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Y. Fang, B. C. Franczak, and S. Subedi. Tackling the infinite likelihood problem when fitting mixtures of shifted asymmetric Laplace distributions, Mar. 2023. URL http://arxiv.org/abs/2303.14211.

M. A. T. Figueiredo and A. K. Jain. Unsupervised learning of finite mixture models. *IEEE Transactions on pattern analysis and machine intelligence*, 24(3):381–396, 2002. URL https://ieeexplore.ieee.org/document/990138.

M. Fop, T. B. Murphy, and L. Scrucca. Model-based Clustering with Sparse Covariance Matrices. *Statistics and Computing*, 29(4):791–819, 2019. URL https://doi.org/10.1007/s11222-018-9838-y.

A. Foss, M. Markatou, B. Ray, and A. Heching. A semiparametric method for clustering mixed data. *Machine Learning*, 105(3):419–458, Dec. 2016. URL http://link.springer.com/10.1007/s10994-016-5575-7.

A. H. Foss and M. Markatou. Kamila: Clustering Mixed-Type Data in R and Hadoop. *Journal of Statistical Software*, 83:1–44, Feb. 2018. URL https://doi.org/10.18637/jss.v083.i13.

A. H. Foss, M. Markatou, and B. Ray. Distance Metrics and Clustering Methods for Mixed-type Data. *International Statistical Review*, 87(1):80–109, Apr. 2019. URL https://onlinelibrary.wiley.com/doi/10.1111/insr.12274.

C. Fraley and A. E. Raftery. Model-Based Clustering, Discriminant Analysis, and Density Estimation. *Journal of the American Statistical Association*, 97(458):611–631, 2002. URL https://www.jstor.org/stable/3085676.

C. Fraley and A. E. Raftery. Bayesian Regularization for Normal Mixture Estimation and Model-Based Clustering. *Journal of Classification*, 24(2):155–181, 2007. doi: 10.1007/s00357-007-0004-5. URL https://link.springer.com/article/10.1007/s00357-007-0004-5.

B. C. Franczak, R. P. Browne, and P. D. McNicholas. Mixtures of Shifted Asymmetric Laplace Distributions. *IEEE Transactions on Pattern Analysis and Machine Intelligence*, 36(6):1149–1157, June 2014. URL https://ieeexplore.ieee.org/document/6654117.

B. C. Franczak, C. Tortora, R. P. Browne, and P. D. McNicholas. Unsupervised learning via mixtures of skewed distributions with hypercube contours. *Pattern Recognition Letters*, 58:69–76, June 2015. URL https://linkinghub.elsevier.com/retrieve/pii/S0167865515000598.

Z. Ghahramani and G. E. Hinton. The EM Algorithm for Mixtures of Factor Analyzers. Technical Report CRG-TR-96-1, University of Toronto, Toronto, Canada, May 1996. URL https://citeseerx.ist.psu.edu/document? resid=rep1&type=pdf&doi=766f4465747394d304d162197e091f1ae8f7f577.

L. A. Goodman. Exploratory Latent Structure Analysis Using Both Identifiable and Unidentifiable Models. *Biometrika*, 61(2):215–231, 1974. URL https://www.jstor.org/stable/2334349.

H. Holzmann, A. Munk, and T. Gneiting. Identifiability of Finite Mixtures of Elliptical Distributions. *Scandinavian Journal of Statistics*, 33(4):753–763, Dec. 2006. URL https://onlinelibrary.wiley.com/doi/10.1111/j. 1467-9469.2006.00505.x.

Z. Huang. Clustering Large Data Sets With Mixed Numeric And Categorical Values. In *Proceedings of the First Pacific Asia Knowledge Discovery and Data Mining Conference*, pages 21–34, Singapore, 1997a. World Scientific. ISBN 978-981-02-3072-2. URL https://citeseerx.ist.psu.edu/document? resid=rep1&type=pdf&doi= d42bb5ad2d03be6d8fefa63d25d02c0711d19728.

Z. Huang. A Fast Clustering Algorithm to Cluster Very Large Categorical Data Sets in Data Mining. In *Proceedings of the SIGMOD Workshop on Research Issues on Data Mining and Knowledge Discovery*, The University of British Columbia, 1997b.

Z. Huang. Extensions to the k-Means Algorithm for Clustering Large Data Sets with Categorical Values. *Data Mining and Knowledge Discovery*, 2(3):283–304, 1998. URL http://link.springer.com/10.1023/A: 1009769707641.
E. Derman and E. L. Pennec. Clustering and Model Selection via Penalized Likelihood for Different-sized Categorical Data Vectors, 2017. URL https://arxiv.org/abs/1709.02294.

Y. Fang, B. C. Franczak, and S. Subedi. Tackling the infinite likelihood problem when fitting mixtures of shifted asymmetric Laplace distributions, Mar. 2023. URL http://arxiv.org/abs/2303.14211.

M. A. T. Figueiredo and A. K. Jain. Unsupervised learning of finite mixture models. *IEEE Transactions on pattern analysis and machine intelligence*, 24(3):381–396, 2002. URL https://ieeexplore.ieee.org/document/990138.

M. Fop, T. B. Murphy, and L. Scrucca. Model-based Clustering with Sparse Covariance Matrices. *Statistics and Computing*, 29(4):791–819, 2019. URL https://doi.org/10.1007/s11222-018-9838-y.

A. Foss, M. Markatou, B. Ray, and A. Heching. A semiparametric method for clustering mixed data. *Machine Learning*, 105(3):419–458, Dec. 2016. URL http://link.springer.com/10.1007/s10994-016-5575-7.

A. H. Foss and M. Markatou. Kamila: Clustering Mixed-Type Data in R and Hadoop. *Journal of Statistical Software*, 83:1–44, Feb. 2018. URL https://doi.org/10.18637/jss.v083.i13.

A. H. Foss, M. Markatou, and B. Ray. Distance Metrics and Clustering Methods for Mixed-type Data. *International Statistical Review*, 87(1):80–109, Apr. 2019. URL https://onlinelibrary.wiley.com/doi/10.1111/insr.12274.

C. Fraley and A. E. Raftery. Model-Based Clustering, Discriminant Analysis, and Density Estimation. *Journal of the American Statistical Association*, 97(458):611–631, 2002. URL https://www.jstor.org/stable/3085676.

C. Fraley and A. E. Raftery. Bayesian Regularization for Normal Mixture Estimation and Model-Based Clustering. *Journal of Classification*, 24(2):155–181, 2007. doi: 10.1007/s00357-007-0004-5. URL https://link.springer.com/article/10.1007/s00357-007-0004-5.

B. C. Franczak, R. P. Browne, and P. D. McNicholas. Mixtures of Shifted Asymmetric Laplace Distributions. *IEEE Transactions on Pattern Analysis and Machine Intelligence*, 36(6):1149–1157, June 2014. URL https://ieeexplore.ieee.org/document/6654117.

B. C. Franczak, C. Tortora, R. P. Browne, and P. D. McNicholas. Unsupervised learning via mixtures of skewed distributions with hypercube contours. *Pattern Recognition Letters*, 58:69–76, June 2015. URL https://linkinghub.elsevier.com/retrieve/pii/S0167865515000598.

Z. Ghahramani and G. E. Hinton. The EM Algorithm for Mixtures of Factor Analyzers. Technical Report CRG-TR-96-1, University of Toronto, Toronto, Canada, May 1996. URL https://citeseerx.ist.psu.edu/document? resid=rep1&type=pdf&doi=766f4465747394d304d162197e091f1ae8f7f577.

L. A. Goodman. Exploratory Latent Structure Analysis Using Both Identifiable and Unidentifiable Models. *Biometrika*, 61(2):215–231, 1974. URL https://www.jstor.org/stable/2334349.

H. Holzmann, A. Munk, and T. Gneiting. Identifiability of Finite Mixtures of Elliptical Distributions. *Scandinavian Journal of Statistics*, 33(4):753–763, Dec. 2006. URL https://onlinelibrary.wiley.com/doi/10.1111/j. 1467-9469.2006.00505.x.

Z. Huang. Clustering Large Data Sets With Mixed Numeric And Categorical Values. In *Proceedings of the First Pacific Asia Knowledge Discovery and Data Mining Conference*, pages 21–34, Singapore, 1997a. World Scientific. ISBN 978-981-02-3072-2. URL https://citeseerx.ist.psu.edu/document? resid=rep1&type=pdf&doi= d42bb5ad2d03be6d8fefa63d25d02c0711d19728.

Z. Huang. A Fast Clustering Algorithm to Cluster Very Large Categorical Data Sets in Data Mining. In *Proceedings of the SIGMOD Workshop on Research Issues on Data Mining and Knowledge Discovery*, The University of British Columbia, 1997b.

Z. Huang. Extensions to the k-Means Algorithm for Clustering Large Data Sets with Categorical Values. *Data Mining and Knowledge Discovery*, 2(3):283–304, 1998. URL http://link.springer.com/10.1023/A: 1009769707641.
L. A. Hunt and M. A. Jorgensen. Mixture Model Clustering of Data Sets with Categorical and Continuous Variables. In *Information, Statistics and Induction in Science*, volume 96, pages 375-284, July 1996. ISBN 978-981-4547-26-0.

M. C. Jones and M. J. Faddy. A skew extension of the t-distribution, with applications. *Journal of the Royal Statistical Society: Series B (Statistical Methodology)*, 65(1):159-174, 2003. URL https://onlinelibrary.wiley.com/doi/abs/10.1111/1467-9868.00378.

