IDEAS home Printed from https://ideas.repec.org/a/spr/advdac/v12y2018i2d10.1007_s11634-017-0293-y.html
   My bibliography  Save this article

Eigenvalues and constraints in mixture modeling: geometric and computational issues

Author

Listed:
  • Luis Angel García-Escudero

    (University of Valladolid)

  • Alfonso Gordaliza

    (University of Valladolid)

  • Francesca Greselin

    (Milano-Bicocca University)

  • Salvatore Ingrassia

    (University of Catania)

  • Agustín Mayo-Iscar

    (University of Valladolid)

Abstract

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.

Suggested Citation

  • Luis Angel García-Escudero & Alfonso Gordaliza & Francesca Greselin & Salvatore Ingrassia & Agustín Mayo-Iscar, 2018. "Eigenvalues and constraints in mixture modeling: geometric and computational issues," Advances in Data Analysis and Classification, Springer;German Classification Society - Gesellschaft für Klassifikation (GfKl);Japanese Classification Society (JCS);Classification and Data Analysis Group of the Italian Statistical Society (CLADAG);International Federation of Classification Societies (IFCS), vol. 12(2), pages 203-233, June.
    • Handle: RePEc:spr:advdac:v:12:y:2018:i:2:d:10.1007_s11634-017-0293-y
    DOI: 10.1007/s11634-017-0293-y
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s11634-017-0293-y
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s11634-017-0293-y?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Sanjeena Subedi & Antonio Punzo & Salvatore Ingrassia & Paul McNicholas, 2015. "Cluster-weighted $$t$$ t -factor analyzers for robust model-based clustering and dimension reduction," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 24(4), pages 623-649, November.
    2. Sanjeena Subedi & Antonio Punzo & Salvatore Ingrassia & Paul McNicholas, 2013. "Clustering and classification via cluster-weighted factor analyzers," Advances in Data Analysis and Classification, Springer;German Classification Society - Gesellschaft für Klassifikation (GfKl);Japanese Classification Society (JCS);Classification and Data Analysis Group of the Italian Statistical Society (CLADAG);International Federation of Classification Societies (IFCS), vol. 7(1), pages 5-40, March.
    3. Biernacki, Christophe & Chrétien, Stéphane, 2003. "Degeneracy in the maximum likelihood estimation of univariate Gaussian mixtures with EM," Statistics & Probability Letters, Elsevier, vol. 61(4), pages 373-382, February.
    4. Salvatore Ingrassia, 2004. "A likelihood-based constrained algorithm for multivariate normal mixture models," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 13(2), pages 151-166, September.
    5. Kiefer, Nicholas M, 1978. "Discrete Parameter Variation: Efficient Estimation of a Switching Regression Model," Econometrica, Econometric Society, vol. 46(2), pages 427-434, March.
    6. Gabriela Ciuperca & Andrea Ridolfi & Jérôme Idier, 2003. "Penalized Maximum Likelihood Estimator for Normal Mixtures," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 30(1), pages 45-59, March.
    7. Fritz, Heinrich & García-Escudero, Luis A. & Mayo-Iscar, Agustín, 2013. "A fast algorithm for robust constrained clustering," Computational Statistics & Data Analysis, Elsevier, vol. 61(C), pages 124-136.
    8. Andrews, Jeffrey L. & McNicholas, Paul D. & Subedi, Sanjeena, 2011. "Model-based classification via mixtures of multivariate t-distributions," Computational Statistics & Data Analysis, Elsevier, vol. 55(1), pages 520-529, January.
    9. L. García-Escudero & A. Gordaliza & A. Mayo-Iscar, 2014. "A constrained robust proposal for mixture modeling avoiding spurious solutions," Advances in Data Analysis and Classification, Springer;German Classification Society - Gesellschaft für Klassifikation (GfKl);Japanese Classification Society (JCS);Classification and Data Analysis Group of the Italian Statistical Society (CLADAG);International Federation of Classification Societies (IFCS), vol. 8(1), pages 27-43, March.
    10. Francesca Greselin & Salvatore Ingrassia & Antonio Punzo, 2011. "Assessing the pattern of covariance matrices via an augmentation multiple testing procedure," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 20(2), pages 141-170, June.
