Multivariate stable distribution

multivariate stable
Probability density function Heatmap showing a Multivariate (bivariate) stable distribution with α = 1.1
Parameters	$\alpha \in (0,2]$ — exponent $\delta \in \mathbb{R}^d$ - shift/location vector $\Lambda(s)$ - a spectral finite measure on the sphere
Support	$u \in \mathbb{R}^d$
PDF	(no analytic expression)
CDF	(no analytic expression)
Variance	Infinite when $\alpha < 2$
CF	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. 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 $\mathbb{S}$ be the unit sphere in $\mathbb R^d : \mathbb{S} = \{u \in \mathbb R^d : |u| = 1\}$ . A random vector, $X$ , has a multivariate stable distribution - denoted as $X \sim S(\alpha, \Lambda, \delta)$ -, if the joint characteristic function of $X$ is^[1]

\operatorname{E} \exp(i u^T X) = \exp \left\{-\int \limits_{s \in \mathbb S}\left\{|u^Ts|^\alpha + i \nu (u^Ts, \alpha) \right\} \, \Lambda(ds) + i u^T\delta\right\}

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

\nu(y,\alpha) =\begin{cases} -\mathbf{sign}(y) \tan(\pi \alpha / 2)|y|^\alpha & \alpha \ne 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 $\Lambda$ (a finite measure on $\mathbb S$ ) and a shift vector $\delta \in \mathbb R^d$ .

Parametrization using projections

Another way to describe a stable random vector is in terms of projections. For any vector $u$ , the projection $u^TX$ is univariate $\alpha-$ stable with some skewness $\beta(u)$ , scale $\gamma(u)$ and some shift $\delta(u)$ . The notation $X \sim S(\alpha,\beta(\cdot),\gamma(\cdot),\delta(\cdot))$ is used if X is stable with $u^TX \sim s(\alpha,\beta(\cdot),\gamma(\cdot),\delta(\cdot))$ for every $u \in \mathbb R^d$ . 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 special cases where the multivariate characteristic function takes a simpler form. Define the characteristic function of a stable marginal as

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

Isotropic multivariate stable distribution

The characteristic function is $E \exp(i u^T X)=\exp\{-\gamma_0|u|^\alpha+i u^T \delta)\}$ The spectral measure is continuous and uniform, leading to radial/isotropic symmetry.^[3] For the multinormal case $\alpha=2$ , this corresponds to independent components, but so is not the case when $\alpha=2$ . Isotropy is a special case of ellipticity (see the next paragraph) – just take $\Sigma$ to be a multiple of the identity matrix.

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 $E \exp(i u^T X)=\exp\{-(u^T\Sigma u)^{\alpha/2}+i u^T \delta)\}$ for some shift vector $\delta \in R^d$ (equal to the mean when it exists) and some positive definite matrix $\Sigma$ (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: $E \exp(i u^T X)=\exp\{-(u^T\Sigma u)+i u^T \delta)\}$ obtained when α = 2.

Independent components

The marginals are independent with $X_j \sim S(\alpha, \beta_j, \gamma_j, \delta_j)$ , 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\}

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 $\lambda_j$ 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. $\alpha$ -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 $X_i$ . $Y_i = S(\alpha, \beta_{y_i}, \gamma_{y_i}, \delta_{y_i})$ . The inference task is to compute the most probable $X_i$ , given the linear relation matrix A and the observations $Y_i$ . This task can be computed in closed-form in O(n³).

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

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: http://www.cs.cmu.edu/~bickson/stable

Notes

↑ J. Nolan, Multivariate stable densities and distribution functions: general and elliptical case, BundesBank Conference, Eltville, Germany, 11 November 2005. See also http://academic2.american.edu/~jpnolan/stable/stable.html
↑ Feldheim, E. (1937). Etude de la stabilité des lois de probabilité . Ph. D. thesis, Faculté des Sciences de Paris, Paris, France.
↑ User manual for STABLE 5.1 Matlab version, Robust Analysis Inc., http://www.RobustAnalysis.com
↑ 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/

Probability distributions

List of probability distributions

Discrete univariate with finite support	Benford Bernoulli Beta-binomial binomial categorical hypergeometric Poisson binomial Rademacher discrete uniform Zipf Zipf–Mandelbrot

Discrete univariate with infinite support	beta negative binomial Borel Conway–Maxwell–Poisson discrete phase-type Delaporte extended negative binomial Gauss–Kuzmin geometric logarithmic negative binomial parabolic fractal Poisson Skellam Yule–Simon zeta

Continuous univariate supported on a bounded interval	Arcsine ARGUS Balding–Nichols Bates Beta Beta rectangular Irwin–Hall Kumaraswamy logit-normal Noncentral beta raised cosine Reciprocal Triangular U-quadratic uniform Wigner semicircle

Continuous univariate supported on a semi-infinite interval	Benini Benktander 1st kind Benktander 2nd kind Beta prime Burr chi-squared chi Dagum Davis exponential-logarithmic Erlang exponential F folded normal Flory-Schulz Fréchet Gamma Gamma/Gompertz generalized inverse Gaussian Gompertz half-logistic half-normal Hotelling's T-squared hyper-Erlang hyperexponential hypoexponential inverse chi-squared scaled inverse chi-squared inverse Gaussian inverse gamma Kolmogorov Lévy log-Cauchy log-Laplace log-logistic log-normal matrix-exponential Maxwell–Boltzmann Maxwell–Jüttner Mittag-Leffler Nakagami noncentral chi-squared Pareto phase-type Poly-Weibull Rayleigh relativistic Breit–Wigner Rice shifted Gompertz truncated normal type-2 Gumbel Weibull Wilks's lambda

Continuous univariate supported on the whole real line	Cauchy exponential power Fisher's z generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson S_U Landau Laplace asymmetric Laplace logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel Tracy–Widom variance-gamma Voigt

Continuous univariate with support whose type varies	generalized extreme value generalized Pareto Tukey lambda q-Gaussian q-exponential q-Weibull shifted log-logistic

Mixed continuous-discrete univariate	rectified Gaussian

Multivariate (joint)	Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet generalized Dirichlet multivariate normal multivariate stable multivariate t normal-inverse-gamma normal-gamma Matrix-valued inverse matrix gamma inverse-Wishart matrix normal matrix t matrix gamma normal-inverse-Wishart normal-Wishart Wishart

Directional	Univariate (circular) directional Circular uniform univariate von Mises wrapped normal wrapped Cauchy wrapped exponential wrapped asymmetric Laplace wrapped Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham

Degenerate and singular	Degenerate Dirac delta function Singular Cantor

Families	Circular compound Poisson elliptical exponential natural exponential location-scale maximum entropy mixture Pearson Tweedie wrapped

This article is issued from Wikipedia - version of the Wednesday, March 02, 2016. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.