L. Kaufman and P. J. Rousseeuw, editors. *Finding Groups in Data*. Wiley Series in Probability and Statistics. John Wiley & Sons, Inc., Hoboken, NJ, USA, Mar. 1990. ISBN 978-0-470-31680-1 978-0-471-87876-6. URL http://doi.wiley.com/10.1002/9780470316801.

I. Kosmidis and D. Karlis. Model-based clustering using copulas with applications. *Statistics and Computing*, 26 (5):1079-1099, Sept. 2016. URL http://arxiv.org/abs/1404.4077.

S. Kotz, T. J. Kozubowski, and K. Podgórski. *The Laplace Distribution and Generalizations*. Birkhäuser Boston, Boston, MA, 2001. ISBN 978-1-4612-6646-4 978-1-4612-0173-1. URL http://link.springer.com/10.1007/978-1-4612-0173-1.

K. L. Lange, R. J. A. Little, and J. M. G. Taylor. Robust Statistical Modeling Using the t Distribution. *Journal of the American Statistical Association*, 84(408):881-896, 1989. URL https://www.jstor.org/stable/2290063.

T. Lartigue, S. Durrleman, and S. Allassonnière. Deterministic Approximate EM Algorithm; Application to the Riemann Approximation EM and the Tempered EM. *Algorithms*, 15(3):78, Feb. 2022. URL https://www.mdpi.com/1999-4893/15/3/78.

M. H. Law, M. A. Figueiredo, and A. K. Jain. Simultaneous feature selection and clustering using mixture models. *IEEE transactions on pattern analysis and machine intelligence*, 26(9):1154-1166, 2004. URL https://ieeexplore.ieee.org/abstract/document/1316850.

K. J. Lee and R.-B. Chen. Bayesian variable selection in a finite mixture of linear mixed-effects models. *Journal of Statistical Computation and Simulation*, 89(13):2434-2453, May 2019. URL https://doi.org/10.1080/00949655.2019.1620746.

S. Lee and G. J. McLachlan. Finite mixtures of multivariate skew t-distributions: Some recent and new results. *Statistics and Computing*, 24(2):181-202, Mar. 2014. URL http://link.springer.com/10.1007/s11222-012-9362-4.

S. X. Lee and G. J. McLachlan. On the fitting of mixtures of multivariate skew t-distributions via the EM algorithm, Sept. 2012. URL http://arxiv.org/abs/1109.4706.

T. I. Lin. Maximum likelihood estimation for multivariate skew normal mixture models. *Journal of Multivariate Analysis*, 100(2):257-265, Feb. 2009. URL https://www.sciencedirect.com/science/article/pii/S0047259X08001152.

T.-I. Lin. Robust mixture modeling using multivariate skew t distributions. *Statistics and Computing*, 20(3): 343-356, July 2010. URL http://link.springer.com/10.1007/s11222-009-9128-9.

T. I. Lin, J. C. Lee, and S. Y. Yen. Finite Mixture Modelling Using the Skew Normal Distribution. *Statistica Sinica*, 17(3):909-927, 2007. URL https://www.jstor.org/stable/24307705.

B. G. Lindsay. *Mixture Models: Theory, Geometry and Applications*, volume 5 of *NSF-CBMS Regional Conference Series in Probability and Statistics*. Institute of Mathematical Statistics and American Statistical Association, 1995. ISBN 0-940600-32-3. URL http://www.jstor.org/stable/4153184.

C. Liu and D. B. Rubin. ML Estimation of the t distribution using EM and its extensions, ECM and ECME. *Statistica Sinica*, 5(1):21, 1995. URL http://www.jstor.org/stable/24305551.

M. Marbac, C. Biernacki, and V. Vandewalle. Model-based clustering of Gaussian copulas for mixed data. *Communications in Statistics - Theory and Methods*, 46(23):11635-11656, Dec. 2017. URL https://www.tandfonline.com/doi/full/10.1080/03610926.2016.1277753.
L. A. Hunt and M. A. Jorgensen. Mixture Model Clustering of Data Sets with Categorical and Continuous Variables. In *Information, Statistics and Induction in Science*, volume 96, pages 375-284, July 1996. ISBN 978-981-4547-26-0.

M. C. Jones and M. J. Faddy. A skew extension of the t-distribution, with applications. *Journal of the Royal Statistical Society: Series B (Statistical Methodology)*, 65(1):159-174, 2003. URL https://onlinelibrary.wiley.com/doi/abs/10.1111/1467-9868.00378.

L. Kaufman and P. J. Rousseeuw, editors. *Finding Groups in Data*. Wiley Series in Probability and Statistics. John Wiley & Sons, Inc., Hoboken, NJ, USA, Mar. 1990. ISBN 978-0-470-31680-1 978-0-471-87876-6. URL http://doi.wiley.com/10.1002/9780470316801.

I. Kosmidis and D. Karlis. Model-based clustering using copulas with applications. *Statistics and Computing*, 26 (5):1079-1099, Sept. 2016. URL http://arxiv.org/abs/1404.4077.

S. Kotz, T. J. Kozubowski, and K. Podgórski. *The Laplace Distribution and Generalizations*. Birkhäuser Boston, Boston, MA, 2001. ISBN 978-1-4612-6646-4 978-1-4612-0173-1. URL http://link.springer.com/10.1007/978-1-4612-0173-1.

K. L. Lange, R. J. A. Little, and J. M. G. Taylor. Robust Statistical Modeling Using the t Distribution. *Journal of the American Statistical Association*, 84(408):881-896, 1989. URL https://www.jstor.org/stable/2290063.

T. Lartigue, S. Durrleman, and S. Allassonnière. Deterministic Approximate EM Algorithm; Application to the Riemann Approximation EM and the Tempered EM. *Algorithms*, 15(3):78, Feb. 2022. URL https://www.mdpi.com/1999-4893/15/3/78.

M. H. Law, M. A. Figueiredo, and A. K. Jain. Simultaneous feature selection and clustering using mixture models. *IEEE transactions on pattern analysis and machine intelligence*, 26(9):1154-1166, 2004. URL https://ieeexplore.ieee.org/abstract/document/1316850.

K. J. Lee and R.-B. Chen. Bayesian variable selection in a finite mixture of linear mixed-effects models. *Journal of Statistical Computation and Simulation*, 89(13):2434-2453, May 2019. URL https://doi.org/10.1080/00949655.2019.1620746.

S. Lee and G. J. McLachlan. Finite mixtures of multivariate skew t-distributions: Some recent and new results. *Statistics and Computing*, 24(2):181-202, Mar. 2014. URL http://link.springer.com/10.1007/s11222-012-9362-4.

S. X. Lee and G. J. McLachlan. On the fitting of mixtures of multivariate skew t-distributions via the EM algorithm, Sept. 2012. URL http://arxiv.org/abs/1109.4706.

T. I. Lin. Maximum likelihood estimation for multivariate skew normal mixture models. *Journal of Multivariate Analysis*, 100(2):257-265, Feb. 2009. URL https://www.sciencedirect.com/science/article/pii/S0047259X08001152.

T.-I. Lin. Robust mixture modeling using multivariate skew t distributions. *Statistics and Computing*, 20(3): 343-356, July 2010. URL http://link.springer.com/10.1007/s11222-009-9128-9.

T. I. Lin, J. C. Lee, and S. Y. Yen. Finite Mixture Modelling Using the Skew Normal Distribution. *Statistica Sinica*, 17(3):909-927, 2007. URL https://www.jstor.org/stable/24307705.

B. G. Lindsay. *Mixture Models: Theory, Geometry and Applications*, volume 5 of *NSF-CBMS Regional Conference Series in Probability and Statistics*. Institute of Mathematical Statistics and American Statistical Association, 1995. ISBN 0-940600-32-3. URL http://www.jstor.org/stable/4153184.

C. Liu and D. B. Rubin. ML Estimation of the t distribution using EM and its extensions, ECM and ECME. *Statistica Sinica*, 5(1):21, 1995. URL http://www.jstor.org/stable/24305551.

M. Marbac, C. Biernacki, and V. Vandewalle. Model-based clustering of Gaussian copulas for mixed data. *Communications in Statistics - Theory and Methods*, 46(23):11635-11656, Dec. 2017. URL https://www.tandfonline.com/doi/full/10.1080/03610926.2016.1277753.
F. Mbuga and C. Tortora. Spectral Clustering of Mixed-Type Data. *Stats*, 5(1):1–11, Dec. 2021. URL https://www.mdpi.com/2571-905X/5/1/1.

B. McCane and M. Albert. Distance functions for categorical and mixed variables. *Pattern Recognition Letters*, 29 (7):986–993, May 2008. URL https://linkinghub.elsevier.com/retrieve/pii/S0167865508000524.

G. McLachlan, D. Peel, and R. Bean. Modelling high-dimensional data by mixtures of factor analyzers. *Computational Statistics & Data Analysis*, 41(3-4):379–388, Jan. 2003. URL https://linkinghub.elsevier.com/retrieve/pii/S0167947302001834.

G. J. McLachlan and T. Krishnan. *The EM Algorithm and Extensions*. Wiley Series in Probability and Statistics. Wiley, Hoboken, 2nd edition, 2008. ISBN 978-0-471-20170-0.

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G. J. McLachlan and D. Peel. *Finite Mixture Models*. Wiley Series in Probability and Statistics Applied Probability and Statistics Section. Wiley, New York, 2000. ISBN 978-0-471-00626-8.

