Mixtures of shifted asymmetric laplace distributions

IEEE Transactions on Pattern Analysis and Machine Intelligence

Volumn 36, Issue 6, 2014, Pages 1149-1157

  Franczak, Brian C ^a   Browne, Ryan P ^a   McNicholas, Paul D ^a  

^a UNIVERSITY OF GUELPH   (Canada)

Author keywords

multivariate statistics; Statistical computing

Indexed keywords

ALGORITHMS; LAPLACE TRANSFORMS;

ASYMMETRIC LAPLACE DISTRIBUTION; EM ALGORITHMS; GENERALIZED INVERSE GAUSSIAN DISTRIBUTIONS; MIXTURE-MODELLING; MODEL-BASED CLUSTERING; MULTIVARIATE STATISTICS; NON-GAUSSIAN; STATISTICAL COMPUTING;

MIXTURES;

EID: 84901837489     PISSN: 01628828     EISSN: None     Source Type: Journal    
DOI: 10.1109/TPAMI.2013.216     Document Type: Article

References (63)

1. A.C. Aitken, "On Bernoulli's Numerical Solution of Algebraic Equations," Proc. Royal Soc. of Edinburgh, vol. 46, pp. 289-305, 1926.
2. M.M. Ali, J. Woo, and S. Nadarajah, "Some Skew Symmetric Inverse Reflected Distributions," Brazilian J. Probability and Statistics, vol. 24, no. 1, pp. 1-23, 2010.
3. J.L. Andrews and P.D. McNicholas, "Mixtures of Modified t-Factor Analyzers for Model-Based Clustering, Classification, and Discriminant Analysis," J. Statistical Planning and Inference, vol. 141, no. 4, pp. 1479-1486, 2011.
4. J. Baek and G.J. McLachlan, "Mixtures of Common t-Factor Analyzers for Clustering High-Dimensional Microarray Data," Bioinformatics, vol. 27, pp. 1269-1276, 2011.
5. J. Baek, G.J. McLachlan, and L.K. Flack, "Mixtures of Factor Analyzers with Common Factor Loadings: Applications to the Clustering and Visualization of High-Dimensional Data," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 32, no. 7, pp. 1298- 1309, July 2010.
6. J.D. Banfield and A.E. Raftery, "Model-Based Gaussian and Non-Gaussian Clustering," Biometrics, vol. 49, no. 3, pp. 803-821, 1993. (Pubitemid 23298358)
7. O. Barndorff-Nielsen and C. Halgreen, "Infinite Divisibility of the Hyperbolic and Generalized Inverse Gaussian Distributions," Probability Theory and Related Fields, vol. 38, no. 4, pp. 309-311, 1977.
8. O.E. Barndorff-Nielsen, "Normal Inverse Gaussian Distributions and Stochastic Volatility Modelling," Scandinavian J. Statistics, vol. 24, no. 1, pp. 1-13, 1997. (Pubitemid 127362002)
9. J.-P. Baudry, A.E. Raftery, G. Celeux, K. Lo, and R. Gottardo, "Combining Mixture Components for Clustering," J. Computational and Graphical Statistics, vol. 19, pp. 332-353, 2010.
10. D. Bhowmick, A.C. Davison, D.R. Goldstein, and Y. Ruffieux, "A Laplace Mixture Model for Identification of Differential Expression in Microarray Experiments," Biostatistics, vol. 7, no. 4, pp. 630-641, 2006. (Pubitemid 44545541)
11. C. Biernacki, G. Celeux, and G. Govaert, "Assessing a Mixture Model for Clustering with the Integrated Completed Likelihood," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 22, no. 7, pp. 719-725, July 2000.
12. P. Blæsild, "The Shape of the Generalized Inverse Gaussian and Hyperbolic Distributions," Research Report 37, Dept. of Theoretical Statistics, Aarhus Univ., 1978.
13. D. Böhning, E. Dietz, R. Schaub, P. Schlattmann, and B. Lindsay, "The Distribution of the Likelihood Ratio for Mixtures of Densities from the One-Parameter Exponential Family," Annals of the Inst. of Statistical Math., vol. 46, pp. 373-388, 1994.
14. D. Bolin, "Spatial Matern Fields Driven by Non-Gaussian Noise," arXiv:1206.0622, 2011.
15. R.P. Browne, P.D. McNicholas, and M.D. Sparling, "Model-Based Learning Using a Mixture of Mixtures of Gaussian and Uniform Distributions," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 34, no. 4, pp. 814-817, Apr. 2012.
16. G. Celeux and G. Govaert, "Gaussian Parsimonious Clustering Models," Pattern Recognition, vol. 28, pp. 781-793, 1995.
