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Multivariate stable distribution
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<h2 id="definition">Definition</h2> <p>Let S {\displaystyle \mathbb {S} } be the unit sphere in R d : S = { u ∈ R d : | u | = 1 } {\displaystyle \mathbb {R} ^{d}\colon \mathbb {S} =\{u\in \mathbb {R} ^{d}\colon |u|=1\}} . A <a href="/facts/Random_vector/qMfooyVf">random vector</a>, X {\displaystyle X} , has a multivariate stable distribution - denoted as X ∼ S ( α , Λ , δ ) {\displaystyle X\sim S(\alpha ,\Lambda ,\delta )} -, if the joint characteristic function of X {\displaystyle X} is<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a> </p> E exp ( i u T X ) = exp { − ∫ s ∈ S { | u T s | α + i ν ( u T s , α ) } Λ ( d s ) + i u T δ } {\displaystyle \operatorname {E} \exp(iu^{T}X)=\exp \left\{-\int \limits _{s\in \mathbb {S} }\left\{|u^{T}s|^{\alpha }+i\nu (u^{T}s,\alpha )\right\}\,\Lambda (ds)+iu^{T}\delta \right\}} <p>where 0 < <i>α</i> < 2, and for y ∈ R {\displaystyle y\in \mathbb {R} } </p> ν ( y , α ) = { − s i g n ( y ) tan ( π α / 2 ) | y | α α ≠ 1 , ( 2 / π ) y ln | y | α = 1. {\displaystyle \nu (y,\alpha )={\begin{cases}-\mathbf {sign} (y)\tan(\pi \alpha /2)|y|^{\alpha }&\alpha \neq 1,\\(2/\pi )y\ln |y|&\alpha =1.\end{cases}}} <p>This is essentially the result of Feldheim,<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> that any stable random vector can be characterized by a spectral measure Λ {\displaystyle \Lambda } (a finite measure on S {\displaystyle \mathbb {S} } ) and a shift vector δ ∈ R d {\displaystyle \delta \in \mathbb {R} ^{d}} . </p> <h2 id="parametrization-using-projections">Parametrization using projections</h2> <p>Another way to describe a stable random vector is in terms of projections. For any vector u {\displaystyle u} , the projection u T X {\displaystyle u^{T}X} is univariate α − {\displaystyle \alpha -} stable with some skewness β ( u ) {\displaystyle \beta (u)} , scale γ ( u ) {\displaystyle \gamma (u)} and some shift δ ( u ) {\displaystyle \delta (u)} . The notation X ∼ S ( α , β ( ⋅ ) , γ ( ⋅ ) , δ ( ⋅ ) ) {\displaystyle X\sim S(\alpha ,\beta (\cdot ),\gamma (\cdot ),\delta (\cdot ))} is used if X is stable with u T X ∼ s ( α , β ( ⋅ ) , γ ( ⋅ ) , δ ( ⋅ ) ) {\displaystyle u^{T}X\sim s(\alpha ,\beta (\cdot ),\gamma (\cdot ),\delta (\cdot ))} for every u ∈ R d {\displaystyle u\in \mathbb {R} ^{d}} . This is called the projection parameterization. </p><p>The spectral measure determines the projection parameter functions by: </p> γ ( u ) = ( ∫ s ∈ S | u T s | α Λ ( d s ) ) 1 / α {\displaystyle \gamma (u)={\Bigl (}\int _{s\in \mathbb {S} }|u^{T}s|^{\alpha }\Lambda (ds){\Bigr )}^{1/\alpha }} β ( u ) = ∫ s ∈ S | u T s | α s i g n ( u T s ) Λ ( d s ) / γ ( u ) α {\displaystyle \beta (u)=\int _{s\in \mathbb {S} }|u^{T}s|^{\alpha }\mathbf {sign} (u^{T}s)\Lambda (ds)/\gamma (u)^{\alpha }} δ ( u ) = { u T δ α ≠ 1 u T δ − ∫ s ∈ S π 2 u T s ln | u T s | Λ ( d s ) α = 1 {\displaystyle \delta (u)={\begin{cases}u^{T}\delta &\alpha \neq 1\\u^{T}\delta -\int _{s\in \mathbb {S} }{\tfrac {\pi }{2}}u^{T}s\ln |u^{T}s|\Lambda (ds)&\alpha =1\end{cases}}} <h2 id="special-cases">Special cases</h2> <p>There are special cases where the multivariate <a href="/facts/Characteristic_function_(probability_theory)/rl3uAGKM">characteristic function</a> takes a simpler form. Define the characteristic function of a