P. McNicholas, T. Murphy, A. McDaid, and D. Frost. Serial and parallel implementations of model-based clustering via parsimonious Gaussian mixture models. *Computational Statistics & Data Analysis*, 54(3):711–723, Mar. 2010. URL https://linkinghub.elsevier.com/retrieve/pii/S0167947309000632.

D. McParland and I. C. Gormley. Model based clustering for mixed data: clustMD. *Advances in Data Analysis and Classification*, 10(2):155–169, June 2016. URL http://link.springer.com/10.1007/s11634-016-0238-x.

D. McParland, I. C. Gormley, T. H. McCormick, S. J. Clark, C. W. Kabudula, and M. A. Collinson. Clustering South African Households Based on Their Asset Status Using Latent Variable Models. *The Annals of Applied Statistics*, 8(2):747–776, 2014. URL https://www.jstor.org/stable/24522075.

D. McParland, C. M. Phillips, L. Brennan, H. M. Roche, and I. C. Gormley. Clustering high-dimensional mixed data to uncover sub-phenotypes: Joint analysis of phenotypic and genotypic data. *Statistics in Medicine*, 36(28): 4548–4569, Dec. 2017. URL https://onlinelibrary.wiley.com/doi/10.1002/sim.7371.

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J. S. Murray, D. B. Dunson, L. Carin, and J. E. Lucas. Bayesian Gaussian Copula Factor Models for Mixed Data. *Journal of the American Statistical Association*, 108(502):656–665, June 2013. URL https://doi.org/10.1080/01621459.2012.762328.

D. Peel and G. J. McLachlan. Robust mixture modelling using the t distribution. *Statistics and Computing*, 10: 339–348, 2000. URL https://link.springer.com/article/10.1023/A:1008981510081.

N. Pocuca, R. P. Browne, and P. D. McNicholas. Mixture: Mixture Models for Clustering and Classification, Sept. 2022. URL https://cran.r-project.org/web/packages/mixture/index.html.

S. Pruilh, A.-S. Jannot, and S. Allassonnière. Spatio-temporal mixture process estimation to detect dynamical changes in population. *Artificial Intelligence in Medicine*, 126:102258, Apr. 2022. URL https://linkinghub.elsevier.com/retrieve/pii/S0933365722000239.

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O. Sahin and C. Czado. Vine copula mixture models and clustering for non-Gaussian data. *Econometrics and Statistics*, 22:136–158, Apr. 2022. URL https://www.sciencedirect.com/science/article/pii/S2452306221001052.

G. Schwarz. Estimating the dimension of a model. *The Annals of Statistics*, 6(2):461–464, 1978. URL https://www.jstor.org/stable/2958889.
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B. McCane and M. Albert. Distance functions for categorical and mixed variables. *Pattern Recognition Letters*, 29 (7):986–993, May 2008. URL https://linkinghub.elsevier.com/retrieve/pii/S0167865508000524.

G. McLachlan, D. Peel, and R. Bean. Modelling high-dimensional data by mixtures of factor analyzers. *Computational Statistics & Data Analysis*, 41(3-4):379–388, Jan. 2003. URL https://linkinghub.elsevier.com/retrieve/pii/S0167947302001834.

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G. J. McLachlan and D. Peel. *Finite Mixture Models*. Wiley Series in Probability and Statistics Applied Probability and Statistics Section. Wiley, New York, 2000. ISBN 978-0-471-00626-8.

P. McNicholas, T. Murphy, A. McDaid, and D. Frost. Serial and parallel implementations of model-based clustering via parsimonious Gaussian mixture models. *Computational Statistics & Data Analysis*, 54(3):711–723, Mar. 2010. URL https://linkinghub.elsevier.com/retrieve/pii/S0167947309000632.

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N. Pocuca, R. P. Browne, and P. D. McNicholas. Mixture: Mixture Models for Clustering and Classification, Sept. 2022. URL https://cran.r-project.org/web/packages/mixture/index.html.

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I. Vrbik and P. McNicholas. Analytic calculations for the EM algorithm for multivariate skew-mixture models. *Statistics & Probability Letters*, 82(6):1169–1174, June 2012. URL https://linkinghub.elsevier.com/retrieve/pii/S0167715212000673.

H. Wang, B. Luo, Q. bing Zhang, and S. Wei. Estimation for the number of components in a mixture model using stepwise split-and-merge EM algorithm. *Pattern Recognition Letters*, 25(16):1799–1809, 2004.

A. Willse and R. J. Boik. Identifiable finite mixtures of location models for clustering mixed-mode data. *Statistics and Computing*, 9:111–121, 1999. URL https://link.springer.com/article/10.1023/A:1008842432747.

S. J. Yakowitz and J. D. Spragins. On the Identifiability of Finite Mixtures. *The Annals of Mathematical Statistics*, 39(1):209–214, 1968. URL https://www.jstor.org/stable/2238925.

M.-S. Yang, C.-Y. Lai, and C.-Y. Lin. A robust EM clustering algorithm for Gaussian mixture models. *Pattern Recognition*, 45(11):3950–3961, Nov. 2012. URL https://linkinghub.elsevier.com/retrieve/pii/S0031320312002117.

B. Zhang, C. Zhang, and X. Yi. Competitive EM algorithm for finite mixture models. *Pattern recognition*, 37(1): 131–144, 2004. URL https://www.sciencedirect.com/science/article/abs/pii/S0031320303001407.

Y. Zhao and M. Udell. Missing Value Imputation for Mixed Data via Gaussian Copula. In *Proceedings of the 26th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining*, pages 636–646. ACM, Aug. 2020. ISBN 978-1-4503-7998-4. URL https://dl.acm.org/doi/10.1145/3394486.3403106.

# A Appendix

## A.1 Material on continuous distributions

### A.1.1 Gaussian distribution

A g-dimensional random variable X following a multivariate Gaussian distribution has the following density

$$p_g(\mathbf{x}_i; \boldsymbol{\mu}, \boldsymbol{\Sigma}) = \frac{1}{|\boldsymbol{\Sigma}|^{1/2} (2\pi)^{g/2}} \exp\left(-\frac{1}{2}(\mathbf{x}_i - \boldsymbol{\mu})^\top \boldsymbol{\Sigma}^{-1} (\mathbf{x}_i - \boldsymbol{\mu})\right) \quad (29)$$

### A.1.2 Student distribution

A g-dimensional random variable X following a multivariate t-distribution has the following density

$$p_g(\mathbf{x}; \boldsymbol{\mu}, \boldsymbol{\Sigma}, \nu) = \frac{\Gamma(\frac{\nu+q}{2}) |\boldsymbol{\Sigma}|^{-1/2}}{(\pi \nu)^{g/2} \Gamma(\frac{\nu}{2}) \{1 + \delta(\mathbf{x}, \boldsymbol{\mu}; \boldsymbol{\Sigma}) / \nu\}^{(\frac{\nu+q}{2})}}, \quad (30)$$
M. S. Smith and M. A. Khaled. Estimation of Copula Models With Discrete Margins via Bayesian Data Augmentation. *Journal of the American Statistical Association*, 107(497):290–303, Mar. 2012. URL http://www.tandfonline.com/doi/abs/10.1080/01621459.2011.644501.

H. Teicher. Identifiability of Finite Mixtures. *The Annals of Mathematical Statistics*, 34(4):1265–1269, 1963. URL https://doi.org/10.1214/aoms/1177703862.

R. D. Telford and R. B. Cunningham. Sex, sport, and body-size dependency of hematology in highly trained athletes. *Medicine and Science in Sports and Exercise*, 23(7):788–794, July 1991. URL https://journals.lww.com/acsm-msse/Abstract/1991/07000/Sex,_sport,_and_body_size_dependency_of_hematology.4.aspx.

N. Ueda and R. Nakano. Mixture density estimation via EM algorithm with deterministic annealing. In *Proceedings of IEEE Workshop on Neural Networks for Signal Processing*, pages 69–77, Ermioni, Greece, Sept. 1994. IEEE. ISBN 0-7803-2026-3. URL https://ieeexplore.ieee.org/abstract/document/366062.

N. Ueda and R. Nakano. Deterministic annealing EM algorithm. *Neural Networks*, 11(2):271–282, Mar. 1998. URL https://www.sciencedirect.com/science/article/abs/pii/S0893608097001330?via%3Dihub.

I. Vrbik and P. McNicholas. Analytic calculations for the EM algorithm for multivariate skew-mixture models. *Statistics & Probability Letters*, 82(6):1169–1174, June 2012. URL https://linkinghub.elsevier.com/retrieve/pii/S0167715212000673.

H. Wang, B. Luo, Q. bing Zhang, and S. Wei. Estimation for the number of components in a mixture model using stepwise split-and-merge EM algorithm. *Pattern Recognition Letters*, 25(16):1799–1809, 2004.

A. Willse and R. J. Boik. Identifiable finite mixtures of location models for clustering mixed-mode data. *Statistics and Computing*, 9:111–121, 1999. URL https://link.springer.com/article/10.1023/A:1008842432747.

S. J. Yakowitz and J. D. Spragins. On the Identifiability of Finite Mixtures. *The Annals of Mathematical Statistics*, 39(1):209–214, 1968. URL https://www.jstor.org/stable/2238925.

M.-S. Yang, C.-Y. Lai, and C.-Y. Lin. A robust EM clustering algorithm for Gaussian mixture models. *Pattern Recognition*, 45(11):3950–3961, Nov. 2012. URL https://linkinghub.elsevier.com/retrieve/pii/S0031320312002117.

