Parsimonious mixture of mean-mixture of normal distributions with missing data

Document Type : Special Issue Dedicated to memory of Prof. Mahbanoo Tata

Authors

1 Department of Statistics, University of Kashan, Kashan, Iran

2 Farhangian University Of Kerman, Kerman, Iran

Abstract

Clustering multivariate data based on mixture distributions is a usual method to characterize groups and label data sets. Mixture models have recently been received considerable attention to accommodate asymmetric and missing data via exploiting skewed and heavy-tailed distributions. In this paper, a mixture of multivariate mean-mixture of normal distributions is considered for handling missing data. The EM-type algorithms are carried out to determine maximum likelihood of parameters estimations. We analyzed the real data sets and conducted simulation studies to demonstrate the superiority of the proposed methodology.

Keywords

Main Subjects


[1] Arellano-Valle, RB., Azzalini, A., Ferreira, CS., & Santoro, K. (2020). A two-piece normal measurement error model, Computational Statistics and Data Analysis, 144, 106863. https://doi.org/10.1016/j.csda.2019.106863
[2] Aitken, A. (1925). On Bernoulli's numerical solution of algebraic equations. Proceedings of the Royal Society of Edinburgh, 46, 289 􀀀 305. https://doi.org/10.1017/S0370164600022070
[3] Akaike, H. (1998). Information theory and an extension of the maximum likelihood principle. In Selected papers of hirotugu akaike, New York, NY: Springer New York, 199 􀀀 213. https://doi.org/10.1007/978-1-4612-1694-0 15
[4] Azzalini, A. (1985). A class of distributions which includes the normal ones. Scandinavian journal of statistics, 171 􀀀 178. https://www.jstor.org/stable/4615982
[5] Bai, X., Chen, K., & Yao, W. (2016). Mixture of linear mixed models using multivariate t distribution. Journal of Statistical Computation and Simulation, 86(4), 771 􀀀 787. https://doi.org/10.1080/00949655.2015.1036431
[6] Ban eld, JD., & Raftery, AE. (1993). Model-based Gaussian and non-Gaussian clustering. Biometrics, 803 􀀀 821. https://doi.org/10.2307/2532201
[7] Biernacki, C., Celeux, G., & Govaert, G. (2000). Assessing a mixture model for clustering with the integrated completed likelihood. IEEE transactions on pattern analysis and machine intelligence, 22(7), 719 􀀀 725. https://doi.org/10.1109/34.865189
[8] Clark, KM., & McNicholas, PD. (2023). Clustering Three-Way Data with Outliers. arXiv preprint arXiv:2310.05288.
[9] Celeux, G., & Govaert, G. (1995). Gaussian parsimonious clustering models. Pattern recognition, 28(5), 781 􀀀 793. https://doi.org/10.1016/0031-3203(94)00125-6
[10] Diaconis, P., & Efron, B. (1983). Computer-intensive methods in statistics. Scienti c American, 248(5), 116-131. https://www.jstor.org/stable/24968902
[11] Dempster, AP., Laird, NM., & Rubin, DB. (1977). Maximum likelihood from incomplete data via the EM algorithm. Journal of the royal statistical society: series B (methodological), 39(1), 1 􀀀 22. https://doi.org/10.1111/j.2517-6161.1977.tb01600.x
[12] Hashemi, F., Naderi, M., & Mashinchi, M. (2019). Clustering right-skewed data stream via Birnbaum{Saunders mixture models: A  exible approach based on fuzzy clustering algorithm. Applied Soft Computing, 82, 105539. https://doi.org/10.1016/j.asoc.2019.105539
[13] Hashemi, F., Naderi, M., Jamalizadeh, A., & Bekker, A. (2021). A exible factor analysis based on the class of mean-mixture of normal distributions. Computational statistics & data analysis, 157, 107162. https://doi.org/10.1016/j.csda.2020.107162
[14] Hubert, L., & Arabie, P. (1985). Comparing partitions. Journal of classi cation, 2, 193-218. https://doi.org/10.1007/BF01908075
