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Subspace clustering for the finite mixture of generalized hyperbolic distributions

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Abstract

The finite mixture of generalized hyperbolic distributions is a flexible model for clustering, but its large number of parameters for estimation, especially in high dimensions, can make it computationally expensive to work with. In light of this issue, we provide an extension of the subspace clustering technique developed for finite Gaussian mixtures to that of generalized hyperbolic distribution. The methodology will be demonstrated with numerical experiments.

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Funding

Funding was provided by Natural Sciences and Engineering Research Council of Canada (NSERC), (Grant No. RGPIN-424130-2012 71617).

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Correspondence to Ryan Browne.

Appendix

Appendix

1.1 Technical details of limiting cases

Here we show how to directly obtain certain limiting cases using the following parametrization,

$$\begin{aligned} f_{GHD}(\varvec{x}\mid {\varvec{\theta }})= & {} \left[ \frac{ \omega + \delta \left( \varvec{x}, \varvec{\mu }| \varvec{\Sigma }\right) }{ \omega + \varvec{\beta }'\varvec{\Sigma }^{-1}\varvec{\beta }} \right] ^{(\lambda -{p}/{2})/2}\nonumber \\&\times \frac{ K_{\lambda - {p}/{2}}\Big (\sqrt{\big [ \omega + \varvec{\beta }'\varvec{\Sigma }^{-1}\varvec{\beta }\big ]\big [\omega +\delta \left( \varvec{x}, \varvec{\mu }| \varvec{\Sigma }\right) \big ]}\Big )}{ \left( 2\pi \right) ^{{p}/{2}} \left| \varvec{\Sigma }\right| ^{{1}/{2}} K_{\lambda }\left( \omega \right) \exp \big \{-\left( \varvec{x}-\varvec{\mu }\right) '\varvec{\Sigma }^{-1}\varvec{\beta }\big \}}, \end{aligned}$$
(2)

where GHD denotes the generalized hyperbolic distribution. So that \(\mathbf {X}\sim GH_p(\varvec{\mu }, \varvec{\beta }, \varvec{\Sigma }, \lambda , \omega )\) or equivalently \(\mathbf {X}\sim GH_p({\varvec{\theta }})\) and the parameters are

$$\begin{aligned} {\varvec{\theta }}= \left( \varvec{\mu }, \varvec{\beta }, \varvec{\Sigma }, \lambda , \omega \right) . \end{aligned}$$
(3)

To find the limiting cases of the GHD we require the following asymptotic relation (see Abramowitz and Stegun 1964)

$$\begin{aligned} K_\lambda (z) \sim \frac{1}{2} \Gamma (\lambda ) \left( \frac{z}{2} \right) ^{-\lambda } \qquad \text{ for } \qquad z \rightarrow 0 \qquad \text{ when } \qquad \lambda > 0. \end{aligned}$$
(4)

A similar relation holds when \(\lambda <0\) as the modified Bessel function has the following symmetry property \(K_{-\lambda }(z) = K_{\lambda }(z)\).

1.1.1 Variance-Gamma (Generalized Laplace)

Kozubowski et al. (2013); Barndorff-Nielsen (1978); McNicholas et al. (2017) provide the density function of multivariate Variance-Gamma (or Generalized Laplace) distribution

$$\begin{aligned} f_{VG}(\varvec{x})= & {} \frac{2 e^{ \left( \varvec{x}- \varvec{\mu }\right) ^t\varvec{\Psi }^{-1}\varvec{\alpha }} }{(2\pi )^{p/2} \Gamma (\lambda ) |\varvec{\Psi } |^{1/2}} \left( \frac{ \delta \left( \varvec{x}, \varvec{\mu } | \varvec{\varvec{\Psi } }\right) }{2 + \varvec{\alpha }'\varvec{\Psi } ^{-1}\varvec{\alpha }} \right) ^{ \left( \lambda - p/2\right) / 2}\nonumber \\&\times K_{\lambda - p/2}\left( \sqrt{ \delta \left( \varvec{x}, \varvec{\mu }_g | \varvec{\Psi }\right) (2 + \varvec{\alpha }'\varvec{\Psi } ^{-1}\varvec{\alpha })} \right) , \end{aligned}$$
(5)

we denote the multivariate Variance-Gamma (VG) distribution by \(VG_p(\varvec{\mu }, \varvec{\alpha }, \varvec{\Psi } , \lambda )\).