B. Zhang, C. Zhang, and X. Yi. Competitive EM algorithm for finite mixture models. *Pattern recognition*, 37(1): 131–144, 2004. URL https://www.sciencedirect.com/science/article/abs/pii/S0031320303001407.

Y. Zhao and M. Udell. Missing Value Imputation for Mixed Data via Gaussian Copula. In *Proceedings of the 26th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining*, pages 636–646. ACM, Aug. 2020. ISBN 978-1-4503-7998-4. URL https://dl.acm.org/doi/10.1145/3394486.3403106.

# A Appendix

## A.1 Material on continuous distributions

### A.1.1 Gaussian distribution

A g-dimensional random variable X following a multivariate Gaussian distribution has the following density

$$p_g(\mathbf{x}_i; \boldsymbol{\mu}, \boldsymbol{\Sigma}) = \frac{1}{|\boldsymbol{\Sigma}|^{1/2}(2\pi)^{g/2}} \exp(-\frac{1}{2}(\mathbf{x}_i - \boldsymbol{\mu})^\top \boldsymbol{\Sigma}^{-1} (\mathbf{x}_i - \boldsymbol{\mu})) \quad (29)$$

### A.1.2 Student distribution

A g-dimensional random variable X following a multivariate t-distribution has the following density

$$p_g(\mathbf{x}; \boldsymbol{\mu}, \boldsymbol{\Sigma}, \nu) = \frac{\Gamma(\frac{\nu+q}{2}) |\boldsymbol{\Sigma}|^{-1/2}}{(\pi \nu)^{g/2} \Gamma(\frac{\nu}{2}) \{1 + \delta(\mathbf{x}, \boldsymbol{\mu}; \boldsymbol{\Sigma}) / \nu\}^{\frac{\nu+q}{2}}}, \quad (30)$$
with center $\mu$, positive definite inner product matrix $\Sigma$, degrees of freedom $\nu \in (0; \infty]$ and $\delta(x, \mu; \Sigma) = (x - \mu)^{\top}\Sigma^{-1}(x - \mu)$ is the Mahalanobis distance.

Given a scaling variable $U \in \mathbb{R}$, $X$ has a multivariate normal distribution, and $U$ is $\Gamma$ with

$$ X|U \sim \mathcal{N}(\mu, \Sigma|U) \text{ and } U \sim \Gamma(\frac{1}{2}\nu, \frac{1}{2}\nu). \qquad (31) $$

### A.1.3 Shifted Asymmetric Laplace distribution

The probability density function of a $g$-dimensional random variable $X$ distributed according to a Shifted Asymmetric Laplace distribution is

$$ p_g(x; \mu, \Sigma, \alpha) = \frac{2 \exp \{(x - \mu)^{\top} \Sigma^{-1} \alpha\}}{(2\pi)^{g/2} |\Sigma|^{1/2}} \times \left( \frac{\delta(x, \mu | \Sigma)}{2 + \alpha^{\top} \Sigma^{-1} \alpha} \right)^{\nu/2} K_{\nu}(u), \qquad (32) $$

with $\alpha \in \mathbb{R}^g$ skewness parameter, $\mu \in \mathbb{R}^g$ shift parameter, $\Sigma \in \mathbb{R}^{g \times g}$ scale matrix, $K_\nu$ modified Bessel function of third kind, with index $\nu = \frac{2-g}{2}$, $\delta$ is the Mahalanobis distance, and $u = \sqrt{(2 + \alpha^{\top} \Sigma^{-1} \alpha)\delta(x, \mu; \Sigma)}$.

From Kotz et al. (2001) for Asymmetric Laplace distributions and Franczak et al. (2014) for Shifted AL distributions, the random variable $X$ admits the representation

$$ X = \mu + W\alpha + \sqrt{W}Y, \quad W \sim \mathcal{E}(1), \quad Y \sim \mathcal{N}_g(0, \Sigma), \qquad (33) $$

with $W$ a random variable from an exponential distribution with rate 1, and $Y \in \mathbb{R}^{g \times g}$ a random variable from a Normal distribution with mean $0^g$ and covariance matrix $\Sigma$. And so, $X|W = w \sim \mathcal{N}_g(\mu + w\alpha, w\Sigma)$. Conditional on the data, $W$ follows a Generalized Inverse Gaussian distribution.

## A.2 Estimation of continuous distributions in mixture models
with center $\mu$, positive definite inner product matrix $\Sigma$, degrees of freedom $\nu \in (0; \infty]$ and $\delta(x, \mu; \Sigma) = (x - \mu)^{\top}\Sigma^{-1}(x - \mu)$ is the Mahalanobis distance.

Given a scaling variable $U \in \mathbb{R}$, $X$ has a multivariate normal distribution, and $U$ is $\Gamma$ with

$$X|U \sim \mathcal{N}(\mu, \Sigma|U) \text{ and } U \sim \Gamma(\frac{1}{2}\nu, \frac{1}{2}\nu). \qquad (31)$$

### A.1.3 Shifted Asymmetric Laplace distribution

The probability density function of a $g$-dimensional random variable $X$ distributed according to a Shifted Asymmetric Laplace distribution is

$$p_g(x; \mu, \Sigma, \alpha) = \frac{2 \exp \{(x - \mu)^{\top} \Sigma^{-1} \alpha\}}{(2\pi)^{g/2} |\Sigma|^{1/2}} \times \left( \frac{\delta(x, \mu|\Sigma)}{2 + \alpha^{\top} \Sigma^{-1} \alpha} \right)^{\nu/2} K_{\nu}(u), \qquad (32)$$

with $\alpha \in \mathbb{R}^g$ skewness parameter, $\mu \in \mathbb{R}^g$ shift parameter, $\Sigma \in \mathbb{R}^{g \times g}$ scale matrix, $K_\nu$ modified Bessel function of third kind, with index $\nu = \frac{2-g}{2}$, $\delta$ is the Mahalanobis distance, and $u = \sqrt{(2 + \alpha^{\top} \Sigma^{-1} \alpha)\delta(x, \mu; \Sigma)}$.

From Kotz et al. (2001) for Asymmetric Laplace distributions and Franczak et al. (2014) for Shifted AL distributions, the random variable $X$ admits the representation

$$X = \mu + W\alpha + \sqrt{W}Y, \quad W \sim \mathcal{E}(1), \quad Y \sim \mathcal{N}_g(0, \Sigma), \qquad (33)$$

with $W$ a random variable from an exponential distribution with rate 1, and $Y \in \mathbb{R}^{g \times g}$ a random variable from a Normal distribution with mean $0^g$ and covariance matrix $\Sigma$. And so, $X|W = w \sim \mathcal{N}_g(\mu + w\alpha, w\Sigma)$. Conditional on the data, $W$ follows a Generalized Inverse Gaussian distribution.

## A.2 Estimation of continuous distributions in mixture models
<table><caption>Table 12: Equations to compute conditional expectations of latent variables at E-step of an EM-type algorithm.</caption><thead><tr><th>Law</th><th>Expectations of latent variables</th><th>Updating equation</th></tr></thead><tbody><tr><td>Gaussian</td><td>τ<sub>ik</sub></td><td><div style="display: inline-block; vertical-align: middle; text-align: center;"><div style="border-bottom: 1px solid black; padding: 0 5px;">π<sub>k</sub>p<sub>Gaussian</sub>(x<sub>i</sub>; μ<sub>k</sub>, Σ<sub>k</sub>)</div><div style="padding: 0 5px;">∑<sup>K</sup><sub>j=1</sub> [π<sub>j</sub>p<sub>Gaussian</sub>(x<sub>i</sub>; μ<sub>j</sub>, Σ<sub>j</sub>)]</div></div> (34)</td></tr><tr><td rowspan="2">Student</td><td>τ<sub>ik</sub></td><td><div style="display: inline-block; vertical-align: middle; text-align: center;"><div style="border-bottom: 1px solid black; padding: 0 5px;">π<sub>k</sub>p<sub>Student</sub>(x<sub>i</sub>; μ<sub>k</sub>, Σ<sub>k</sub>, ν<sub>k</sub>)</div><div style="padding: 0 5px;">∑<sup>K</sup><sub>j=1</sub> [π<sub>j</sub>p<sub>Student</sub>(x<sub>i</sub>; μ<sub>j</sub>, Σ<sub>j</sub>, ν<sub>j</sub>)]</div></div> (35)</td></tr><tr><td>E<sub>u,ik</sub></td><td><div style="display: inline-block; vertical-align: middle; text-align: center;"><div style="border-bottom: 1px solid black; padding: 0 5px;">ν<sub>k</sub> + g</div><div style="padding: 0 5px;">ν<sub>k</sub> + δ(x<sub>i</sub>|μ<sub>k</sub>, Σ<sub>k</sub>)</div></div> (36)</td></tr><tr><td rowspan="3">SAL</td><td>τ<sub>ik</sub></td><td><div style="display: inline-block; vertical-align: middle; text-align: center;"><div style="border-bottom: 1px solid black; padding: 0 5px;">π<sub>k</sub>p<sub>SAL</sub>(x<sub>i</sub>; μ<sub>k</sub>, Σ<sub>k</sub>, α<sub>k</sub>)</div><div style="padding: 0 5px;">∑<sup>K</sup><sub>j=1</sub> [π<sub>j</sub>p<sub>SAL</sub>(x<sub>i</sub>; μ<sub>j</sub>, Σ<sub>j</sub>, α<sub>j</sub>)]</div></div> (37)</td></tr><tr><td>E<sub>1ik</sub></td><td><div style="display: inline-block; vertical-align: middle; text-align: center;"><div style="border-bottom: 1px solid black; padding: 0 5px;">√<span style="text-decoration: overline;">b<sub>ik</sub></span></div><div style="padding: 0 5px;">a<sub>k</sub></div></div> R<sub>ν</sub>(√a<sub>k</sub>b<sub>ik</sub>) (38)</td></tr><tr><td>E<sub>2ik</sub></td><td><div style="display: inline-block; vertical-align: middle; text-align: center;"><div style="border-bottom: 1px solid black; padding: 0 5px;">√<span style="text-decoration: overline;">a<sub>k</sub></span></div><div style="padding: 0 5px;">b<sub>ik</sub></div></div> R<sub>ν</sub>(√a<sub>k</sub>b<sub>ik</sub>) - <div style="display: inline-block; vertical-align: middle; text-align: center;"><div style="border-bottom: 1px solid black; padding: 0 5px;">2ν</div><div style="padding: 0 5px;">b<sub>ik</sub></div></div> (39)</td></tr></tbody></table>

