## Avoiding spurious local maximizers in mixture modeling.(English)Zbl 1331.62100

Summary: The maximum likelihood estimation in the finite mixture of distributions setting is an ill-posed problem that is treatable, in practice, through the EM algorithm. However, the existence of spurious solutions (singularities and non-interesting local maximizers) makes difficult to find sensible mixture fits for non-expert practitioners. In this work, a constrained mixture fitting approach is presented with the aim of overcoming the troubles introduced by spurious solutions. Sound mathematical support is provided and, which is more relevant in practice, a feasible algorithm is also given. This algorithm allows for monitoring solutions in terms of the constant involved in the restrictions, which yields a natural way to discard spurious solutions and a valuable tool for data analysts.

### MSC:

 62F10 Point estimation

TCLUST
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### References:

 [1] Anderson, E.: The irises of the Gaspe Peninsula. Bull. Am. Iris Soc. 59, 2-5 (1935) · Zbl 0704.62103 [2] Banfield, JD; Raftery, AE, Model-based gaussian and non-Gaussian clustering, Biometrics, 49, 803-821, (1993) · Zbl 0794.62034 [3] Celeux, G., Govaert, G.: Gaussian parsimonious clustering models. Pattern Recognition 28, 781-793 (1995) · Zbl 0891.62020 [4] Celeux, G., Diebolt, J.: The SEM algorithm: a probabilistic teacher algorithm derived from the EM algorithm for the mixture problem. Comput. Stat. Quater. 2, 73-82 (1985) [5] Chen, J; Tan, X, Inference for multivariate normal mixtures, J. Multivar. Anal., 100, 1367-1383, (2009) · Zbl 1162.62052 [6] Ciuperca, G; Ridolfi, A; Idier, J, Penalized maximum likelihood estimator for normal mixtures, Scand. J. Stat., 30, 45-59, (2003) · Zbl 1034.62018 [7] Coretto, P; Hennig, C, A simulations study to compare robust clustering methods based on mixtures, Adv. Data Anal. Classif., 4, 111-135, (2010) · Zbl 1284.62366 [8] Day, NE, Estimating the components of a mixture of two normal distributions, Biometrika, 56, 463-474, (1969) · Zbl 0183.48106 [9] Dempster, AP; Laird, NM; Rubin, DB, Maximum likelihood from incomplete data via the EM algorithm, J. R. Stat. Soc., 39, 1-38, (1977) · Zbl 0364.62022 [10] Dennis, JE; Powell, MJD (ed.), Algorithms for nonlinear Fitting, (1982), Cambridge · Zbl 0545.65007 [11] Dykstra, R.L.: An algorithm for restricted least squares regression. J. Am. Stat. Assoc. 78, 837-842 (1983) · Zbl 0535.62063 [12] Fisher, RA, The use of multiple measurements in taxonomic problems, Ann. Eugen., 7, 179-188, (1936) [13] Fraley, C; Raftery, AE, Bayesian regularization for normal mixture estimation and model-based clustering, J. Classif., 24, 155-181, (2007) · Zbl 1159.62302 [14] Fritz, H; García-Escudero, LA; Mayo-Iscar, A, A fast algorithm for robust constrained clustering, Comput. Stat. Data Anal., 61, 124-136, (2013) · Zbl 1349.62264 [15] Gallegos, MT; Ritter, G, Trimming algorithms for clustering contaminated grouped data and their robustness, Adv. Data Anal. Classif., 3, 135-167, (2009) · Zbl 1284.62372 [16] Gallegos, MT; Ritter, G, Trimmed ML estimation of contaminated mixtures, Sankhya, 71, 164-220, (2009) · Zbl 1193.62021 [17] García-Escudero, LA; Gordaliza, A; Matrán, C; Mayo-Iscar, A, A general trimming approach to robust cluster analysis, Ann. Stat., 36, 1324-1345, (2008) · Zbl 1360.62328 [18] Greselin, F; Ingrassia, S; Fink, A (ed.); Lausen, B (ed.); Seidel, W (ed.); Ultsch, A (ed.), Weakly homoscedastic constraints for mixtures of $$t$$-distributions, 219-228, (2010), Berlin [19] Hathaway, R.J.: Constrained maximum-likelihood estimation for a mixture of $$m$$ univariate normal distributions. Ph.D. Dissertation, Department of Mathematical Sciences, Rice University (1983) [20] Hathaway, R.J.: A constrained formulation of maximum likelihood estimation for normal mixture distributions. Ann. Stat. 13, 795-800 (1985) · Zbl 0576.62039 [21] Hathaway, RJ, A constrained EM algorithm for univariate normal mixtures, J. Stat. Comput. Simul., 23, 211-230, (1986) [22] Hennig, C, Breakdown points for maximum likelihood estimators of location-scale mixtures, Ann. Stat., 32, 1313-1340, (2004) · Zbl 1047.62063 [23] Ingrassia, S; Rocci, R, Constrained monotone EM algorithms for finite mixture of multivariate gaussians, Comput. Stat. Data Anal., 51, 5339-5351, (2007) · Zbl 1445.62116 [24] Maitra, R.: Initializing partition-optimization algorithms. IEEE/ACM Trans. Comput. Biol. Bioinf. 6, 144-157 (2009) · Zbl 1193.62021 [25] McLachlan, G; Peel, D, Contribution to the discussion of paper by S. Richardson and P.J. Green, J. R. Stat. Soc., 59, 772-773, (1997) [26] McLachlan, G., Peel, D.: Finite Mixture Models. Wiley, New York (2000) · Zbl 0963.62061 [27] Neykov, N., Filzmoser, P., Dimova, R., Neytchev, P.: Robust fitting of mixtures using the trimmed likelihood estimator. Comput. Stat. Data Anal. 17, 299-308 (2007) · Zbl 1328.62033 [28] Richardson, S; Green, PJ, On the Bayesian analysis of mixtures with an unknown number of components (with discussion), J. R. Stat. Soc., 59, 731-792, (1997) · Zbl 0891.62020 [29] Roeder, K, Density estimation with confidence sets exemplified by superclusters and voids in galaxies, J. Am. Stat. Assoc., 85, 617-624, (1990) · Zbl 0704.62103 [30] Titterington, D.M., Smith, A.F., Makov, U.E.: Statistical Analysis of Finite Mixture Distributions. Wiley, New York (1985) · Zbl 0646.62013 [31] Van der Vaart, A.W., Wellner, J.A.: Weak Convergence and Empirical Processes. Wiley, New York (1996) · Zbl 0862.60002
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