Multivariate stable distribution

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multivariate stable
Probability density function Heatmap showing a Multivariate (bivariate) stable distribution with α = 1.1
Parameters	$\alpha \in (0,2]$ — exponent $\delta \in \mathbb{R}^d$ - shift/location vector $Λ(s)$ - a spectral finite measure on the sphere
Support	$u \in \mathbb{R}^d$
PDF	(no analytic expression)
CDF	(no analytic expression)
Variance	Infinite when $α < 2$
CF	see text

The multivariate stable distribution is a multivariate probability distribution that is a multivariate generalisation of the univariate stable distribution. The multivariate stable distribution defines linear relations between stable distribution marginals.^{[clarification needed]} In the same way as for the univariate case, the distribution is defined in terms of its characteristic function.

The multivariate stable distribution can also be thought as an extension of the multivariate normal distribution. It has parameter, α, which is defined over the range 0 < α ≤ 2, and where the case α = 2 is equivalent to the multivariate normal distribution. It has an additional skew parameter that allows for non-symmetric distributions, where the multivariate normal distribution is symmetric.

Contents

1 Definition
2 Parametrization using projections
3 Special cases
4 Linear properties
5 Inference in the independent component model
6 Resources
7 Notes

Definition

Let S be the unit sphere in $R^d : \mathbb{S} = \{u \in R^d : |u| = 1\}$ . For a random variable, X, it has a multivariate stable distribution and the notation $X \sim S (α,Λ,δ)$ is used, if the joint characteristic function of $X$ is^[1]

$\operatorname{E} \exp(u^T X) = \exp \left\{-\int \limits_{S \in R^d}\left\{|u^Ts|^\alpha + i \nu (u^Ts, \alpha) \right\} \, \Lambda(ds) + u^T\delta\right\}$

where 0 < α < 2, and

$\nu(u,\alpha) =\begin{cases} -\mathbf{sign}(u) \tan(\pi \alpha / 2)|u|^\alpha & \alpha \ne 1, \\ (2/\pi)u \ln |u| & \alpha=1. \end{cases}$

This is essentially the result of Feldheim,^[2] that any stable random vector can be characterized by a spectral measure $Λ$ (a finite measure on $S$ ) and a shift vector $\delta \in R^d$ .

Parametrization using projections

Another way to describe a stable random vector is in terms of projections. For any vector u, the projection $u T X$ is univariate $α -$ stable with some skewness $β(u)$ , scale $γ(u)$ and some shift $δ(u)$ . The notation $X \sim S(\alpha,\beta(\cdot),\gamma(\cdot),\delta(\cdot))$ is used if $u T X$ is stable with

$u^TX \sim s(\alpha,\beta(\cdot),\gamma(\cdot),\delta(\cdot))$ for every $u \in R^d$ . This is called the projection parameterization.

The spectral measure determines the projection parameter functions by:

$\gamma(u) = \int_{s \in \mathbb{S}} |u^Ts|^\alpha \Lambda(ds)$

$\beta(u) = \int_{s \in \mathbb{S}}|u^Ts|^\alpha \mathbf{sign}(u^Ts)\Lambda(ds)$

$\delta(u)=\begin{cases}u^T \delta & \alpha \ne 1\\u^T \delta -\int_{s \in \mathbb{S}}\tfrac{\pi}{2} u^Ts \ln|u^Ts|\Lambda(ds)&\alpha=1\end{cases}$

Special cases

There are four special cases where the multivariate characteristic function takes a simpler form. Define the characteristic function of a stable marginal as

$\omega(u|\alpha,\beta) = \begin{cases}|u|^\alpha\left[1-i \beta(\tan \tfrac{\pi\alpha}{2})\mathbf{sign}(u)\right]& \alpha \ne 1\\ |u|\left[1+i \beta \tfrac{2}{\pi} \mathbf{sign}(u)\ln |u|\right] & \alpha = 1\end{cases}$

Isotropic multivariate stable distribution

The characteristic function is $E \exp(i u^T X)=\exp\{-\gamma_0^\alpha+i u^T \delta)\}$ The spectral measure is continuous and uniform, leading to radial/isotropic symmetry.^[3]

Elliptically contoured multivariate stable distribution

Elliptically contoured m.v. stable distribution is a special symmetric case of the multivariate stable distribution. If X is $α$ -stable and elliptically contoured, then it has joint characteristic function $E exp(i u T X) = exp{ - (u T Σ u) α / 2 + i u T δ)}$ for some positive definite matrix $Σ$ and shift vector $\delta \in R^d$ . Note the relation to characteristic function of the multivariate normal distribution: $E exp(i u T X) = exp{ - (u T Σ u) + i u T δ)}$ . In other words, when α = 2 we get the characteristic function of the multivariate normal distribution.

Independent components

The marginals are independent with $X j \sim S (α,β j,γ j,δ j)$ , then the characteristic function is

$E \exp(i u^T X) = \exp\left\{-\sum_{j=1}^m \omega(u_j|\alpha,\beta_j)\gamma_j^\alpha +i u^T \delta)\right\}$

Heatmap showing a multivariate (bivariate) independent stable distribution with α = 1

Heatmap showing a multivariate (bivariate) independent stable distribution with α = 2.

