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| | | I'm a 50 years old and work at the high school ([http://www.encyclopedia.com/searchresults.aspx?q=Religious Religious] Studies).<br>In my [http://Www.ehow.com/search.html?s=spare+time spare time] I teach myself Japanese. I've been twicethere and look forward to go there sometime near future. I love to read, preferably on my kindle. I like to watch Arrested Development and Arrested Development as well as documentaries about anything technological. I enjoy Conlanging.<br><br>Feel free to visit my web blog :: [https://www.scribd.com/doc/243681617/Daftar-SMS-Nusa-085739687809 smsnusa money game] |
| In [[statistics]], the '''matrix t-distribution''' (or '''matrix variate t-distribution''') is the generalization of the [[multivariate t-distribution]] from vectors to [[matrix (mathematics)|matrices]].<ref>Zhu, Shenghuo and Kai Yu and Yihong Gong (2007). [http://books.nips.cc/papers/files/nips20/NIPS2007_0896.pdf "Predictive Matrix-Variate ''t'' Models."] In J.C. Platt, D. Koller, Y. Singer, and S. Roweis, editors, ''NIPS '07: Advances in Neural Information Processing Systems'' 20, pages 1721-1728. MIT Press, Cambridge, MA, 2008. The notation is changed a bit in this article for consistency with the [[matrix normal distribution]] article.</ref> The matrix t-distribution shares the same relationship with the multivariate t-distribution that the [[matrix normal distribution]] shares with the [[multivariate normal distribution]].{{clarify|date=May 2012}} For example, the matrix t-distribution is the [[compound distribution]] that results from sampling from a matrix normal distribution having sampled the covariance matrix of the matrix normal from an [[inverse Wishart distribution]].{{cn|date=July 2012}}
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| In a [[Bayesian multivariate linear regression|Bayesian analysis]] of a [[multivariate linear regression]] model based on the matrix normal distribution, the matrix t-distribution is the [[posterior predictive distribution]].
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| ==Definition==
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| {{Probability distribution|
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| name =Matrix t|
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| type =density|
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| pdf_image =|
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| cdf_image =|
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| notation =<math>{\rm T}_{n,p}(\nu,\mathbf{M},\boldsymbol\Sigma, \boldsymbol\Omega)</math>|
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| parameters =
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| <math>\mathbf{M}</math> [[location parameter|location]] ([[real number|real]] <math>n\times p</math> [[matrix (mathematics)|matrix]])<br/>
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| <math>\boldsymbol\Omega</math> [[scale matrix|scale]] ([[positive-definite matrix|positive-definite]] [[real number|real]] <math>p\times p</math> [[matrix (mathematics)|matrix]])<br/>
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| <math>\boldsymbol\Sigma</math> [[scale matrix|scale]] ([[positive-definite matrix|positive-definite]] real <math>n\times n</math> [[matrix (mathematics)|matrix]]) <br/>
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| <math>\nu</math> [[degrees of freedom (statistics)|degrees of freedom]] |
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| support =<math>\mathbf{X} \in\mathbb{R}^{n\times p}</math>|
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| pdf =<math>
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| \frac{\Gamma_p\left(\frac{\nu+n+p-1}{2}\right)}{(\pi)^\frac{np}{2} \Gamma_p\left(\frac{\nu+p-1}{2}\right)} |\boldsymbol\Omega|^{-\frac{n}{2}} |\boldsymbol\Sigma|^{-\frac{p}{2}}</math>
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| :<math>\times \left|\mathbf{I}_n + \boldsymbol\Sigma^{-1}(\mathbf{X} - \mathbf{M})\boldsymbol\Omega^{-1}(\mathbf{X}-\mathbf{M})^{\rm T}\right|^{-\frac{\nu+n+p-1}{2}}
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| </math>
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| cdf =No analytic expression|
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| mean =<math>\mathbf{M}</math> if <math>\nu + p - n > 1</math>, else undefined|
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| mode =<math>\mathbf{M}</math>|
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| variance =<math>\frac{\boldsymbol\Sigma \otimes \boldsymbol\Omega}{\nu+p-n-2}</math> if <math>\nu + p - n > 2</math>, else undefined|
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| kurtosis =|
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| entropy =|
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| mgf =|
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| char =see below|
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| }}
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| For a matrix t-distribution, the [[probability density function]] at the point <math>\mathbf{X}</math> of an <math>n\times p</math> space is
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| :<math> f(\mathbf{X} ; \nu,\mathbf{M},\boldsymbol\Sigma, \boldsymbol\Omega) = K
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| \times \left|\mathbf{I}_n + \boldsymbol\Sigma^{-1}(\mathbf{X} - \mathbf{M})\boldsymbol\Omega^{-1}(\mathbf{X}-\mathbf{M})^{\rm T}\right|^{-\frac{\nu+n+p-1}{2}},
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| </math>
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| where the constant of integration ''K'' is given by
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| :<math> K =
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| \frac{\Gamma_p\left(\frac{\nu+n+p-1}{2}\right)}{(\nu\pi)^\frac{np}{2} \Gamma_p\left(\frac{\nu+p-1}{2}\right)} |\boldsymbol\Omega|^{-\frac{n}{2}} |\boldsymbol\Sigma|^{-\frac{p}{2}}.</math>
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| Here <math>\Gamma_p</math> is the [[multivariate gamma function]].
