A, Ingrassia S, McNicholas PD (2013) Clustering and classification via cluster-weighted factor analyzers. Adv Data Anal Classif 7:5–40 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(4):623–649 Tanaka K (2009) Strong consistency of the maximum likelihood estimator for finite mixtures of location-scale distributions when penalty is imposed on the ratios of the scale parameters. Scand J Stat 36(1):171–184 Tanaka K, Takemura A (2006) Strong consistency of the maximum likelihood estimator for finite mixtures of location-scale distributions when the scale parameters are exponentially small. Bernoulli 12(6):1003–1017 Tarone RD, Gruenhage G (1975) A note on the uniqueness of the roots of the likelihood equations for vector-valued parameters. J Am Stat Assoc 70(352):903–904 Tarone RD, Gruenhage G (1979) Corrigenda: a note on the uniqueness of the roots of the likelihood equations for vector-valued parameters. J Am Stat Assoc 74(367):744 Theobald C (1975) An inequality with application to multivariate analysis. Biometrika 62(2):461–466 Theobald C (1976) Corrections and amendments: an inequality with application to multivariate analysis. Biometrika 63(3):685 Tipping M, Bishop CM (1999) Mixtures of probabilistic principal component mixtures of probabilistic principal component analysers. Neural Comput 11(2):443–482 Titterington DM, Smith AFM, Makov UE (1985) Statistical analysis of finite mixture distributions. Wiley, New York van Laarhoven PJM, Aarts EHL (1988) Simulated annealing: theory and practice. D. Reidel, Dordecht Wu CFJ (1983) On convergence properties of the EM algorithm. Ann Stat 11:95–103