17. A. Cord, C. Ambroise, and J.-P. Cocquerez, "Feature Selection in Robust Clustering Based on Laplace Mixture," Pattern Recognition Letters, vol. 27, no. 6, pp. 627-635, 2006. (Pubitemid 43254606)
18. A.P. Dempster, N.M. Laird, and D.B. Rubin, "Maximum Likelihood from Incomplete Data via the EM Algorithm," J. Royal Statistical Soc.: Series B, vol. 39, no. 1, pp. 1-38, 1977.
19. T. Eltoft, T. Kim, and T. Lee, "On the Multivariate Laplace Distribution," IEEE Signal Processing Letters, vol. 13, no. 5, pp. 300-303, May 2006.
20. C. Fraley and A.E. Raftery, "Model-Based Clustering, Discriminant Analysis, and Density Estimation," J. Am. Statistical Assoc., vol. 97, no. 458, pp. 611-631, 2002.
21. M.T. Gallegos and G. Ritter, "A Robust Method for Cluster Analysis," Annals of Statistics, vol. 33, pp. 347-380, 2005. (Pubitemid 40653793)
22. Z. Ghahramani and G.E. Hinton, "The EM Algorithm for Factor Analyzers," Technical Report CRG-TR-96-1, Univ. of Toronto, 1997.
23. C. Halgreen, "Self-Decomposibility of the Generalized Inverse Gaussian and Hyperbolic Distributions," Probability Theory and Related Fields, vol. 47, no. 1, pp. 13-18, 1979.
24. T. Hastie and R. Tibshirani, "Discriminant Analysis by Gaussian Mixtures," J. Royal Statistical Soc.: Series B, vol. 58, no. 1, pp. 155- 176, 1996.
25. C. Hennig, "Methods for Merging Gaussian Mixture Components," Advances in Data Analysis and Classification, vol. 4, pp. 3-34, 2010.
26. L. Hubert and P. Arabie, "Comparing Partitions," J. Classification, vol. 2, pp. 193-218, 1985.
27. B. Jørgensen, Statistical Properties of the Generalized Inverse Gaussian Distribution. Springer-Verlag, 1982.
28. D. Karlis and L. Meligkotsidou, "Finite Mixtures of Multivariate Poisson Distributions with Application," J. Statistical Planning and Inference, vol. 137, no. 6, pp. 1942-1960, 2007.
29. D. Karlis and A. Santourian, "Model-Based Clustering with Non- Elliptically Contoured Distributions," Statistics and Computing, vol. 19, no. 1, pp. 73-83, 2009.
30. S. Kotz, T.J. Kozubowski, and K. Podgorski, The Laplace Distribution and Generalizations: A Revisit with Applications to Communications, Economics, Eng., and Finance. first ed., Burkhäuser, 2001.
31. S. Lee and G.J. McLachlan, "On the Fitting of Mixtures of Multivariate Skew t-Distributions via the EM Algorithm," arXiv:1109. 4706, 2011.
32. T.-I. Lin, "Maximum Likelihood Estimation for Multivariate Skew Normal Mixture Models," J. Multivariate Analysis, vol. 100, pp. 257-265, 2009.
33. T.-I. Lin, "Robust Mixture Modeling Using Multivariate Skew t- Distributions," Statistics and Computing, vol. 20, pp. 343-356, 2010.
34. B.G. Lindsay, Mixture Models: Theory, Geometry and Applications, vol. 5, NSF-CBMS Regional Conf. Series in Probability and Statistics, Inst. of Math. Statistics, 1995.
35. C. Maugis, G. Celeux, and M.-L. Martin-Magniette, "Variable Selection for Clustering with Gaussian Mixture Models," Biometrics, vol. 65, no. 3, pp. 701-709, 2009.
36. G.J. McLachlan, Discriminant Analysis and Statistical Pattern Recognition. John Wiley & Sons, 1992.
37. G.J. McLachlan, R.W. Bean, and L. Ben-Tovim Jones, "Extension of the Mixture of Factor Analyzers Model to Incorporate the Multivariate t-Distribution," Computational Statistics and Data Analysis, vol. 51, no. 11, pp. 5327-5338, 2007. (Pubitemid 46694022)
38. G.J. McLachlan and D. Peel, "Robust Cluster Analysis via Mixtures of Multivariate t-Distributions," Advances in Pattern Recognition, pp. 658-666, Springer-Verlag, 1998. (Pubitemid 128117168)
39. G.J. McLachlan and D. Peel, "Mixtures of Factor Analyzers," Proc. Seventh Int'l Conf. Machine Learning, pp. 599-606, 2000.
40. P.D. McNicholas, "Model-Based Classification Using Latent Gaussian Mixture Models," J. Statistical Planning and Inference, vol. 140, no. 5, pp. 1175-1181, 2010.
