Multivariate stable distribution

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

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 XS(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 uTX 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 uTX 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 Eexp(iuTX) = exp{ − (uTΣu)α / 2 + iuTδ)} for some positive definite matrix Σ and shift vector $\delta \in R^d$. Note the relation to characteristic function of the multivariate normal distribution: Eexp(iuTX) = exp{ − (uTΣu) + iuTδ)}. In other words, when α = 2 we get the characteristic function of the multivariate normal distribution.

Independent components

The marginals are independent with XjS(α,βjjj), 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 λj 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 Xi. $Y_i = S(\alpha, \beta_{y_i}, \gamma_{y_i}, \delta_{y_i})$. The inference task is to compute the most probable Xi, given the linear relation matrix A and the observations Yi. This task can be computed in closed-form in O(n3).

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

Notes

1. ^ Multivariate stable densities and distribution functions: general and elliptical case, BundesBank Conference, Eltville, Germany, 11 November 2005
2. ^ Feldheim, E. (1937). Etude de la stabilit´e des lois de probabilit´e . Ph. D. thesis, Facult´e des Sciences de Paris, Paris, France.
3. ^ User manual for STABLE 5.1 Matlab version, Robust Analytics inc.[Full citation needed]
4. ^ 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/

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