## A.3 Pseudocodes

### A.3.1 Generic EM-MD and β computation pseudocodes

<table><thead><tr><th colspan="2">Algorithm 3: Generic EM algorithm</th></tr></thead><tbody><tr><td>Input</td><td>ε &gt; 0, K, t<sup>max</sup>, dataset X = [X<sup>c</sup>, X<sup>D</sup>] with X<sub>i</sub> &isin; R<sup>g</sup> &times; X</td></tr><tr><td>Initialization:</td><td>Compute τ<sub>ik</sub><sup>0</sup> &larr; K-Means(K, X<sup>c</sup>, maxiter=1)</td></tr><tr><td>Compute</td><td>π<sub>k</sub><sup>0</sup> with (6)</td></tr><tr><td>Compute</td><td>μ<sub>k</sub><sup>0</sup></td></tr><tr><td>Compute</td><td>Σ<sub>k</sub><sup>0</sup></td></tr><tr><td>Compute other continuous parameters</td><td></td></tr><tr><td>Compute</td><td>p<sub>k</sub><sup>d,0</sup> &forall; d &isin; [1, ..., D] with (25), (26), (27)</td></tr><tr><td>t &larr; 1</td><td></td></tr><tr><td>1 while</td><td>|l<sub>&infin;</sub><sup>t</sup> - l<sub>&infin;</sub><sup>t-1</sup>| &gt; ε and t &lt; t<sup>max</sup> do</td><td>/* Aitken's convergence */</td></tr><tr><td></td><td><b>E-Step</b></td><td></td></tr><tr><td></td><td>Compute τ<sub>ik</sub><sup>t</sup> with (20)</td><td></td></tr><tr><td></td><td>Compute other latent variables</td><td></td></tr><tr><td></td><td><b>M-Step</b></td><td></td></tr><tr><td></td><td>Compute π<sub>k</sub><sup>t</sup> with (6)</td><td></td></tr><tr><td></td><td>Compute μ<sub>k</sub><sup>t</sup></td><td></td></tr><tr><td></td><td>Compute Σ<sub>k</sub><sup>t</sup></td><td></td></tr><tr><td></td><td>Compute other continuous parameters</td><td></td></tr><tr><td></td><td>Compute discrete probabilities p<sub>k</sub><sup>d,t</sup> with (25), (26), (27)</td><td></td></tr><tr><td></td><td>t &larr; t + 1</td><td></td></tr><tr><td></td><td>end</td><td></td></tr></tbody></table>

### A.3.2 DEM-MD pseudocodes
<table><caption>Table 12: Equations to compute conditional expectations of latent variables at E-step of an EM-type algorithm.</caption><thead><tr><th>Law</th><th>Expectations of latent variables</th><th>Updating equation</th></tr></thead><tbody><tr><td>Gaussian</td><td>τ<sub>ik</sub></td><td><div style="display: inline-block; vertical-align: middle; text-align: center;"><div style="border-bottom: 1px solid black; padding: 0 5px;">π<sub>k</sub>p<sub>Gaussian</sub>(x<sub>i</sub>; μ<sub>k</sub>, Σ<sub>k</sub>)</div><div style="padding: 0 5px;">∑<sup>K</sup><sub>j=1</sub> [π<sub>j</sub>p<sub>Gaussian</sub>(x<sub>i</sub>; μ<sub>j</sub>, Σ<sub>j</sub>)]</div></div> (34)</td></tr><tr><td rowspan="2">Student</td><td>τ<sub>ik</sub></td><td><div style="display: inline-block; vertical-align: middle; text-align: center;"><div style="border-bottom: 1px solid black; padding: 0 5px;">π<sub>k</sub>p<sub>Student</sub>(x<sub>i</sub>; μ<sub>k</sub>, Σ<sub>k</sub>, ν<sub>k</sub>)</div><div style="padding: 0 5px;">∑<sup>K</sup><sub>j=1</sub> [π<sub>j</sub>p<sub>Student</sub>(x<sub>i</sub>; μ<sub>j</sub>, Σ<sub>j</sub>, ν<sub>j</sub>)]</div></div> (35)</td></tr><tr><td>E<sub>u,ik</sub></td><td><div style="display: inline-block; vertical-align: middle; text-align: center;"><div style="border-bottom: 1px solid black; padding: 0 5px;">ν<sub>k</sub> + g</div><div style="padding: 0 5px;">ν<sub>k</sub> + δ(x<sub>i</sub>|μ<sub>k</sub>, Σ<sub>k</sub>)</div></div> (36)</td></tr><tr><td rowspan="3">SAL</td><td>τ<sub>ik</sub></td><td><div style="display: inline-block; vertical-align: middle; text-align: center;"><div style="border-bottom: 1px solid black; padding: 0 5px;">π<sub>k</sub>p<sub>SAL</sub>(x<sub>i</sub>; μ<sub>k</sub>, Σ<sub>k</sub>, α<sub>k</sub>)</div><div style="padding: 0 5px;">∑<sup>K</sup><sub>j=1</sub> [π<sub>j</sub>p<sub>SAL</sub>(x<sub>i</sub>; μ<sub>j</sub>, Σ<sub>j</sub>, α<sub>j</sub>)]</div></div> (37)</td></tr><tr><td>E<sub>1ik</sub></td><td><div style="display: inline-block; vertical-align: middle; text-align: center;"><div style="border-bottom: 1px solid black; padding: 0 5px;">√<span style="text-decoration: overline;">b<sub>ik</sub></span></div><div style="padding: 0 5px;">a<sub>k</sub></div></div>R<sub>ν</sub>(√a<sub>k</sub>b<sub>ik</sub>) (38)</td></tr><tr><td>E<sub>2ik</sub></td><td><div style="display: inline-block; vertical-align: middle; text-align: center;"><div style="border-bottom: 1px solid black; padding: 0 5px;">√<span style="text-decoration: overline;">a<sub>k</sub></span></div><div style="padding: 0 5px;">b<sub>ik</sub></div></div>R<sub>ν</sub>(√a<sub>k</sub>b<sub>ik</sub>) - <div style="display: inline-block; vertical-align: middle; text-align: center;"><div style="border-bottom: 1px solid black; padding: 0 5px;">2ν</div><div style="padding: 0 5px;">b<sub>ik</sub></div></div> (39)</td></tr></tbody></table>

## A.3 Pseudocodes

### A.3.1 Generic EM-MD and β computation pseudocodes

<table><thead><tr><th colspan="2">Algorithm 3: Generic EM algorithm</th></tr></thead><tbody><tr><td>Input</td><td>ε &gt; 0, K, t<sup>max</sup>, dataset X = [X<sup>c</sup>, X<sup>D</sup>] with X<sub>i</sub> &isin; R<sup>g</sup> &times; X</td></tr><tr><td>Initialization:</td><td>Compute τ<sub>ik</sub><sup>0</sup> &larr; K-Means(K, X<sup>c</sup>, maxiter=1)</td></tr><tr><td>Compute</td><td>π<sub>k</sub><sup>0</sup> with (6)</td></tr><tr><td>Compute</td><td>μ<sub>k</sub><sup>0</sup></td></tr><tr><td>Compute</td><td>Σ<sub>k</sub><sup>0</sup></td></tr><tr><td>Compute other continuous parameters</td><td></td></tr><tr><td>Compute</td><td>p<sub>k</sub><sup>d,0</sup> &forall; d &isin; [1, ..., D] with (25), (26), (27)</td></tr><tr><td>t &larr; 1</td><td></td></tr><tr><td>1 while</td><td>|l<sub>&infin;</sub><sup>t</sup> - l<sub>&infin;</sub><sup>t-1</sup>| &gt; ε and t &lt; t<sup>max</sup> do</td><td>/* Aitken's convergence */</td></tr><tr><td></td><td><b>E-Step</b></td><td></td></tr><tr><td></td><td>Compute τ<sub>ik</sub><sup>t</sup> with (20)</td><td></td></tr><tr><td></td><td>Compute other latent variables</td><td></td></tr><tr><td></td><td><b>M-Step</b></td><td></td></tr><tr><td></td><td>Compute π<sub>k</sub><sup>t</sup> with (6)</td><td></td></tr><tr><td></td><td>Compute μ<sub>k</sub><sup>t</sup></td><td></td></tr><tr><td></td><td>Compute Σ<sub>k</sub><sup>t</sup></td><td></td></tr><tr><td></td><td>Compute other continuous parameters</td><td></td></tr><tr><td></td><td>Compute discrete probabilities p<sub>k</sub><sup>d,t</sup> with (25), (26), (27)</td><td></td></tr><tr><td></td><td>t &larr; t + 1</td><td></td></tr><tr><td></td><td>end</td><td></td></tr></tbody></table>

