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A mixture of generalized hyperbolic factor analyzers

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Abstract

The mixture of factor analyzers model, which has been used successfully for the model-based clustering of high-dimensional data, is extended to generalized hyperbolic mixtures. The development of a mixture of generalized hyperbolic factor analyzers is outlined, drawing upon the relationship with the generalized inverse Gaussian distribution. An alternating expectation-conditional maximization algorithm is used for parameter estimation, and the Bayesian information criterion is used to select the number of factors as well as the number of components. The performance of our generalized hyperbolic factor analyzers model is illustrated on real and simulated data, where it performs favourably compared to its Gaussian analogue and other approaches.

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Acknowledgments

The authors are grateful to an associate editor and anonymous reviewers for their very helpful comments and suggestions, the cumulative effect of which has been a stronger manuscript.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, McMaster University, Hamilton, ON, Canada

    Cristina Tortora, Paul D. McNicholas & Ryan P. Browne

Authors
  1. Cristina Tortora
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  2. Paul D. McNicholas
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  3. Ryan P. Browne
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Corresponding author

Correspondence to Cristina Tortora.

Additional information

This work was supported by a grant-in-aid from Compusense Inc. as well as a Collaborative Research and Development grant from the Natural Sciences and Engineering Research Council of Canada.

Appendix: Updates for component covariance parameters

Appendix: Updates for component covariance parameters

At the second stage for our AECM algorithm, the (conditional) expected value of complete-data log-likelihood is given by

$$\begin{aligned} {Q}_2= & {} C-\frac{1}{2}\sum _{i=1}^n \sum _{g=1}^G\hat{z}_{ig}\log |\varvec{\varPsi }_g|-\frac{1}{2}\sum _{i=1}^n \sum _{g=1}^G\hat{z}_{ig} \\&\times \left[ b_{ig}\,\text{ tr }\left\{ (\mathbf {x}_i-\hat{{\varvec{\mu }}}_g)(\mathbf {x}_i-\hat{{\varvec{\mu }}}_g)'\varvec{\varPsi }_g^{-1}\right\} -2\,\text{ tr }\left\{ (\mathbf {x}_i-\hat{{\varvec{\mu }}}_g)\hat{\varvec{\alpha }}_g'\varvec{\varPsi }_g^{-1}\right\} \right. \\&+a_{ig}\,\text{ tr }\left\{ \hat{\varvec{\alpha }}_g\hat{\varvec{\alpha }}_g'\varvec{\varPsi }_g^{-1}\right\} -2\,\text{ tr }\left\{ (\mathbf {x}_i-\hat{{\varvec{\mu }}}_g)'\varvec{\varPsi }_g^{-1}\varvec{\varLambda }_g\varvec{E}_{2ig}\right\} \\&\left. +2\,\text{ tr }\left\{ \hat{\varvec{\alpha }}_g'\varvec{\varPsi }_g^{-1}\varvec{\varLambda }_g\varvec{E}_{1ig}\right\} +\,\text{ tr }\left\{ \varvec{\varLambda }_g\varvec{E}_{3ig}\varvec{\varLambda }_g'\varvec{\varPsi }_g^{-1}\right\} \right] , \end{aligned}$$

where \(C\) is constant with respect to \(\varvec{\varLambda }_g\) and \(\varvec{\varPsi }_g\). Differentiating \({Q}_2\) with respect to \(\varvec{\varLambda }_g\) gives

$$\begin{aligned} S_1(\varvec{\varLambda }_g,\varvec{\varPsi }_g)= & {} \frac{\partial {Q}_2}{\partial \varvec{\varLambda }_g}=-\frac{1}{2}\sum _{i=1}^n\hat{z}_{ig}\left[ -2\varvec{\varPsi }_g^{-1}(\mathbf {x}_i-\hat{{\varvec{\mu }}}_g)\varvec{E}_{2ig}'+2\varvec{\varPsi }_g^{-1}\hat{\varvec{\alpha }}_g\varvec{E}_{1ig}' \right. \\&\left. +\varvec{\varPsi }_g^{-1}\varvec{\varLambda }_g\left( \varvec{E}_{3ig}'+\varvec{E}_{3ig}\right) \right] \end{aligned}$$