stable marginal as </p> ω ( y | α , β ) = { | y | α [ 1 − i β ( tan π α 2 ) s i g n ( y ) ] α ≠ 1 | y | [ 1 + i β 2 π s i g n ( y ) ln | y | ] α = 1 {\displaystyle \omega (y|\alpha ,\beta )={\begin{cases}|y|^{\alpha }\left[1-i\beta (\tan {\tfrac {\pi \alpha }{2}})\mathbf {sign} (y)\right]&\alpha \neq 1\\|y|\left[1+i\beta {\tfrac {2}{\pi }}\mathbf {sign} (y)\ln |y|\right]&\alpha =1\end{cases}}} <h3>Isotropic multivariate stable distribution</h3> <p>The characteristic function is E exp ( i u T X ) = exp { − γ 0 α | u | α + i u T δ ) } {\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.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> For the multinormal case α = 2 {\displaystyle \alpha =2} , this corresponds to independent components, but so is not the case when α < 2 {\displaystyle \alpha <2} . Isotropy is a special case of ellipticity (see the next paragraph) – just take Σ {\displaystyle \Sigma } to be a multiple of the identity matrix. </p> <h3>Elliptically contoured multivariate stable distribution</h3> <p>The <a href="/facts/Elliptical_distribution/IrHSRhQ8">elliptically contoured</a> multivariate stable distribution is a special symmetric case of the multivariate stable distribution. If <i>X</i> is <i>α</i>-stable and elliptically contoured, then it has joint <a href="/facts/Characteristic_function_(probability_theory)/rl3uAGKM">characteristic function</a> E exp ( i u T X ) = exp { − ( u T Σ u ) α / 2 + i u T δ ) } {\displaystyle E\exp(iu^{T}X)=\exp\{-(u^{T}\Sigma u)^{\alpha /2}+iu^{T}\delta )\}} for some shift vector δ ∈ R d {\displaystyle \delta \in R^{d}} (equal to the mean when it exists) and some positive definite matrix Σ {\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 <a href="/facts/Multivariate_normal_distribution/2Xfegqz2">multivariate normal distribution</a>: E exp ( i u T X ) = exp { − ( u T Σ u ) + i u T δ ) } {\displaystyle E\exp(iu^{T}X)=\exp\{-(u^{T}\Sigma u)+iu^{T}\delta )\}} obtained when <i>α</i> = 2. </p> <h3>Independent components</h3> <p>The marginals are independent with X j ∼ S ( α , β j , γ j , δ j ) {\displaystyle X_{j}\sim S(\alpha ,\beta _{j},\gamma _{j},\delta _{j})} , then the characteristic function is </p> E exp ( i u T X ) = exp { − ∑ j = 1 m ω ( u j | α , β j ) γ j α + i u T δ ) } {\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\}} <p>Observe that when <i>α</i> = 2 this reduces again to the multivariate normal; note that the iid case and the isotropic case do not coincide when <i>α</i> < 2. Independent components is a special case of discrete spectral measure (see next paragraph), with the spectral measure supported by the standard unit vectors. </p> <table><tbody><tr><td></td><td></td></tr></tbody></table> <h3>Discrete</h3> <p>If the spectral measure is discrete with mass λ j {\displaystyle \lambda _{j}} at s j ∈ S , j = 1 , … , m {\displaystyle s_{j}\in \mathbb {S} ,j=1,\ldots ,m} the characteristic function is </p> E exp ( i u T X ) = exp { − ∑ j = 1 m ω ( u T s j | α , 1 ) λ j α + i u T δ ) } {\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\}} <h2 id="linear-properties">Linear properties</h2> <p>If X ∼ S ( α , β ( ⋅ ) , γ ( ⋅ ) , δ ( ⋅ ) ) {\displaystyle X\sim S(\alpha ,\beta (\cdot ),\gamma (\cdot ),\delta (\cdot ))} is <i>d</i>-dimensional, <i>A</i> is an <i>m</i> x <i>d</i> matrix, and b ∈ R m , {\displaystyle b\in \mathbb {R} ^{m},} then <i>AX + b</i> is <i>m</i>-dimensional α {\displaystyle \alpha } -stable with scale function γ ( A T ⋅ ) , {\displaystyle \gamma (A^{T}\cdot ),} skewness function β ( A T ⋅ ) , {\displaystyle \beta (A^{T}\cdot ),} and location function δ ( A T ⋅ ) + b T . {\displaystyle \delta (A^{T}\cdot )+b^{T}.