### A.3.2 DEM-MD pseudocodes
**Algorithm 4:** Computation of parameter $\beta$

$$
\begin{array}{ll}
\textbf{Input} : & \pi^{EM}, \pi^{(new)}, \pi^{(old)}, K, n \\
& \pi_{(1)}^{EM} \leftarrow \max_{1 \le k \le K} \pi_k^{EM}, \pi_{(1)}^{(old)} \leftarrow \max_{1 \le k \le K} \pi_k^{(old)} \\
& E \leftarrow \sum_{k=1}^{K} \pi_k^{(old)} \log(\pi_k^{(old)}) \\
& \beta \leftarrow \min \left\{ \frac{\sum_{k=1}^{K} \exp(-\eta n |\pi_k^{(new)} - \pi_k^{(old)}|)}{K}, \frac{(1-\pi_{(1)}^{EM})}{(-\pi_{(1)}^{(old)} E)} \right\} \\
\textbf{Output:} & \beta
\end{array}
$$

**Algorithm 5:** DEM-MD for Gaussian Mixtures

**Input:** $\varepsilon > 0$, $\gamma$, dataset $\mathbf{X} = [\mathbf{X}^c, \mathbf{X}^D]$ with $\mathbf{X}_i \in \mathbb{R}^g \times \mathcal{X}$
**Initialization:** $K^0 \leftarrow n$, $\beta^0 \leftarrow 1$
$\pi_k^0 \leftarrow 1/n \quad \forall k$, $\mu^0 \leftarrow \mathbf{X}^c$
$\Sigma_k^0 \leftarrow D_k(\lceil \sqrt{K^{\text{initial}}} \rceil) \mathbf{I}_g$ with $D_k = \text{sort} \left\{ d_{ki}^2 = \|x_i - \mu_k\|_2^2 : d_{ki}^2 > 0, \ i \neq k, \ 1 \le i \le n \right\}$
Initialize $p_k^{d,0} \quad \forall d \in [1, \dots, D]$ with (24)
$t \leftarrow 1$
Compute $\tau_{ik}^t$ with (20)
1 & while $|l_{\infty}^t - l_{\infty}^{t-1}| > \varepsilon$ do & /* Aitken's convergence */
& M-Step
& Compute $\pi_k^t$ with (21)
& Compute $\mu_k^t$ with (7)
& $\beta^t \leftarrow$ Algorithm 4
case delete classes with $\pi_k^t < 1/n$ do
& update class number $K^t$, adjust $\pi_k^t$ and $\tau_{ik}^t$
& $t_{\text{component}} \leftarrow 1$ /* variable to keep in memory the number of iterations with a stable number of components */
otherwise do
& $K^t \leftarrow K^{t-1}$
end
& if $t \ge t_{\text{min}}$ and $t_{\text{component}} \ge 100$ then
& if no superimposed clusters then
& $\beta^t = 0$
& else if superimposed clusters and $t_{\text{component}} < 200$ then /* give more time to the algorithm to converge */
& $t_{\text{min}} \leftarrow t_{\text{min}} + 50$
& else merge superimposed clusters
& adjust $\pi_k^t$, $\mu_k^t$, $\Sigma_k^t$ and $\tau_k^t$
end
end
& Compute $\Sigma_k^{EM}$ with (8) and $\Sigma_k^t = (1-\gamma)\Sigma_k^{EM} + \gamma\mathbf{P}$ with
& $\mathbf{P} = d_{\text{min}}^2 \mathbf{I}_g$, $d_{\text{min}}^2 = \min\{d_{ij}^2 : d_{ij}^2 = \|x_i - x_j\|_2^2 > 0, 1 \le i, j \le n\}$
& Compute discrete probabilities $p_k^{d,t}$ with (25), (26), (27)
E-Step
& Compute $\tau_{ik}^{t+1}$ with (20)
& $t \leftarrow t+1$
& $t_{\text{component}} \leftarrow t_{\text{component}} + 1$
end
**Algorithm 4:** Computation of parameter $\beta$

$$
\begin{array}{ll}
\textbf{Input} : & \pi^{EM}, \pi^{(new)}, \pi^{(old)}, K, n \\
& \pi_{(1)}^{EM} \leftarrow \max_{1 \le k \le K} \pi_k^{EM}, \pi_{(1)}^{(old)} \leftarrow \max_{1 \le k \le K} \pi_k^{(old)} \\
& E \leftarrow \sum_{k=1}^{K} \pi_k^{(old)} \log(\pi_k^{(old)}) \\
& \beta \leftarrow \min \left\{ \frac{\sum_{k=1}^{K} \exp(-\eta n |\pi_k^{(new)} - \pi_k^{(old)}|)}{K}, \frac{(1-\pi_{(1)}^{EM})}{(-\pi_{(1)}^{(old)} E)} \right\} \\
\textbf{Output:} & \beta
\end{array}
$$