We can obtain the multivariate VG distribution as a limiting case of the GHD by forcing \(\lambda > 0\) and then letting \(\omega \), \(\varvec{\beta }\) and \(\varvec{\Sigma }\) become small in a controlled way. i.e. we let

$$\begin{aligned} {\varvec{\theta }}_{VG} = \left( \varvec{\mu }, \gamma \varvec{\alpha }, \gamma \varvec{\Psi } , \lambda , 2 \gamma \right) , \end{aligned}$$
(6)

Substituting these parameter values into the GHD density we have

$$\begin{aligned} f_{GHD}(\varvec{x}\mid {\varvec{\theta }}_{VG} )= & {} \left[ \frac{ 2 \gamma + \gamma ^{-1}\delta \left( \varvec{x}, \varvec{\mu }| \varvec{\Psi } \right) }{ 2 \gamma + \gamma \varvec{\alpha }'\varvec{\Psi } ^{-1}\varvec{\alpha }} \right] ^{(\lambda -{p}/{2})/2}\\&\times \frac{ K_{\lambda - {p}/{2}}\Big (\sqrt{\big [ 2 \gamma + \gamma \varvec{\alpha }'\varvec{\Psi } ^{-1}\varvec{\alpha }\big ]\big [2 \gamma + \gamma ^{-1} \delta \left( \varvec{x}, \varvec{\mu }| \varvec{\Psi } \right) \big ]}\Big )}{ \left( 2\pi \right) ^{{p}/{2}} \left| \gamma \varvec{\Psi } \right| ^{{1}/{2}} K_{\lambda }\left( 2 \gamma \right) \exp \big \{-\left( \varvec{x}-\varvec{\mu }\right) '\varvec{\Psi } ^{-1}\varvec{\alpha }\big \}} \end{aligned}$$

which after some manipulation becomes

$$\begin{aligned} f_{GHD}(\varvec{x}\mid {\varvec{\theta }}_{VG} )= & {} \frac{ \gamma ^{-\lambda } }{K_{\lambda }\left( 2 \gamma \right) } \left[ \frac{ 2 \gamma ^2 + \delta \left( \varvec{x}, \varvec{\mu }| \varvec{\Psi } \right) }{ 2 + \varvec{\alpha }'\varvec{\Psi } ^{-1}\varvec{\alpha }} \right] ^{(\lambda -{p}/{2})/2}\\&\times \frac{ K_{\lambda - {p}/{2}}\Big (\sqrt{\big [ 2 + \varvec{\alpha }'\varvec{\Psi } ^{-1}\varvec{\alpha }\big ]\big [2 \gamma ^2 + \delta \left( \varvec{x}, \varvec{\mu }| \varvec{\Psi } \right) \big ]}\Big )}{ \left( 2\pi \right) ^{{p}/{2}} \left| \varvec{\Psi } \right| ^{{1}/{2}} \exp \big \{-\left( \varvec{x}-\varvec{\mu }\right) '\varvec{\Psi } ^{-1}\varvec{\alpha }\big \}}. \end{aligned}$$

Now letting \(\gamma \rightarrow 0\) and using the asymptotic relation in (4) while \(\lambda > 0\) we obtain

$$\begin{aligned} \frac{2}{ \Gamma (\lambda ) } \left[ \frac{\delta \left( \varvec{x}, \varvec{\mu }| \varvec{\Psi } \right) }{ 2 + \varvec{\alpha }'\varvec{\Psi } ^{-1}\varvec{\alpha }} \right] ^{(\lambda -{p}/{2})/2} \frac{ K_{\lambda - {p}/{2}}\Big (\sqrt{\big [ 2 + \varvec{\alpha }'\varvec{\Psi } ^{-1}\varvec{\alpha }\big ]\delta \left( \varvec{x}, \varvec{\mu }| \varvec{\Psi } \right) }\Big )}{ \left( 2\pi \right) ^{{p}/{2}} \left| \varvec{\Psi } \right| ^{{1}/{2}} \exp \big \{-\left( \varvec{x}-\varvec{\mu }\right) '\varvec{\Psi } ^{-1}\varvec{\alpha }\big \}}, \end{aligned}$$

which is the density of \(VG_p(\varvec{\mu }, \varvec{\alpha }, \varvec{\Psi } , \lambda )\).

1.1.2 Asymmetric Laplace

The density of the shifted asymmetric Laplace is

$$\begin{aligned} f_{SAL}(\varvec{x}| \varvec{\mu }, \varvec{\alpha }, \varvec{\Psi } )= & {} \frac{2 e^{ \left( \varvec{x}- \varvec{\mu }\right) ^t\varvec{\Psi }^{-1}\varvec{\alpha }} }{ \left( 2 \pi \right) ^{p/2} \left| \varvec{\Psi } \right| ^{1/2}} \left[ \frac{ \delta \left( \varvec{x}, \varvec{\mu }_g | \varvec{\Psi }\right) }{ 2+ \varvec{\alpha }^t\varvec{\Psi }^{-1}\varvec{\alpha }} \right] ^{v/2} \nonumber \\&\times K_v \left( \sqrt{ \left[ 2+ \varvec{\alpha }^t\varvec{\Psi }^{-1}\varvec{\alpha } \right] \delta \left( \varvec{x}, \varvec{\mu }_g | \varvec{\Psi }\right) } \right) \end{aligned}$$
(7)

where \(v=(2-p)/2\). The asymmetric Laplace is a special case of the \(VG_p(\varvec{\mu }, \varvec{\alpha }, \varvec{\Psi } , \lambda )\) when \(\lambda =1\) and therefore a limiting case of the GHD.

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Kim, NH., Browne, R. Subspace clustering for the finite mixture of generalized hyperbolic distributions. Adv Data Anal Classif 13, 641–661 (2019). https://doi.org/10.1007/s11634-018-0333-2

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