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The determination of uncertainty levels in robust clustering of subjects with longitudinal observations using the Dirichlet process mixture

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Abstract

In this paper we introduce a new method to the cluster analysis of longitudinal data focusing on the determination of uncertainty levels for cluster memberships. The method uses the Dirichlet-t distribution which notably utilizes the robustness feature of the student-t distribution in the framework of a Bayesian semi-parametric approach together with robust clustering of subjects evaluates the uncertainty level of subjects memberships to their clusters. We let the number of clusters and the uncertainty levels be unknown while fitting Dirichlet process mixture models. Two simulation studies are conducted to demonstrate the proposed methodology. The method is applied to cluster a real data set taken from gene expression studies.

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Notes

  1. The probability density function of a Dirichlet random vector \(\mathbf {u}=(u_1,\ldots ,u_k)^\prime \) with \(u_1,\ldots ,u_k\ge 0\), and  \(\sum _{i=1}^k u_i=1\), is given by \(f(\mathbf {u})\propto \prod _{i=1}^k u_i^{\alpha _i-1}\) where \(\alpha _1,\ldots ,\alpha _k>0\), (see, e.g., Sorensen and Gianola 2002).

  2. The probability density function of an inverse-Wishart random matrix \(\mathbf {U}\) of order T is given by \(f(\mathbf {U})\propto |\mathbf {U}|^{-\frac{\tau +T+1}{2} } e^{-\frac{1}{2}tr\left( \varPsi \mathbf {U}^{-1}\right) }\) where \(\tau \) and \(\varPsi \) are the shape and the scale parameters, respectively (see, e.g., Sorensen and Gianola 2002).

  3. The probability density function of an Inverse-Gamma random variable u is given by \(f(u)\propto u^{-(a+1) } e^{-b/u}\) where a and b are the shape and the scale parameters, respectively (see, e.g., Sorensen and Gianola 2002).

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Acknowledgments

The authors gratefully acknowledge two reviewers for their valuable comments and suggestions.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Isfahan University of Technology, Isfahan, 84156-83111, Iran

    Reyhaneh Rikhtehgaran

  2. Department of Statistics, University of Isfahan, Isfahan, 81746, Iran

    Iraj Kazemi

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  1. Reyhaneh Rikhtehgaran
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  2. Iraj Kazemi
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Correspondence to Reyhaneh Rikhtehgaran.

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Rikhtehgaran, R., Kazemi, I. The determination of uncertainty levels in robust clustering of subjects with longitudinal observations using the Dirichlet process mixture. Adv Data Anal Classif 10, 541–562 (2016). https://doi.org/10.1007/s11634-016-0262-x

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  • DOI: https://doi.org/10.1007/s11634-016-0262-x

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