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| The [[characteristic function (probability theory)|characteristic function]] and various other properties can be derived from the generalized matrix t-distribution (see below).
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| == Generalized matrix t-distribution ==
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| {{Probability distribution|
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| name =Generalized matrix t|
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| type =density|
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| pdf_image =|
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| cdf_image =|
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| notation =<math>{\rm T}_{n,p}(\alpha,\beta,\mathbf{M},\boldsymbol\Sigma, \boldsymbol\Omega)</math>|
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| parameters =
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| <math>\mathbf{M}</math> [[location parameter|location]] ([[real number|real]] <math>n\times p</math> [[matrix (mathematics)|matrix]])<br/>
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| <math>\boldsymbol\Omega</math> [[scale matrix|scale]] ([[positive-definite matrix|positive-definite]] [[real number|real]] <math>p\times p</math> [[matrix (mathematics)|matrix]])<br/>
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| <math>\boldsymbol\Sigma</math> [[scale matrix|scale]] ([[positive-definite matrix|positive-definite]] [[real number|real]] <math>n\times n</math> [[matrix (mathematics)|matrix]])<br/>
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| <math>\alpha > (p-1)/2</math> [[shape parameter]]<br />
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| <math>\beta > 0</math> [[scale parameter]] |
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| support =<math>\mathbf{X} \in\mathbb{R}^{n\times p}</math>|
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| pdf =<math>\frac{\Gamma_p(\alpha+n/2)}{(2\pi/\beta)^\frac{np}{2} \Gamma_p(\alpha)} |\boldsymbol\Omega|^{-\frac{n}{2}} |\boldsymbol\Sigma|^{-\frac{p}{2}}</math>
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| :<math>\times \left|\mathbf{I}_n + \frac{\beta}{2}\boldsymbol\Sigma^{-1}(\mathbf{X} - \mathbf{M})\boldsymbol\Omega^{-1}(\mathbf{X}-\mathbf{M})^{\rm T}\right|^{-(\alpha+n/2)}</math>
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| *<math>\Gamma_p</math> is the [[multivariate gamma function]].
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| cdf =No analytic expression|
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| mean =<math>\mathbf{M}</math>|
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| median =|
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| mode =|
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| variance =<math>\frac{2(\boldsymbol\Sigma \otimes \boldsymbol\Omega)}{\beta(2\alpha-n-1)}</math>|
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| skewness =|
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| kurtosis =|
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| entropy =|
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| mgf =|
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| char =see below|
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| }}
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| The '''generalized matrix t-distribution''' is a generalization of the matrix t-distribution with two parameters ''α'' and ''β'' in place of ''ν''.<ref name="iranmanesha">Iranmanesha, Anis, M. Arashib and S. M. M. Tabatabaeya (2010). [http://www.ijmsi.ir/browse.php?a_id=139&slc_lang=en&sid=1&ftxt=1 "On Conditional Applications of Matrix Variate Normal Distribution"]. ''Iranian Journal of Mathematical Sciences and Informatics'', 5:2, pp. 33–43.</ref>
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| This reduces to the standard matrix t-distribution with <math>\beta=2, \alpha=\frac{\nu+p-1}{2}.</math>
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| The generalized matrix t-distribution is the [[compound distribution]] that results from an infinite [[mixture density|mixture]] of a matrix normal distribution with an [[inverse multivariate gamma distribution]] placed over either of its covariance matrices.