Discrete

If the spectral measure is discrete with mass $λ j$ at $s_j \in \mathbb{S},j=1,\ldots,m$ the characteristic function is

$E \exp(i u^T X)= \exp\left\{-\sum_{j=1}^m \omega(u^Ts_j|\alpha,1)\gamma_j^\alpha +i u^T \delta)\right\}$

Linear properties

if $X \sim S(\alpha, \beta(\cdot), \gamma(\cdot), \delta(\cdot))$ is d-dim, and A is a m x d matrix, $b \in \mathbb{R}^m$ then AX + b is m dim. $α$ -stable with scale function $\gamma(A^T\cdot)$ , skewness function $\beta(A^T\cdot)$ , and location function $\delta(A^T\cdot) + b^T\cdot$

Inference in the independent component model

Recently^[4] it was shown how to compute inference in closed-form in a linear model (or equivalently a factor analysis model),involving independent component models.

More specifically, let $X_i \sim S(\alpha, \beta_{x_i}, \gamma_{x_i}, \delta_{x_i}), i=1,\ldots,n$ be a set of i.i.d. unobserved univariate drawn from a stable distribution. Given a known linear relation matrix A of size $n \times n$ , the observation $Y_i = \sum_{i=1}^n A_{ij}X_j$ are assumed to be distributed as a convolution of the hidden factors $X i$ . $Y_i = S(\alpha, \beta_{y_i}, \gamma_{y_i}, \delta_{y_i})$ . The inference task is to compute the most probable $X i$ , given the linear relation matrix A and the observations $Y i$ . This task can be computed in closed-form in O(n³).

An application for this construction is multiuser detection with stable, non-Gaussian noise.

Resources

Mark Veillette's stable distribution matlab package http://math.bu.edu/people/mveillet/research.html
The plots in this page where plotted using Danny Bickson's inference in linear-stable model Matlab package: http://www.cs.cmu.edu/~bickson/stable

Notes

^ J. Nolan, Multivariate stable densities and distribution functions: general and elliptical case, BundesBank Conference, Eltville, Germany, 11 November 2005. See also http://academic2.american.edu/~jpnolan/stable/stable.html
^ Feldheim, E. (1937). Etude de la stabilit´e des lois de probabilit´e . Ph. D. thesis, Facult´e des Sciences de Paris, Paris, France.
^ User manual for STABLE 5.1 Matlab version, Robust Analysis Inc., http://www.RobustAnalysis.com
^ D. Bickson and C. Guestrin. Inference in linear models with multivariate heavy-tails. In Neural Information Processing Systems (NIPS) 2010, Vancouver, Canada, Dec. 2010. http://www.cs.cmu.edu/~bickson/stable/

Probability distributions

Discrete univariate with finite support

Benford Bernoulli Beta-binomial binomial categorical hypergeometric Poisson binomial Rademacher discrete uniform Zipf Zipf-Mandelbrot

Discrete univariate with infinite support

beta negative binomial Boltzmann Conway–Maxwell–Poisson discrete phase-type extended negative binomial Gauss–Kuzmin geometric logarithmic negative binomial parabolic fractal Poisson Skellam Yule–Simon zeta

Continuous univariate supported on a bounded interval, e.g. [0,1]

Arcsine ARGUS Balding-Nichols Bates Beta Noncentral beta Irwin–Hall Kumaraswamy logit-normal raised cosine triangular U-quadratic uniform Wigner semicircle

Continuous univariate supported on a semi-infinite interval, usually [0,∞)

generalized inverse Gaussian

half-logistic

half-normal

Hotelling's T-squared

hyper-exponential

hypoexponential

inverse chi-squared (scaled-inverse-chi-squared)

noncentral chi-squared

Pareto

phase-type

Rayleigh

relativistic Breit–Wigner

Continuous univariate supported on the whole real line (−∞, ∞)

Cauchy exponential power Fisher's z generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Landau Laplace Linnik logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel variance-gamma Voigt

Continuous univariate with support whose type varies

generalized extreme value generalized Pareto Tukey lambda q-Gaussian q-exponential shifted log-logistic

Mixed continuous-discrete univariate distributions

rectified Gaussian

Multivariate (joint)

Discrete Ewens multinomial multivariate Pólya negative multinomial Continuous Dirichlet Generalized Dirichlet multivariate normal Multivariate stable multivariate Student normal-scaled inverse gamma normal-gamma Matrix-valued inverse-Wishart matrix normal Wishart

Directional

Univariate (circular) directional Circular uniform univariate von Mises wrapped normal wrapped Cauchy wrapped exponential wrapped Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham

Degenerate and singular

Degenerate discrete degenerate Dirac delta function Singular Cantor

Families

Circular compound Poisson elliptical exponential natural exponential location-scale maximum entropy mixture Pearson Tweedie wrapped

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