} </p> <h2 id="inference-in-the-independent-component-model">Inference in the independent component model</h2> <p>Recently<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> it was shown how to compute inference in closed-form in a linear model (or equivalently a <a href="/facts/Factor_analysis/LT6B9I7D">factor analysis</a> model), involving independent component models. </p><p>More specifically, let X i ∼ S ( α , β x i , γ x i , δ x i ) , i = 1 , … , n {\displaystyle 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 <a href="/facts/Stable_distribution/rvtRmhlz">stable distribution</a>. Given a known linear relation matrix A of size n × n {\displaystyle n\times n} , the observation Y i = ∑ i = 1 n A i j X j {\displaystyle Y_{i}=\sum _{i=1}^{n}A_{ij}X_{j}} are assumed to be distributed as a convolution of the hidden factors X i {\displaystyle X_{i}} . Y i = S ( α , β y i , γ y i , δ y i ) {\displaystyle Y_{i}=S(\alpha ,\beta _{y_{i}},\gamma _{y_{i}},\delta _{y_{i}})} . The inference task is to compute the most probable X i {\displaystyle X_{i}} , given the linear relation matrix A and the observations Y i {\displaystyle Y_{i}} . This task can be computed in closed-form in O(<i>n</i>3). </p><p>An application for this construction is <a href="/facts/Multiuser_detection/CQtNs500">multiuser detection</a> with stable, non-Gaussian noise. </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Multivariate_Cauchy_distribution/3HS6D3Xk">Multivariate Cauchy distribution</a></li> <li><a href="/facts/Multivariate_normal_distribution/2Xfegqz2">Multivariate normal distribution</a></li></ul> <h2 id="resources">Resources</h2> <ul><li>Mark Veillette's stable distribution matlab package <a href="http://www.mathworks.com/matlabcentral/fileexchange/37514">http://www.mathworks.com/matlabcentral/fileexchange/37514</a></li> <li>The plots in this page where plotted using Danny Bickson's inference in linear-stable model Matlab package: <a href="https://www.cs.cmu.edu/~bickson/stable">https://www.cs.cmu.edu/~bickson/stable</a></li></ul> <h2 id="notes">Notes</h2> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>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 <a href="http://academic2.american.edu/~jpnolan/stable/stable.html" target="_blank">http://academic2.american.edu/~jpnolan/stable/stable.html</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Feldheim, E. (1937). Etude de la stabilité des lois de probabilité . Ph. D. thesis, Faculté des Sciences de Paris, Paris, France. <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>User manual for STABLE 5.1 Matlab version, Robust Analysis Inc., http://www.RobustAnalysis.com <a href="http://www.RobustAnalysis.com" target="_blank">http://www.RobustAnalysis.com</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>D. Bickson and C. Guestrin. Inference in linear models with multivariate heavy-tails. In Neural Information Processing Systems (NIPS) 2010, Vancouver, Canada, Dec. 2010. https://www.cs.cmu.edu/~bickson/stable/ <a href="https://www.cs.cmu.edu/~bickson/stable/" target="_blank">https://www.cs.cmu.edu/~bickson/stable/</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> </ol>