The multivariate stable distribution is a multivariate probability distribution

Probability distribution

In probability theory, a probability mass, probability density, or probability distribution is a function that describes the probability of a random variable taking certain values....

 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

Characteristic function

In mathematics, characteristic function can refer to any of several distinct concepts:* The most common and universal usage is as a synonym for indicator function, that is the function* In probability theory, the characteristic function of any probability distribution on the real line is given by...

.

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 S be the unit sphere in . For a random variable

Random variable

In probability and statistics, a random variable or stochastic variable is, roughly speaking, a variable whose value results from a measurement on some type of random process. Formally, it is a function from a probability space, typically to the real numbers, which is measurable functionmeasurable...

, X, it has a multivariate stable distribution and the notation is used, if the joint characteristic function of is

where 0 < α < 2, and

This is essentially the result of Feldheim, that any stable random vector can be characterized by a spectral measure (a finite measure on ) and a shift vector .

Parametrization using projections

Another way to describe a stable random vector is in terms of projections. For any vector u, the projection is univariate stable with some skewness , scale and some shift . The notation is used if is stable with


for every . This is called the projection parameterization.

The spectral measure determines the projection parameter functions by:

Special cases

There are four special cases where the multivariate characteristic function

Characteristic function

In mathematics, characteristic function can refer to any of several distinct concepts:* The most common and universal usage is as a synonym for indicator function, that is the function* In probability theory, the characteristic function of any probability distribution on the real line is given by...

 takes a simpler form. Define the characteristic function of a stable marginal as

Isotropic multivariate stable distribution

The characteristic function is

The spectral measure is continuous and uniform, leading to radial/isotropic symmetry.

Elliptically contoured multivariate stable distribution

Elliptically contoured m.v. stable distribution is a special symmetric case of the multivariate stable distribution.
If X is -stable and elliptically contoured, then it has joint characteristic function

Characteristic function

In mathematics, characteristic function can refer to any of several distinct concepts:* The most common and universal usage is as a synonym for indicator function, that is the function* In probability theory, the characteristic function of any probability distribution on the real line is given by...

for some positive definite matrix and shift vector .
Note the relation to characteristic function of the multivariate normal distribution: .
In other words, when α = 2 we get the characteristic function of the multivariate normal distribution.

Independent components

The marginals are independent with , then the
characteristic function is

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 at
the characteristic function is

Linear properties

if is d-dim, and A is a m x d matrix,
then AX + b is m dim. -stable with scale function
, skewness function ,
and location function

Inference in the independent component model

Recently it was shown how to compute inference in closed-form in a linear model (or equivalently a factor analysis

Factor analysis

Factor analysis is a statistical method used to describe variability among observed, correlated variables in terms of a potentially lower number of unobserved, uncorrelated variables called factors. In other words, it is possible, for example, that variations in three or four observed variables...

 model),involving independent component models.

More specifically, let be a set of i.i.d. unobserved univariate drawn from a stable distribution. Given a known linear relation matrix A of size , the observation are assumed to be distributed as a convolution of the hidden factors . . The inference task is to compute the most probable , given the linear relation matrix A and the observations . This task can be computed in closed-form in O(n³).

An application for this construction is multiuser detection

Multiuser detection

Multiuser detection is one of the receiver design technology for detecting desired signal from interference and noise. Traditionally single-user receiver is known suffering from the so-called near-far problem, where a near-by or strong signal source may block the signal receiption of far-away or...

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

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