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 was provided by Natural Sciences and Engineering Research Council of Canada (NSERC), (Grant No. RGPIN-424130-2012 71617).
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Appendix
Appendix
1.1 Technical details of limiting cases
Here we show how to directly obtain certain limiting cases using the following parametrization,
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
To find the limiting cases of the GHD we require the following asymptotic relation (see Abramowitz and Stegun 1964)
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
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
Substituting these parameter values into the GHD density we have
which after some manipulation becomes
Now letting \(\gamma \rightarrow 0\) and using the asymptotic relation in (4) while \(\lambda > 0\) we obtain
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
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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DOI: https://doi.org/10.1007/s11634-018-0333-2