**Algorithm 5:** DEM-MD for Gaussian Mixtures

**Input:** $\varepsilon > 0$, $\gamma$, dataset $\mathbf{X} = [\mathbf{X}^c, \mathbf{X}^D]$ with $\mathbf{X}_i \in \mathbb{R}^g \times \mathcal{X}$
**Initialization:** $K^0 \leftarrow n$, $\beta^0 \leftarrow 1$
$\pi_k^0 \leftarrow 1/n \quad \forall k$, $\boldsymbol{\mu}^0 \leftarrow \mathbf{X}^c$
$\boldsymbol{\Sigma}_k^0 \leftarrow D_{k(\lceil \sqrt{K^{\text{initial}}} \rceil)} \mathbf{I}_g$ with $D_k = \text{sort} \left\{ d_{ki}^2 = \|x_i - \mu_k\|_2^2 : d_{ki}^2 > 0, \ i \neq k, \ 1 \le i \le n \right\}$
Initialize $p_k^{d,0} \quad \forall d \in [1, \dots, D]$ with (24)
$t \leftarrow 1$
Compute $\tau_{ik}^t$ with (20)
1 & while $|l_{\infty}^t - l_{\infty}^{t-1}| > \varepsilon$ do & /* Aitken's convergence */
& M-Step
& Compute $\pi_k^t$ with (21)
& Compute $\boldsymbol{\mu}_k^t$ with (7)
& $\beta^t \leftarrow$ Algorithm 4
case delete classes with $\pi_k^t < 1/n$ do
& update class number $K^t$, adjust $\pi_k^t$ and $\tau_{ik}^t$
& $t_{\text{component}} \leftarrow 1$ /* variable to keep in memory the number of iterations with a stable number of components */
otherwise do
& $K^t \leftarrow K^{t-1}$
end
& if $t \ge t_{\text{min}}$ and $t_{\text{component}} \ge 100$ then
& if no superimposed clusters then
& $\beta^t = 0$
& else if superimposed clusters and $t_{\text{component}} < 200$ then /* give more time to the algorithm to converge */
& $t_{\text{min}} \leftarrow t_{\text{min}} + 50$
& else merge superimposed clusters
& adjust $\pi_k^t$, $\boldsymbol{\mu}_k^t$, $\boldsymbol{\Sigma}^t$ and $\boldsymbol{\tau}^t$
end
end
& Compute $\boldsymbol{\Sigma}_k^{EM}$ with (8) and $\boldsymbol{\Sigma}_k^t = (1-\gamma)\boldsymbol{\Sigma}_k^{EM} + \gamma \mathbf{P}$ with
& $\mathbf{P} = d_{\text{min}}^2 \mathbf{I}_g$, $d_{\text{min}}^2 = \min\{d_{ij}^2 : d_{ij}^2 = \|x_i - x_j\|_2^2 > 0, 1 \le i, j \le n\}$
& Compute discrete probabilities $p_k^{d,t}$ with (25), (26), (27)
E-Step
& Compute $\tau_{ik}^{t+1}$ with (20)
& $t \leftarrow t+1$
& $t_{\text{component}} \leftarrow t_{\text{component}} + 1$
end
<figure><img src="image_4.png" /><figcaption>Figure 3: Boxplots of relative discrete distribution parameter errors for settings $C_{221}^B$, $C_{221}^P$, $C_{221}^M$ and $C_{451}^M$, for the three continuous distributions on both DEM-MD (green) and EM-MD (blue). Each column corresponds to a continuous distribution, in this order: Gaussian, Student, SAL. Each row corresponds to a setting, in this order: $C_{221}^B$, $C_{221}^M$, $C_{451}^M$ and $C_{221}^P$.</figcaption></figure>
<figure><img src="image_4.png" /><figcaption>Figure 3: Boxplots of relative discrete distribution parameter errors for settings $C_{221}^B$, $C_{221}^P$, $C_{221}^M$ and $C_{451}^M$, for the three continuous distributions on both DEM-MD (green) and EM-MD (blue). Each column corresponds to a continuous distribution, in this order: Gaussian, Student, SAL. Each row corresponds to a setting, in this order: $C_{221}^B$, $C_{221}^M$, $C_{451}^M$ and $C_{221}^P$.</figcaption></figure>
<figure><img src="image_5.png" /><figcaption>Figure 4: Boxplots of relative discrete distribution parameter errors for settings $C_{524}$ (a), $C_{423}$ (b) and $C_{223}^P$ (c) on both DEM-MD (green) and EM-MD (blue). In each subplot, a row corresponds to a continuous distribution (Gaussian, Student, SAL in this order).</figcaption></figure>
<figure><img src="image_5.png" /><figcaption>Figure 4: Boxplots of relative discrete distribution parameter errors for settings $C_{524}$ (a), $C_{423}$ (b) and $C_{223}^P$ (c) on both DEM-MD (green) and EM-MD (blue). In each subplot, a row corresponds to a continuous distribution (Gaussian, Student, SAL in this order).</figcaption></figure>
<figure><img src="image_6.png" /><figcaption>Figure 4: Boxplots of relative discrete distribution parameter errors for settings $C_{524}$ (a), $C_{423}$ (b) and $C_{223}^P$ (c) on both DEM-MD (green) and EM-MD (blue). In each subplot, a row corresponds to a continuous distribution (Gaussian, Student, SAL in this order).</figcaption></figure>
<figure><img src="image_6.png" /><figcaption>Figure 4: Boxplots of relative discrete distribution parameter errors for settings $C_{524}$ (a), $C_{423}$ (b) and $C_{223}^P$ (c) on both DEM-MD (green) and EM-MD (blue). In each subplot, a row corresponds to a continuous distribution (Gaussian, Student, SAL in this order).</figcaption></figure>
<figure><img src="image_7.png" /><figcaption>Figure 4: Boxplots of relative discrete distribution parameter errors for settings $C_{524}$ (a), $C_{423}$ (b) and $C_{223}^P$ (c) on both DEM-MD (green) and EM-MD (blue). In each subplot, a row corresponds to a continuous distribution (Gaussian, Student, SAL in this order).</figcaption></figure>
<figure><img src="image_7.png" /><figcaption>Figure 4: Boxplots of relative discrete distribution parameter errors for settings $C_{524}$ (a), $C_{423}$ (b) and $C_{223}^P$ (c) on both DEM-MD (green) and EM-MD (blue). In each subplot, a row corresponds to a continuous distribution (Gaussian, Student, SAL in this order).</figcaption></figure>
<figure><img src="image_8.png" /><figcaption>Figure 5: Boxplots of relative discrete distribution parameter errors for settings $C_{223}$ (a) and $C_{343}$ (b), for the three DEM-MD on both DEM-MD (green) and EM-MD (blue). In each subplot, a row corresponds to a continuous distribution (Gaussian, Student, SAL in this order).</figcaption></figure>
<figure><img src="image_8.png" /><figcaption>Figure 5: Boxplots of relative discrete distribution parameter errors for settings $C_{223}$ (a) and $C_{343}$ (b), for the three DEM-MD on both DEM-MD (green) and EM-MD (blue). In each subplot, a row corresponds to a continuous distribution (Gaussian, Student, SAL in this order).</figcaption></figure>
<figure><img src="image_9.png" /><figcaption>Figure 5: Boxplots of relative discrete distribution parameter errors for settings $C_{223}$ (a) and $C_{343}$ (b), for the three DEM-MD on both DEM-MD (green) and EM-MD (blue). In each subplot, a row corresponds to a continuous distribution (Gaussian, Student, SAL in this order).</figcaption></figure>
<figure><img src="image_9.png" /><figcaption>Figure 5: Boxplots of relative discrete distribution parameter errors for settings $C_{223}$ (a) and $C_{343}$ (b), for the three DEM-MD on both DEM-MD (green) and EM-MD (blue). In each subplot, a row corresponds to a continuous distribution (Gaussian, Student, SAL in this order).</figcaption></figure>
<figure><img src="image_10.png" /><figcaption>Figure 6: Relative errors of SAL DEM-MD with/out Fraley regularization, on setting $C_{451}^M$. In blue are given the errors without regularization, from the same eighteen previous runs. In green are the errors with scale regularization. Results are over 72 runs with $n = 600$.</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>Figure 7: Estimated centers and discrete parameters for all DEM-MD on the prostate cancer dataset. Each color is associated with a model (see figure legends) and numbers indicate clusters. Rows indicate variable names in (a) and (c). Rows indicate modality numbers in (b) and (d).</figcaption></figure>
<figure><img src="image_10.png" /><figcaption>Figure 6: Relative errors of SAL DEM-MD with/out Fraley regularization, on setting $C_{451}^M$. In blue are given the errors without regularization, from the same eighteen previous runs. In green are the errors with scale regularization. Results are over 72 runs with $n = 600$.</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>Figure 7: Estimated centers and discrete parameters for all DEM-MD on the prostate cancer dataset. Each color is associated with a model (see figure legends) and numbers indicate clusters. Rows indicate variable names in (a) and (c). Rows indicate modality numbers in (b) and (d).</figcaption></figure>
<caption>**Algorithm 6:** DEM-MD for Student Mixtures</caption>

**Input:** $\varepsilon > 0$, $\gamma$, dataset $\mathbf{X} = [\mathbf{X}^c, \mathbf{X}^D]$ with $\mathbf{X}_i \in \mathbb{R}^g \times \mathcal{X}$
**Initialization:** $K^0 \leftarrow n$, $\beta^0 \leftarrow 1$
$\pi_k^0 \leftarrow 1/n \quad \forall k$, $\boldsymbol{\mu}^0 \leftarrow \mathbf{X}^c$, $\nu_k^0 \leftarrow 10 \quad \forall k$
$\Sigma_k^0 \leftarrow D_k(\lceil \sqrt{K^{\text{initial}}} \rceil) \mathbf{I}_g$ with $D_k = \text{sort} \left\{ d_{ki}^2 = \|x_i - \mu_k\|_2^2 : d_{ki}^2 > 0, \ i \neq k, \ 1 \le i \le n \right\}$
initialize $p_k^{d,0} \quad \forall d \in [1, \dots, D]$ with (24)
$t \leftarrow 1$
Compute $\tau_{ik}^t$ with (20)
Compute $E_{u,ik}^t$ with (36)
1 **while** $|l_\infty^t - l_\infty^{t-1}| > \varepsilon$ **do**
| **M-Step**
| Compute $\pi_k^t$ with (21)
| Compute $\boldsymbol{\mu}_k^t$ with (10)
| $\beta^t \leftarrow$ Algorithm 4
| **case** delete classes with $\pi_k^t < 1/n$ **do**
| | update class number $K^t$, adjust $\pi_k^t$ and $\tau_{ik}^t$
| | $t_{\text{component}} \leftarrow 1$ \textit{* variable to keep in memory the number of iterations with a stable number of components \textit{}}
| | **otherwise** **do**
| | | $K^t \leftarrow K^{t-1}$
| | **end**
| | if $t \ge t_{\min}$ and $t_{\text{component}} \ge 100$ **then**
| | | if no superimposed clusters **then**
| | | | $\beta^t = 0$
| | | else if superimposed clusters and $t_{\text{component}} < 200$ **then** \textit{* give more time to the algorithm to converge \textit{}}
| | | | $t_{\min} \leftarrow t_{\min} + 50$
| | | else merge superimposed clusters
| | | | adjust $\boldsymbol{\pi}^t$, $\boldsymbol{\mu}^t$, $\Sigma^{t-1}$, $\nu^{t-1}$ and $\boldsymbol{\tau}^t$
| | **end**
| | **end**
| | Compute $\Sigma_k^{EM}$ with (11) and $\Sigma_k^t = (1-\gamma)\Sigma_k^{EM} + \gamma\mathbf{P}$ with
| | $\gamma = 0.0001$, $\mathbf{P} = d_{\min}^2 \mathbf{I}_g$, $d_{\min}^2 = \min\{d_{ij}^2 : d_{ij}^2 = \|x_i - x_j\|_2^2 > 0, 1 \le i, j \le n\}$
| Compute $\nu_k^t$ by solving (4) with Brent's method on the interval $[2, 200]$
| Compute discrete probabilities $p_k^{d,t}$ with (25), (26), (27)
| **E-Step**
| Compute $\tau_{ik}^{t+1}$ with (20)
| Compute $E_{u,ik}^{t+1}$ with (36)
| $t \leftarrow t+1$
| $t_{\text{component}} \leftarrow t_{\text{component}} + 1$
| **end**
<caption>**Algorithm 6:** DEM-MD for Student Mixtures</caption>

