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Multivariate hidden Markov regression models: random covariates and heavy-tailed distributions

Author

Listed:
  • Antonio Punzo

    (University of Catania)

  • Salvatore Ingrassia

    (University of Catania)

  • Antonello Maruotti

    (LUMSA
    University of Bergen)

Abstract

Despite recent methodological advances in hidden Markov regression models and a rapid increase in their application in a wide range of empirical settings, complex clustering-based research questions that include the contribution of the covariates set to the classification and the presence of atypical observations are often addressed ignoring the possible effects of wrong model assumptions. Hidden Markov regression models with random covariates (HMRMRCs) have been recently proposed as an improvement over the classical fixed covariates approach, allowing the covariates to contribute to the underlying clustering structure. To make the approach more flexible, when all the considered random variables are continuous, HMRMRCs are here defined focusing on three multivariate elliptical distributions: the normal (reference distribution), the t, and the contaminated normal. The latter two, heavy-tailed generalizations of the normal distribution, are introduced to protect the reference model for the occurrence of mildly atypical points and also allow us their automatic detection. Identifiability conditions are provided, EM-based algorithms are outlined for parameter estimation, and various implementation and operational issues are discussed. Properties of the estimators of the regression coefficients, as well as of the hidden path parameters, are evaluated through Monte Carlo experiments with the aim of showing the consequences of wrong model assumptions on paramaters estimates and inferred clustering. Artificial and real data analyses are provided to investigate models behavior in presence of heterogeneity and atypical observations.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:stpapr:v:62:y:2021:i:3:d:10.1007_s00362-019-01146-3
    DOI: 10.1007/s00362-019-01146-3
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    References listed on IDEAS

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    1. Leroux, Brian G., 1992. "Maximum-likelihood estimation for hidden Markov models," Stochastic Processes and their Applications, Elsevier, vol. 40(1), pages 127-143, February.
    2. 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.
    3. 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.
    4. 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.
    5. 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.
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    7. Antonello Maruotti & Antonio Punzo & Luca Bagnato, 2019. "Hidden Markov and Semi-Markov Models with Multivariate Leptokurtic-Normal Components for Robust Modeling of Daily Returns Series," Journal of Financial Econometrics, Oxford University Press, vol. 17(1), pages 91-117.
    8. Bartolucci, Francesco & Farcomeni, Alessio, 2009. "A Multivariate Extension of the Dynamic Logit Model for Longitudinal Data Based on a Latent Markov Heterogeneity Structure," Journal of the American Statistical Association, American Statistical Association, vol. 104(486), pages 816-831.
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    11. Ingrassia, Salvatore & Minotti, Simona C. & Punzo, Antonio, 2014. "Model-based clustering via linear cluster-weighted models," Computational Statistics & Data Analysis, Elsevier, vol. 71(C), pages 159-182.
    12. F. Bartolucci & A. Farcomeni & F. Pennoni, 2014. "Rejoinder on: Latent Markov models: a review of a general framework for the analysis of longitudinal data with covariates," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 23(3), pages 484-486, September.
    13. 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.
    14. F. Bartolucci & A. Farcomeni & F. Pennoni, 2014. "Latent Markov models: a review of a general framework for the analysis of longitudinal data with covariates," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 23(3), pages 433-465, September.
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    16. Cristophe Croux & Catherine Dehon, 2003. "Estimators of the multiple correlation coefficient: Local robustness and confidence intervals," Statistical Papers, Springer, vol. 44(3), pages 315-334, July.
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    Cited by:

    1. Michael P. B. Gallaugher & Salvatore D. Tomarchio & Paul D. McNicholas & Antonio Punzo, 2022. "Multivariate cluster weighted models using skewed distributions," 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. 16(1), pages 93-124, March.

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