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

Themultivariate stable distribution is a multivariateprobability distribution that is a multivariate generalisation of the univariatestable distribution. The multivariate stable distribution defines linear relations betweenstable distribution marginals.^{[clarification needed]} In the same way as for the univariate case, the distribution is defined in terms of itscharacteristic function.

The multivariate stable distribution can also be thought as an extension of themultivariate 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 themultivariate normal distribution is symmetric.

Let${\displaystyle \mathbb{S}}$ be the Euclidean unit sphere in${\displaystyle {\mathbb{R}}^d}$, that is,${\displaystyle \mathbb{S}=\{u\in {\mathbb{R}}^d:|u|=1\}}$. Arandom vector${\displaystyle X}$ has a multivariate stable distribution—denoted as${\displaystyle X\sim S(\alpha ,\mathrm{\Lambda },\delta )}$—, if the joint characteristic function of${\displaystyle X}$ is^[1]

${\displaystyle \mathrm{E}\exp (i{u}^{T}X)=\exp \left\{-\underset{\mathbb{S}}{\int }\left\{|u^{T}s{|}^{\alpha }+i\nu (u^{T}s,\alpha )\right\}\mathrm{\Lambda }(ds)+i{u}^T\delta \right\}}$,

where 0 < α < 2, and for${\displaystyle y\in \mathbb{R}}$

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

Another way to describe a stable random vector is in terms of projections. For any vector${\displaystyle u}$ the projection${\displaystyle u^{T}X}$ is univariate${\displaystyle \alpha }$-stable with some skewness${\displaystyle \beta (u)}$, scale${\displaystyle \gamma (u)}$, and some shift${\displaystyle \delta (u)}$. The notation${\displaystyle X\sim S(\alpha ,\beta ,\gamma ,\delta )}$ is used ifX is stable with${\displaystyle u^{T}X\sim s(\alpha ,\beta (u),\gamma (u),\delta (u))}$for every${\displaystyle u\in {\mathbb{R}}^d}$. This is called the projection parametrization.

The spectral measure determines the projection parameter functions by:

${\displaystyle \gamma (u)=({\int }_{\mathbb{S}}|u^{T}s{|}^{\alpha }\mathrm{\Lambda }(ds){)}^{1/\alpha }}$

${\displaystyle \beta (u)=\gamma (u{)}^{-\alpha }{\int }_{\mathbb{S}}|u^{T}s{|}^{\alpha }\mathrm{sign}(u^{T}s)\mathrm{\Lambda }(ds)}$

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

Here the characteristic function is${\displaystyle E\exp (i{u}^{T}X)=\exp \{-{\gamma }_0^{\alpha }|u{|}^{\alpha }+i{u}^T\delta )\}}$. The spectral measure is a scalar multiple of the uniform distribution on the sphere, leading to radial/isotropic symmetry.^[3]For the Gaussian case${\displaystyle \alpha =2}$ this corresponds to independent components, but this is not the case when. Isotropy is a special case of ellipticity (see the next paragraph)—just take${\displaystyle \mathrm{\Sigma }}$ to be a multiple of the identity matrix.

Theelliptically contoured multivariate stable distribution is a special symmetric case of the multivariate stable distribution.X isα-stable and elliptically contoured iff it has jointcharacteristic function${\displaystyle E\exp (i{u}^{T}X)=\exp \{-(u^T\mathrm{\Sigma }u{)}^{\alpha /2}+i{u}^T\delta )\}}$ for some shift vector${\displaystyle \delta \in {\mathbb{R}}^d}$ (equal to the mean when it exists) and some positive semidefinite matrix${\displaystyle \mathrm{\Sigma }}$ (akin to a correlation matrix, although the usual definition of correlation fails to be meaningful).Note the relation to the characteristic function of themultivariate normal distribution:${\displaystyle E\exp (i{u}^{T}X)=\exp \{-(u^T\mathrm{\Sigma }u)+i{u}^T\delta )\}}$, obtained whenα = 2.

The marginals are independent with${\displaystyle X_j\sim S(\alpha ,{\beta }_j,{\gamma }_j,{\delta }_j)}$ iff thecharacteristic function is

${\displaystyle E\exp (i{u}^{T}X)=\exp \left\{-\sum _{j=1}^m\omega (u_j|\alpha ,{\beta }_j){\gamma }_j^{\alpha }+i{u}^T\delta \right\}}$.

Observe that whenα = 2 this reduces again to the multivariate normal; note that the i.i.d. case and the isotropic case do not coincide whenα < 2.Independent components is a special case of a discrete spectral measure (see next paragraph), with the spectral measure supported by the standard unit vectors.

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

${\displaystyle E\exp (i{u}^{T}X)=\exp \left\{-\sum _{j=1}^m\omega (u^T{s}_j|\alpha ,1){\lambda }_j^{\alpha }+i{u}^T\delta \right\}}$.

If${\displaystyle X\sim S(\alpha ,\beta (\cdot ),\gamma (\cdot ),\delta (\cdot ))}$ isd-dimensional${\displaystyle \alpha }$-stable,A is anm ×d matrix, and${\displaystyle b\in {\mathbb{R}}^m,}$ thenAX + b ism-dimensional${\displaystyle \alpha }$-stable with scale function${\displaystyle \gamma \circ A^T}$, skewness function${\displaystyle \beta \circ A^T}$, and location function${\displaystyle \delta \circ A^T+b^T}$.

Bickson and Guestrin have shown how to compute inference in closed form in a linear model (or equivalently afactor analysis model), involving independent-component models.^[4]

More specifically, let${\displaystyle X_i\sim S(\alpha ,{\beta }_{x_i},{\gamma }_{x_i},{\delta }_{x_i})}$${\displaystyle (i=1,\dots ,n)}$ be a family of i.i.d. unobserved univariates drawn from astable distribution. Given a known linear relation matrixA of size${\displaystyle n\times n}$, the observations${\displaystyle Y_i=\sum _{i=1}^n{A}_{ij}{X}_j}$ are assumed to be distributed as a convolution of the hidden factors${\displaystyle X_i}$, hence${\displaystyle Y_i=S(\alpha ,{\beta }_{y_i},{\gamma }_{y_i},{\delta }_{y_i})}$. The inference task is to compute the most likely${\displaystyle X_i}$, given the linear relation matrixA and the observations${\displaystyle Y_i}$. This task can be computed in closed form in O(n³).

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

• Multivariate Cauchy distribution
• Multivariate normal distribution

• Mark Veillette's stable distribution matlab packagehttp://www.mathworks.com/matlabcentral/fileexchange/37514
• The plots in this page where plotted using Danny Bickson's inference in linear-stable model Matlab package:https://www.cs.cmu.edu/~bickson/stable

• Multivariate continuous distributions
• Probability distributions with non-finite variance

• Articles with short description
• Short description is different from Wikidata
• Wikipedia articles needing clarification from January 2011