Multivariate stable distribution

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.

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 ${\displaystyle \Lambda (s)}$ - 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 ${\displaystyle \alpha <2}$
CF	see text

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.

Definition

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

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

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

${\displaystyle \nu (y,\alpha )={\begin{cases}-\operatorname {sign} (y)\tan(\pi \alpha /2)|y|^{\alpha }&\alpha \neq 1,\\(2/\pi )y\ln |y|&\alpha =1.\end{cases}}}$

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

Parametrization using projections

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 if X 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:

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

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

${\displaystyle \delta (u)={\begin{cases}u^{T}\delta &\alpha \neq 1\\u^{T}\delta -\int _{\mathbb {S} }{\frac {\pi }{2}}u^{T}s\ln |u^{T}s|\,\Lambda (ds)&\alpha =1\end{cases}}}$

Special cases

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

${\displaystyle \omega (y|\alpha ,\beta )={\begin{cases}|y|^{\alpha }\left[1-i\beta (\tan {\frac {\pi \alpha }{2}})\operatorname {sign} (y)\right]&\alpha \neq 1\\|y|\left[1+i\beta {\tfrac {2}{\pi }}\operatorname {sign} (y)\ln |y|\right]&\alpha =1\end{cases}}}$

Isotropic multivariate stable distribution

The characteristic function is ${\displaystyle E\exp(iu^{T}X)=\exp\{-\gamma _{0}^{\alpha }|u|^{\alpha }+iu^{T}\delta )\}}$ The spectral measure is continuous and uniform, leading to radial/isotropic symmetry.^[3] For the multinormal case ${\displaystyle \alpha =2}$, this corresponds to independent components, but so is not the case when ${\displaystyle \alpha <2}$. Isotropy is a special case of ellipticity (see the next paragraph) – just take ${\displaystyle \Sigma }$ to be a multiple of the identity matrix.

Elliptically contoured multivariate stable distribution

The elliptically contoured multivariate 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 ${\displaystyle E\exp(iu^{T}X)=\exp\{-(u^{T}\Sigma u)^{\alpha /2}+iu^{T}\delta )\}}$ for some shift vector ${\displaystyle \delta \in R^{d}}$ (equal to the mean when it exists) and some positive definite matrix ${\displaystyle \Sigma }$ (akin to a correlation matrix, although the usual definition of correlation fails to be meaningful). Note the relation to characteristic function of the multivariate normal distribution: ${\displaystyle E\exp(iu^{T}X)=\exp\{-(u^{T}\Sigma u)+iu^{T}\delta )\}}$ obtained when α = 2.

Independent components

The marginals are independent with ${\displaystyle X_{j}\sim S(\alpha ,\beta _{j},\gamma _{j},\delta _{j})}$, then the characteristic function is

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

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

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 ${\displaystyle \lambda _{j}}$ at ${\displaystyle s_{j}\in \mathbb {S} ,j=1,\ldots ,m}$ the characteristic function is

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

Linear properties

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

Inference in the independent component model

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

More specifically, let ${\displaystyle X_{i}\sim S(\alpha ,\beta _{x_{i}},\gamma _{x_{i}},\delta _{x_{i}})}$ ${\displaystyle (i=1,\ldots ,n)}$ be a family of i.i.d. unobserved univariates drawn from a stable distribution. Given a known linear relation matrix A 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 matrix A and the observations ${\displaystyle Y_{i}}$. This task can be computed in closed-form in O(n³).

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

See also

• Multivariate Cauchy distribution
• Multivariate normal distribution

Resources

• Mark Veillette's stable distribution matlab package http://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