**Input:** $\varepsilon > 0$, $\gamma$, dataset $\mathbf{X} = [\mathbf{X}^c, \mathbf{X}^D]$ with $\mathbf{X}_i \in \mathbb{R}^g \times \mathcal{X}$
**Initialization:** $K^0 \leftarrow n$, $\beta^0 \leftarrow 1$
$\pi_k^0 \leftarrow 1/n \quad \forall k$, $\boldsymbol{\mu}^0 \leftarrow \mathbf{X}^c$, $\nu_k^0 \leftarrow 10 \quad \forall k$
$\Sigma_k^0 \leftarrow D_k(\lceil \sqrt{K^{\text{initial}}} \rceil) \mathbf{I}_g$ with $D_k = \text{sort} \left\{ d_{ki}^2 = \|x_i - \mu_k\|_2^2 : d_{ki}^2 > 0, \ i \neq k, \ 1 \le i \le n \right\}$
initialize $p_k^{d,0} \quad \forall d \in [1, \dots, D]$ with (24)
$t \leftarrow 1$
Compute $\tau_{ik}^t$ with (20)
Compute $E_{u,ik}^t$ with (36)
1 **while** $|l_\infty^t - l_\infty^{t-1}| > \varepsilon$ **do**
| **M-Step**
| Compute $\pi_k^t$ with (21)
| Compute $\boldsymbol{\mu}_k^t$ with (10)
| $\beta^t \leftarrow$ Algorithm 4
| **case** delete classes with $\pi_k^t < 1/n$ **do**
| | update class number $K^t$, adjust $\pi_k^t$ and $\tau_{ik}^t$
| | $t_{\text{component}} \leftarrow 1$ \textit{ /* variable to keep in memory the number of iterations with a stable number of components */}
| | **otherwise** **do**
| | | $K^t \leftarrow K^{t-1}$
| | **end**
| | if $t \ge t_{\min}$ and $t_{\text{component}} \ge 100$ **then**
| | | if no superimposed clusters **then**
| | | | $\beta^t = 0$
| | | | **else if** superimposed clusters and $t_{\text{component}} < 200$ **then** \textit{ /* give more time to the algorithm to converge */}
| | | | $t_{\min} \leftarrow t_{\min} + 50$
| | | | **else** merge superimposed clusters
| | | | adjust $\boldsymbol{\pi}^t$, $\boldsymbol{\mu}^t$, $\boldsymbol{\Sigma}^{t-1}$, $\nu^{t-1}$ and $\boldsymbol{\tau}^t$
| | | **end**
| | **end**
| | Compute $\boldsymbol{\Sigma}_k^{EM}$ with (11) and $\boldsymbol{\Sigma}_k^t = (1-\gamma)\boldsymbol{\Sigma}_k^{EM} + \gamma \mathbf{P}$ with $\gamma = 0.0001$, $\mathbf{P} = d_{\min}^2 \mathbf{I}_g$, $d_{\min}^2 = \min\{d_{ij}^2 : d_{ij}^2 = \|x_i - x_j\|_2^2 > 0, 1 \le i, j \le n\}$
| | Compute $\nu_k^t$ by solving (4) with Brent's method on the interval $[2, 200]$
| | Compute discrete probabilities $p_k^{d,t}$ with (25), (26), (27)
| | **E-Step**
| | Compute $\tau_{ik}^{t+1}$ with (20)
| | Compute $E_{u,ik}^{t+1}$ with (36)
| | $t \leftarrow t + 1$
| | $t_{\text{component}} \leftarrow t_{\text{component}} + 1$
| | **end**
<caption>**Algorithm 7:** DEM-MD for SAL Mixtures</caption>

**Input:** $\varepsilon > 0$, dataset $\mathbf{X} = [\mathbf{X}^c, \mathbf{X}^D]$ with $\mathbf{X}_i \in \mathbb{R}^g \times \mathcal{X}$, a and b for temperature
**Initialization:** $K^0 \leftarrow n$, $\beta^0 \leftarrow 1$
$\pi_k^0 \leftarrow 1/n \quad \forall k$, $\alpha_k^0 \leftarrow [0, \dots, 0]_g \quad \forall k$
$\boldsymbol{\mu}^0 \leftarrow \mathbf{X}^c + \epsilon$, $\epsilon \sim \mathcal{N}(0, \sigma^2)$
$\boldsymbol{\Sigma}_k^0 \leftarrow D_{k(\lceil \sqrt{K^{\text{initial}}} \rceil)} \mathbf{I}_g$ with $D_k = \text{sort}\{d_{ki}^2 = \|x_i - \mu_k\|_2^2 : d_{ki}^2 > 0, \ i \neq k, \ 1 \le i \le n\}$
Initialize $p_k^{d,0} \quad \forall d \in [1, \dots, D]$ with (24)
$t \leftarrow 1$
Compute $\tau_{ik}^t$ with (23)
Compute $E_{1i}^{k,t}$ with (38)
Compute $E_{2i}^{k,t}$ with (39)
1. while $|l_{\infty}^t - l_{\infty}^{t-1}| > \varepsilon$ do
   /* Aitken's convergence */
   **CM-Step 1**
   Compute $\pi_k^t$ with (21)
   Compute $\boldsymbol{\mu}_k^t$ with (13)
   $\beta^t \leftarrow$ Algorithm 4
   **case delete classes with $\pi_k^t < 1/n$ do**
   update class number $K^t$, adjust $\pi_k^t$ and $\tau_{ik}^t$
   $t_{\text{component}} \leftarrow 1$ /* variable to keep in memory the number of iterations with a stable number of components */
   **otherwise do**
   $K^t \leftarrow K^{t-1}$
   **end**
   **if** $t \ge t_{\text{min}}$ and $t_{\text{component}} \ge 100$ **then**
   **if** no superimposed clusters **then**
   $\beta^t = 0$
   **else if** superimposed clusters and $t_{\text{component}} < 200$ **then** /* give more time to the algorithm to converge */
   $t_{\text{min}} \leftarrow t_{\text{min}} + 50$
   **else** merge superimposed clusters
   adjust $\boldsymbol{\pi}^t$, $\boldsymbol{\mu}^t$, $\alpha^{t-1}$, $\boldsymbol{\Sigma}^{t-1}$ and $\boldsymbol{\tau}^t$
   **end**
   **end**
   Compute discrete probabilities $p_k^{d,t}$ with (25), (26), (27)
   **Check** $\boldsymbol{\mu}^t = x_i^c$ and compute $\alpha_k^t$ with Algorithm 1
   **Intermediate E-step**
   Compute $\tilde{\tau}_{ik}^t$, $\tilde{E}_{1i}^{k,t}$ and $\tilde{E}_{2i}^{k,t}$ with respectively (23), (38) and (39)
   **CM-Step 2**
   Compute $\boldsymbol{\Sigma}_k^t$ with (14)
   **E-Step**
   Compute $\tau_{ik}^{t+1}$ with (23)
   Compute $E_{1i}^{k,t+1}$ with (38)
   Compute $E_{2i}^{k,t+1}$ with (39)
   $t \leftarrow t + 1$
   $t_{\text{component}} \leftarrow t_{\text{component}} + 1$
   **end**
<caption>**Algorithm 7:** DEM-MD for SAL Mixtures</caption>

**Input:** $\varepsilon > 0$, dataset $\mathbf{X} = [\mathbf{X}^c, \mathbf{X}^D]$ with $\mathbf{X}_i \in \mathbb{R}^g \times \mathcal{X}$, a and b for temperature
**Initialization:** $K^0 \leftarrow n$, $\beta^0 \leftarrow 1$
$\pi_k^0 \leftarrow 1/n \quad \forall k$, $\alpha_k^0 \leftarrow [0, \dots, 0]_g \quad \forall k$
$\boldsymbol{\mu}^0 \leftarrow \mathbf{X}^c + \epsilon$, $\epsilon \sim \mathcal{N}(0, \sigma^2)$
$\boldsymbol{\Sigma}_k^0 \leftarrow D_{k(\lceil \sqrt{K^{\text{initial}}} \rceil)} \mathbf{I}_g$ with $D_k = \text{sort}\{d_{ki}^2 = \|x_i - \mu_k\|_2^2 : d_{ki}^2 > 0, \ i \neq k, \ 1 \le i \le n\}$
Initialize $p_k^{d,0} \quad \forall d \in [1, \dots, D]$ with (24)
$t \leftarrow 1$
Compute $\tau_{ik}^t$ with (23)
Compute $E_{1i}^{k,t}$ with (38)
Compute $E_{2i}^{k,t}$ with (39)
1. while $|l_{\infty}^t - l_{\infty}^{t-1}| > \varepsilon$ do
   /* Aitken's convergence */
   **CM-Step 1**
   Compute $\pi_k^t$ with (21)
   Compute $\boldsymbol{\mu}_k^t$ with (13)
   $\beta^t \leftarrow$ Algorithm 4
   **case delete classes with $\pi_k^t < 1/n$ do**
   update class number $K^t$, adjust $\pi_k^t$ and $\tau_{ik}^t$
   $t_{\text{component}} \leftarrow 1$ /* variable to keep in memory the number of iterations with a stable number of components */
   **otherwise do**
   $K^t \leftarrow K^{t-1}$
   **end**
   **if** $t \ge t_{\text{min}}$ and $t_{\text{component}} \ge 100$ **then**
   **if** no superimposed clusters **then**
   $\beta^t = 0$
   **else if** superimposed clusters and $t_{\text{component}} < 200$ **then** /* give more time to the algorithm to converge */
   $t_{\text{min}} \leftarrow t_{\text{min}} + 50$
   **else** merge superimposed clusters
   adjust $\boldsymbol{\pi}^t$, $\boldsymbol{\mu}^t$, $\alpha^{t-1}$, $\boldsymbol{\Sigma}^{t-1}$ and $\boldsymbol{\tau}^t$
   **end**
   **end**
   Compute discrete probabilities $p_k^{d,t}$ with (25), (26), (27)
   **Check** $\boldsymbol{\mu}^t = x_i^c$ and compute $\alpha_k^t$ with Algorithm 1
   **Intermediate E-step**
   Compute $\tilde{\tau}_{ik}^t$, $\tilde{E}_{1i}^{k,t}$ and $\tilde{E}_{2i}^{k,t}$ with respectively (23), (38) and (39)
   **CM-Step 2**
   Compute $\boldsymbol{\Sigma}_k^t$ with (14)
   **E-Step**
   Compute $\tau_{ik}^{t+1}$ with (23)
   Compute $E_{1i}^{k,t+1}$ with (38)
   Compute $E_{2i}^{k,t+1}$ with (39)
   $t \leftarrow t + 1$
   $t_{\text{component}} \leftarrow t_{\text{component}} + 1$
   **end**