    11. Biernacki, Christophe & Celeux, Gilles & Govaert, Gerard, 2003. "Choosing starting values for the EM algorithm for getting the highest likelihood in multivariate Gaussian mixture models," Computational Statistics & Data Analysis, Elsevier, vol. 41(3-4), pages 561-575, January.
    12. Kentaro Tanaka, 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," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 36(1), pages 171-184, March.
    13. García-Escudero, Luis Angel & Gordaliza, Alfonso & Greselin, Francesca & Ingrassia, Salvatore & Mayo-Iscar, Agustín, 2016. "The joint role of trimming and constraints in robust estimation for mixtures of Gaussian factor analyzers," Computational Statistics & Data Analysis, Elsevier, vol. 99(C), pages 131-147.
    14. Ingrassia, Salvatore & Rocci, Roberto, 2007. "Constrained monotone EM algorithms for finite mixture of multivariate Gaussians," Computational Statistics & Data Analysis, Elsevier, vol. 51(11), pages 5339-5351, July.
    15. Chris Fraley & Adrian E. Raftery, 2007. "Bayesian Regularization for Normal Mixture Estimation and Model-Based Clustering," Journal of Classification, Springer;The Classification Society, vol. 24(2), pages 155-181, September.
    16. Ingrassia, Salvatore & Rocci, Roberto, 2011. "Degeneracy of the EM algorithm for the MLE of multivariate Gaussian mixtures and dynamic constraints," Computational Statistics & Data Analysis, Elsevier, vol. 55(4), pages 1715-1725, April.
    17. Seo, Byungtae & Kim, Daeyoung, 2012. "Root selection in normal mixture models," Computational Statistics & Data Analysis, Elsevier, vol. 56(8), pages 2454-2470.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Keefe Murphy & Thomas Brendan Murphy, 2020. "Gaussian parsimonious clustering models with covariates and a noise component," Advances in Data Analysis and Classification, Springer;German Classification Society - Gesellschaft für Klassifikation (GfKl);Japanese Classification Society (JCS);Classification and Data Analysis Group of the Italian Statistical Society (CLADAG);International Federation of Classification Societies (IFCS), vol. 14(2), pages 293-325, June.
    2. Andrea Cappozzo & Francesca Greselin & Thomas Brendan Murphy, 2020. "A robust approach to model-based classification based on trimming and constraints," Advances in Data Analysis and Classification, Springer;German Classification Society - Gesellschaft für Klassifikation (GfKl);Japanese Classification Society (JCS);Classification and Data Analysis Group of the Italian Statistical Society (CLADAG);International Federation of Classification Societies (IFCS), vol. 14(2), pages 327-354, June.
    3. Cong, Lin & Yao, Weixin, 2021. "A Likelihood Ratio Test of a Homoscedastic Multivariate Normal Mixture Against a Heteroscedastic Multivariate Normal Mixture," Econometrics and Statistics, Elsevier, vol. 18(C), pages 79-88.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Roberto Rocci & Stefano Antonio Gattone & Roberto Di Mari, 2018. "A data driven equivariant approach to constrained Gaussian mixture modeling," Advances in Data Analysis and Classification, Springer;German Classification Society - Gesellschaft für Klassifikation (GfKl);Japanese Classification Society (JCS);Classification and Data Analysis Group of the Italian Statistical Society (CLADAG);International Federation of Classification Societies (IFCS), vol. 12(2), pages 235-260, June.
    2. Angelo Mazza & Antonio Punzo, 2020. "Mixtures of multivariate contaminated normal regression models," Statistical Papers, Springer, vol. 61(2), pages 787-822, April.
    3. Roberto Mari & Roberto Rocci & Stefano Antonio Gattone, 2020. "Scale-constrained approaches for maximum likelihood estimation and model selection of clusterwise linear regression models," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 29(1), pages 49-78, March.
    4. Andrews, Jeffrey L., 2018. "Addressing overfitting and underfitting in Gaussian model-based clustering," Computational Statistics & Data Analysis, Elsevier, vol. 127(C), pages 160-171.
    5. Paul D. McNicholas, 2016. "Model-Based Clustering," Journal of Classification, Springer;The Classification Society, vol. 33(3), pages 331-373, October.