41. P.D. McNicholas and R.P. Browne, "Discussion of 'How to Find an Appropriate Clustering for Mixed-Type Variables with Application to Socio-Economic Stratification'," J. Royal Statistical Soc.: Series C, vol. 62, no. 3, pp. 352-353, 2013.
42. P.D. McNicholas and T.B. Murphy, "Parsimonious Gaussian Mixture Models," Statistics and Computing, vol. 18, pp. 285-296, 2008.
43. P.D. McNicholas and T.B. Murphy, "Model-Based Clustering of Microarray Expression Data via Latent Gaussian Mixture Models," Bioinformatics, vol. 26, no. 21, pp. 2705-2712, 2010.
44. P.D. McNicholas, T.B. Murphy, A.F. McDaid, and D. Frost, "Serial and Parallel Implementations of Model-Based Clustering via Parsimonious Gaussian Mixture Models," Computational Statistics and Data Analysis, vol. 54, no. 3, pp. 711-723, 2010.
45. P.D. McNicholas and S. Subedi, "Clustering Gene Expression Time Course Data Using Mixtures of Multivariate t- Distributions," J. Statistical Planning and Inference, vol. 142, no. 5, pp. 1114-1127, 2012.
46. N. Mitianoudis and T. Stathaki, "Overcomplete Source Separation Using Laplacian Mixture Models," IEEE Signal Processing Letters, vol. 12, no. 4, pp. 277-280, Apr. 2005. (Pubitemid 40555672)
47. K. Nakai and M. Kanehisa, "Expert System for Predicting Protein Localization Sites in Gram-Negative Bacteria," Protiens, vol. 11, no. 2, pp. 95-110, 1991.
48. K. Nakai and M. Kanehisa, "A Knowledge Base for Predicting Protein Localization Sites in Eukaryotic Cells," Genomics, vol. 14, no. 4, pp. 897-911, 1992. (Pubitemid 23053027)
49. D. Peel and G.J. McLachlan, "Robust Mixture Modelling Using the t-Distribution," Statistics and Computing, vol. 10, no. 4, pp. 339-348, 2000.
50. A. Punzo and P.D. McNicholas, "Outlier Detection via Parsimonious Mixtures of Contaminated Gaussian Distributions," arXiv: 1305.4669, 2013.
51. S. Pyne, X. Hu, K. Wang, E. Rossin, T.-I. Lin, L.M. Maier, C. Baecher-Allan, G.J. McLachlan, P. Tamayo, D.A. Hafler, P.L. De Jager, and J.P. Mesirow, "Automated High-Dimensional Flow Cytometric Data Analysis," Proc. Nat'l Academy of Sciences, vol. 106, pp. 8519-8524, 2009.
52. R Core Team, R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, 2013.
53. A.E. Raftery and N. Dean, "Variable Selection for Model-Based Clustering," J. Am. Statistical Assoc., vol. 101, no. 473, pp. 168-178, 2006. (Pubitemid 43500033)
54. W.M. Rand, "Objective Criteria for the Evaluation of Clustering Methods," J. Am. Statistical Assoc., vol. 66, pp. 846-850, 1971.
55. S.K. Sahu, D.K. Dey, and M.D. Branco, "A New Class of Multivariate Skew Distributions with Application to Bayesian Regression Models," Canadian J. Statistics, vol. 31, pp. 129-150, 2003. (Pubitemid 37395671)
56. A.J. Scallan, "Maximum Likelihood Estimation for a Normal/Laplace Mixture Distribution," J. Royal Statistical Soc.: Series D, vol. 41, no. 2, pp. 227-231, 1992.
57. G. Schwarz, "Estimating the Dimension of a Model," The Annals of Statistics, vol. 6, no. 2, pp. 461-464, 1978.
58. L. Scrucca, "Dimension Reduction for Model-Based Clustering," Statistics and Computing, vol. 20, no. 4, pp. 471-484, 2010.
59. F. Shi and I.W. Selesnick, "Multivariate Quasi-Laplacian Mixture Models Forwavelet-Based Image Denoising," Proc. IEEE Int'l Conf. Image Processing, pp. 2625-2628, Oct. 2006.
60. T.E. Tipping and C.M. Bishop, "Mixtures of Probabilistic Principal Component Analysers," Neural Computation, vol. 11, no. 2, pp. 443-482, 1999.
61. D.M. Titterington, A.F.M. Smith, and U.E. Makov, Statistical Analysis of Finite Mixture Distributions. John Wiley & Sons, 1985.
62. I. Vrbik and P.D. McNicholas, "Analytic Calculations for the EM Algorithm for Multivariate Skew-t Mixture Models," Statistics and Probability Letters, vol. 82, no. 6, pp. 1169-1174, 2012.
63. H. Zhou and K.L. Lange, "On the Bumpy Road to the Dominant Mode," Scandinavian J. Statistics, vol. 37, no. 4, pp. 612-631, 2010.