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| ===Properties===
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| If <math>\mathbf{X} \sim {\rm T}_{n,p}(\alpha,\beta,\mathbf{M},\boldsymbol\Sigma, \boldsymbol\Omega)</math> then{{citation needed|date=May 2012}}
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| :<math>\mathbf{X}^{\rm T} \sim {\rm T}_{p,n}(\alpha,\beta,\mathbf{M}^{\rm T},\boldsymbol\Omega, \boldsymbol\Sigma).</math>
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| This makes use of the following:{{citation needed|date=May 2012}}
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| :<math>\det\left(\mathbf{I}_n + \frac{\beta}{2}\boldsymbol\Sigma^{-1}(\mathbf{X} - \mathbf{M})\boldsymbol\Omega^{-1}(\mathbf{X}-\mathbf{M})^{\rm T}\right) =</math>
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| ::<math>\det\left(\mathbf{I}_p + \frac{\beta}{2}\boldsymbol\Omega^{-1}(\mathbf{X}^{\rm T} - \mathbf{M}^{\rm T})\boldsymbol\Sigma^{-1}(\mathbf{X}^{\rm T}-\mathbf{M}^{\rm T})^{\rm T}\right) .</math>
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| If <math>\mathbf{X} \sim {\rm T}_{n,p}(\alpha,\beta,\mathbf{M},\boldsymbol\Sigma, \boldsymbol\Omega)</math> and <math>\mathbf{A}(n\times n)</math> and <math>\mathbf{B}(p\times p)</math> are [[nonsingular matrices]] then{{citation needed|date=May 2012}}
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| :<math>\mathbf{AXB} \sim {\rm T}_{n,p}(\alpha,\beta,\mathbf{AMB},\mathbf{A}\boldsymbol\Sigma\mathbf{A}^{\rm T}, \mathbf{B}^{\rm T}\boldsymbol\Omega\mathbf{B}) | |
| .</math>
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| The [[characteristic function (probability theory)|characteristic function]] is<ref name="iranmanesha"/>
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| :<math>\phi_T(\mathbf{Z}) = \frac{\exp({\rm tr}(i\mathbf{Z}'\mathbf{M}))|\boldsymbol\Omega|^\alpha}{\Gamma_p(\alpha)(2\beta)^{\alpha p}} |\mathbf{Z}'\boldsymbol\Sigma\mathbf{Z}|^\alpha B_\alpha\left(\frac{1}{2\beta}\mathbf{Z}'\boldsymbol\Sigma\mathbf{Z}\boldsymbol\Omega\right),</math>
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| where
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| :<math>B_\delta(\mathbf{WZ}) = |\mathbf{W}|^{-\delta} \int_{\mathbf{S}>0} \exp\left({\rm tr}(-\mathbf{SW}-\mathbf{S^{-1}Z})\right)|\mathbf{S}|^{-\delta-\frac12(p+1)}d\mathbf{S},</math> | |
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| and where <math>B_\delta</math> is the type-two [[Bessel function]] of Herz of a matrix argument.
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| == See also ==
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| * [[multivariate t-distribution]].
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| * [[matrix normal distribution]].
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| ==Notes ==
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| {{Reflist}}
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| ==External links==
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| * [https://github.com/zweng/rmg A C++ library for random matrix generator]
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| <!-- ==References ==
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| (fill in) -->
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| {{ProbDistributions|multivariate}}
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| [[Category:Random matrices]]
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| [[Category:Multivariate continuous distributions]]
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| [[Category:Probability distributions]]
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I'm a 50 years old and work at the high school (Religious Studies).
In my spare time I teach myself Japanese. I've been twicethere and look forward to go there sometime near future. I love to read, preferably on my kindle. I like to watch Arrested Development and Arrested Development as well as documentaries about anything technological. I enjoy Conlanging.
Feel free to visit my web blog :: smsnusa money game