    6. Utkarsh J. Dang & Antonio Punzo & Paul D. McNicholas & Salvatore Ingrassia & Ryan P. Browne, 2017. "Multivariate Response and Parsimony for Gaussian Cluster-Weighted Models," Journal of Classification, Springer;The Classification Society, vol. 34(1), pages 4-34, April.
    7. Antonio Punzo & Paul. D. McNicholas, 2017. "Robust Clustering in Regression Analysis via the Contaminated Gaussian Cluster-Weighted Model," Journal of Classification, Springer;The Classification Society, vol. 34(2), pages 249-293, July.
    8. Seo, Byungtae & Kim, Daeyoung, 2012. "Root selection in normal mixture models," Computational Statistics & Data Analysis, Elsevier, vol. 56(8), pages 2454-2470.
    9. Andrea Cappozzo & Francesca Greselin & Thomas Brendan Murphy, 2020. "A robust approach to model-based classification based on trimming and constraints," Advances in Data Analysis and Classification, Springer;German Classification Society - Gesellschaft für Klassifikation (GfKl);Japanese Classification Society (JCS);Classification and Data Analysis Group of the Italian Statistical Society (CLADAG);International Federation of Classification Societies (IFCS), vol. 14(2), pages 327-354, June.
    10. García-Escudero, Luis Angel & Gordaliza, Alfonso & Greselin, Francesca & Ingrassia, Salvatore & Mayo-Iscar, Agustín, 2016. "The joint role of trimming and constraints in robust estimation for mixtures of Gaussian factor analyzers," Computational Statistics & Data Analysis, Elsevier, vol. 99(C), pages 131-147.
    11. Kim, Daeyoung & Seo, Byungtae, 2014. "Assessment of the number of components in Gaussian mixture models in the presence of multiple local maximizers," Journal of Multivariate Analysis, Elsevier, vol. 125(C), pages 100-120.
    12. Maruotti, Antonello & Punzo, Antonio, 2017. "Model-based time-varying clustering of multivariate longitudinal data with covariates and outliers," Computational Statistics & Data Analysis, Elsevier, vol. 113(C), pages 475-496.
    13. Kasa, Siva Rajesh & Rajan, Vaibhav, 2022. "Improved Inference of Gaussian Mixture Copula Model for Clustering and Reproducibility Analysis using Automatic Differentiation," Econometrics and Statistics, Elsevier, vol. 22(C), pages 67-97.
    14. Salvatore Ingrassia & Simona Minotti & Giorgio Vittadini, 2012. "Local Statistical Modeling via a Cluster-Weighted Approach with Elliptical Distributions," Journal of Classification, Springer;The Classification Society, vol. 29(3), pages 363-401, October.
    15. Volodymyr Melnykov, 2013. "Finite mixture modelling in mass spectrometry analysis," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 62(4), pages 573-592, August.
    16. Diani, Cecilia & Galimberti, Giuliano & Soffritti, Gabriele, 2022. "Multivariate cluster-weighted models based on seemingly unrelated linear regression," Computational Statistics & Data Analysis, Elsevier, vol. 171(C).
    17. Derek S. Young & Xi Chen & Dilrukshi C. Hewage & Ricardo Nilo-Poyanco, 2019. "Finite mixture-of-gamma distributions: estimation, inference, and model-based clustering," Advances in Data Analysis and Classification, Springer;German Classification Society - Gesellschaft für Klassifikation (GfKl);Japanese Classification Society (JCS);Classification and Data Analysis Group of the Italian Statistical Society (CLADAG);International Federation of Classification Societies (IFCS), vol. 13(4), pages 1053-1082, December.
    18. Antonio Punzo & Salvatore Ingrassia & Antonello Maruotti, 2021. "Multivariate hidden Markov regression models: random covariates and heavy-tailed distributions," Statistical Papers, Springer, vol. 62(3), pages 1519-1555, June.
    19. Morris, Katherine & Punzo, Antonio & McNicholas, Paul D. & Browne, Ryan P., 2019. "Asymmetric clusters and outliers: Mixtures of multivariate contaminated shifted asymmetric Laplace distributions," Computational Statistics & Data Analysis, Elsevier, vol. 132(C), pages 145-166.
    20. Chi, Eric C. & Lange, Kenneth, 2014. "Stable estimation of a covariance matrix guided by nuclear norm penalties," Computational Statistics & Data Analysis, Elsevier, vol. 80(C), pages 117-128.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:advdac:v:12:y:2018:i:2:d:10.1007_s11634-017-0293-y. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.