[15] Lin, TI. (2014). Learning from incomplete data via parameterized t mixture models through eigenvalue decomposition. Computational statistics & data analysis, 71, 183-195. https://doi.org/10.1016/j.csda.2013.02.020
[16] Lin, TC., & Lin, TI. (2010). Supervised learning of multivariate skew normal mixture models with missing information. Computational Statistics, 25, 183 􀀀 201. https://doi.org/10.1007/s00180-009-0169-5
[17] Little, R. J., & Rubin, D. B. (2019). Statistical analysis with missing data (Vol. 793). John Wiley & Sons.
[18] Liu, C., & Rubin, DB. (1994). The ECME algorithm: a simple extension of EM and ECM with faster monotone convergence. Biometrika, 81(4), 633 􀀀 648. https://doi.org/10.1093/biomet/81.4.633
[19] McNicholas, PD., & Murphy, TB. (2008). Parsimonious Gaussian mixture models. Statistics and Computing, 18, 285 􀀀 296. https://doi.org/10.1007/s11222-008-9056-0
[20] Meng, XL., & Rubin, DB. (1993). Maximum likelihood estimation via the ECM algorithm: A general framework. Biometrika, 80(2), 267 􀀀 278. https://doi.org/10.1093/biomet/80.2.267
[21] Naderi, M., Hung, WL., Lin, TI., & Jamalizadeh, A. (2019). A novel mixture model using the multivariate normal mean{variance mixture of Birnbaum{Saunders distributions and its application to extrasolar planets. Journal of Multivariate Analysis, 171, 126 􀀀 138. https://doi.org/10.1016/j.jmva.2018.11.015
[22] Naderi, M., Hashemi, F., Bekker, A., & Jamalizadeh, A. (2020). Modeling right-skewed  nancial data streams: A likelihood inference based on the generalized Birnbaum-Saunders mixture model. Applied Mathematics and Computation, 376, 125109. https://doi.org/10.1016/j.amc.2020.125109
[23] Naderi, M., Bekker, A., Arashi, M., & Jamalizadeh, A. (2020). A theoretical framework for Landsat data modeling based on the matrix variate mean-mixture of normal model. Plos one, 15(4), e0230773. https://doi.org/10.1371/journal.pone.0230773
[24] Naderi, M., & Nooghabi, MJ. (2024). Clustering asymmetrical data with out-liers: Parsimonious mixtures of contaminated mean-mixture of normal distributions. Journal of Computational and Applied Mathematics, 437, 115433.
https://doi.org/10.1016/j.cam.2023.115433
[25] Negarestani, H., Jamalizadeh, A., Sha ei, S., & Balakrishnan, N. (2019). Mean mixtures of normal distributions: properties, inference and application. Metrika, 82, 501 􀀀 528. https://doi.org/10.1007/s00184-018-0692-x
[26] Rand, WM. (1971). Objective criteria for the evaluation of clustering methods. Journal of the American Statistical association, 66(336), 846 􀀀 850. https://doi.org/10.1080/01621459.1971.10482356
[27] Punzo, A., & McNicholas, PD. (2016). Parsimonious mixtures of multivariate contaminated normal distributions. Biometrical Journal, 58(6), 1506 􀀀 1537. https://doi.org/10.1002/bimj.201500144
[28] Schwarz, G. (1978). Estimating the dimension of a model. The annals of statistics, 461 􀀀 464. https://www.jstor.org/stable/2958889
[29] Sepahdar, A., Madadi, M., Balakrishnan, N., & Jamalizadeh, A. (2022). Parsimonious mixture-of-experts based on mean mixture of multivariate normal distributions. Stat, 11(1), e421. https://doi.org/10.1002/sta4.421
[30] Wang, WL., & Lin, TI. (2015). Robust model-based clustering via mixtures of skew-t distributions with missing information. Advances in Data Analysis and Classi cation, 9, 423 􀀀 445. https://doi.org/10.1007/s11634-015-0221-y

Articles in Press, Accepted Manuscript
Available Online from 29 April 2024
  • Receive Date: 08 December 2023
  • Revise Date: 21 February 2024
  • Accept Date: 28 April 2024