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. 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.

Definition

Let be the Euclidean unit sphere in, that is,. A random vector has a multivariate stable distribution—denoted as —, if the joint characteristic function of is
where 0 < α < 2, and for
This is essentially the result of Feldheim, that any stable random vector can be characterized by a spectral measure and a shift vector.

Parametrization using projections

Another way to describe a stable random vector is in terms of projections. For any vector the projection is univariate -stable with some skewness, scale, and some shift. The notation is used if X is stable with
for every. This is called the projection parametrization.
The spectral measure determines the projection parameter functions by:

Special cases

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

Isotropic multivariate stable distribution

Here the characteristic function is
.
The spectral measure is a scalar multiple of the uniform distribution on the sphere, leading to radial/isotropic symmetry.
For the Gaussian case this corresponds to independent components, but this is not the case when. Isotropy is a special case of ellipticity —just take 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.
X is α-stable and elliptically contoured iff it has joint characteristic function

for some shift vector and some positive semidefinite matrix .
Note the relation to the characteristic function of the multivariate normal distribution:, obtained when α = 2.

Independent components

The marginals are independent with iff the
characteristic function is
Observe that when α = 2 this reduces again to the multivariate normal; note that the i.i.d. case and the isotropic case do not coincide when α < 2.
Independent components is a special case of a discrete spectral measure, with the spectral measure supported by the standard unit vectors.

Discrete

If the spectral measure is discrete with mass at,,
the characteristic function is

Linear properties

If is d-dimensional -stable, A is an m × d matrix, and
then AX + b is m-dimensional -stable with scale function, skewness function, and location function.

Inference in the independent-component model

Bickson and Guestrin have shown how to compute inference in closed form in a linear model, involving independent-component models.
More specifically, let be a family of i.i.d. unobserved univariates drawn from a stable distribution. Given a known linear relation matrix A of size, the observations are assumed to be distributed as a convolution of the hidden factors, hence. The inference task is to compute the most likely, given the linear relation matrix A and the observations. This task can be computed in closed form in O.
An application for this construction is multiuser detection with stable, non-Gaussian noise.

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