Note that \(\varvec{E}_{3ig}\) is a symmetric matrix. Now, solving \(S_1(\hat{\varvec{\varLambda }}_g,\varvec{\varPsi }_g)=\varvec{0}\) gives the update:

$$\begin{aligned} \hat{\varvec{\varLambda }}_g = \left\{ \sum _{i=1}^n\hat{z}_{ig}\left[ (\mathbf {x}_i-\hat{{\varvec{\mu }}}_g)\varvec{E}_{2ig}'-\hat{\varvec{\alpha }}_g\varvec{E}_{1ig}'\right] \right\} \left\{ \sum _{i=1}^n\hat{z}_{ig}\varvec{E}_{3ig}\right\} ^{-1}. \end{aligned}$$

Differentiating \({Q}_2\) with respect to \(\varvec{\varPsi }_g^{-1}\) gives

$$\begin{aligned}&S_2(\varvec{\varLambda }_g,\varvec{\varPsi }_g) =\frac{\partial {Q}_2}{\partial \varvec{\varPsi }_g^{-1}}\!=\!\frac{1}{2}\sum _{i=1}^n\hat{z}_{ig}\varvec{\varPsi }_g \!-\!\frac{1}{2}\sum _{i=1}^n\hat{z}_{ig}\left[ b_{ig}(\mathbf {x}_i\!-\!\hat{{\varvec{\mu }}}_g)(\mathbf {x}_i\!-\!\hat{{\varvec{\mu }}}_g)' \right. \\&\quad \left. -2\hat{\varvec{\alpha }}_g(\mathbf {x}_i-\hat{{\varvec{\mu }}}_g)'+a_{ig}\hat{\varvec{\alpha }}_g\hat{\varvec{\alpha }}_g' -2(\mathbf {x}_i-\hat{{\varvec{\mu }}}_g)\varvec{E}_{2ig}'\varvec{\varLambda }_g' +2\hat{\varvec{\alpha }}_g\varvec{E}_{1ig}'\varvec{\varLambda }_g'+\varvec{\varLambda }_g\varvec{E}_{3ig}\varvec{\varLambda }_g'\right] \!. \end{aligned}$$

Now, solving \(\text {diag}\{S_2(\hat{\varvec{\varLambda }}_g,\hat{\varvec{\varPsi }}_g)\}=\varvec{0}\) gives the update:

$$\begin{aligned} \hat{\varvec{\varPsi }}_g= & {} \frac{1}{n_g}\text {diag}\left\{ \sum _{i=1}^n\hat{z}_{ig}\left[ b_{ig}(\mathbf {x}_i-\hat{{\varvec{\mu }}}_g)(\mathbf {x}_i-\hat{{\varvec{\mu }}}_g)'-2\hat{\varvec{\alpha }}_g(\mathbf {x}_i-\hat{{\varvec{\mu }}}_g)'+a_{ig}\hat{\varvec{\alpha }}_g\hat{\varvec{\alpha }}_g' \right. \right. \\&\left. \left. -2(\mathbf {x}_i-\hat{{\varvec{\mu }}}_g)\varvec{E}_{2ig}'\hat{\varvec{\varLambda }}_g'+2\hat{\varvec{\alpha }}_g\varvec{E}_{1ig}'\hat{\varvec{\varLambda }}_g'+\hat{\varvec{\varLambda }}_g\varvec{E}_{3ig}\hat{\varvec{\varLambda }}_g'\right] \right\} . \end{aligned}$$

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Tortora, C., McNicholas, P.D. & Browne, R.P. A mixture of generalized hyperbolic factor analyzers. Adv Data Anal Classif 10, 423–440 (2016). https://doi.org/10.1007/s11634-015-0204-z

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