Matrix t-distribution

Matrix t

Notation	${\displaystyle {\rm {T}}_{n,p}(\nu ,\mathbf {M} ,{\boldsymbol {\Sigma }},{\boldsymbol {\Omega }})}$
Parameters	${\displaystyle \mathbf {M} }$ location (real ${\displaystyle n\times p}$ matrix) ${\displaystyle {\boldsymbol {\Omega }}}$ scale (positive-definite real ${\displaystyle p\times p}$ matrix) ${\displaystyle {\boldsymbol {\Sigma }}}$ scale (positive-definite real ${\displaystyle n\times n}$ matrix) ${\displaystyle \nu >0}$ degrees of freedom (real)
Support	${\displaystyle \mathbf {X} \in \mathbb {R} ^{n\times p}}$
PDF	${\displaystyle {\frac {\Gamma _{p}\left({\frac {\nu +n+p-1}{2}}\right)}{(\pi )^{\frac {np}{2}}\Gamma _{p}\left({\frac {\nu +p-1}{2}}\right)}}|{\boldsymbol {\Omega }}|^{-{\frac {n}{2}}}|{\boldsymbol {\Sigma }}|^{-{\frac {p}{2}}}}$ ${\displaystyle \times \left|\mathbf {I} _{n}+{\boldsymbol {\Sigma }}^{-1}(\mathbf {X} -\mathbf {M} ){\boldsymbol {\Omega }}^{-1}(\mathbf {X} -\mathbf {M} )^{\rm {T}}\right|^{-{\frac {\nu +n+p-1}{2}}}}$
CDF	No analytic expression
Mean	${\displaystyle \mathbf {M} }$ if ${\displaystyle \nu >1}$, else undefined
Mode	${\displaystyle \mathbf {M} }$
Variance	${\displaystyle \mathrm {cov} (\mathrm {vec} (\mathbf {X} ))={\frac {{\boldsymbol {\Sigma }}\otimes {\boldsymbol {\Omega }}}{\nu -2}}}$ if ${\displaystyle \nu >2}$, else undefined
CF	see below

In statistics, the matrix t-distribution (or matrix variate t-distribution) is the generalization of the multivariate t-distribution from vectors to matrices. ^{[1] [2]}

The matrix t-distribution shares the same relationship with the multivariate t-distribution that the matrix normal distribution shares with the multivariate normal distribution: If the matrix has only one row, or only one column, the distributions become equivalent to the corresponding (vector-)multivariate distribution. The matrix t-distribution is the compound distribution that results from an infinite mixture of a matrix normal distribution with an inverse Wishart distribution placed over either of its covariance matrices, ^[1] and the multivariate t-distribution can be generated in a similar way. ^[2]

In a Bayesian analysis of a multivariate linear regression model based on the matrix normal distribution, the matrix t-distribution is the posterior predictive distribution. ^[3]

Definition

For a matrix t-distribution, the probability density function at the point ${\displaystyle \mathbf {X} }$ of an ${\displaystyle n\times p}$ space is

${\displaystyle f(\mathbf {X}$ ;\nu ,\mathbf {M} ,{\boldsymbol {\Sigma }},{\boldsymbol {\Omega }})=K\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}}},}

where the constant of integration K is given by

${\displaystyle K={\frac {\Gamma _{p}\left({\frac {\nu +n+p-1}{2}}\right)}{(\pi )^{\frac {np}{2}}\Gamma _{p}\left({\frac {\nu +p-1}{2}}\right)}}|{\boldsymbol {\Omega }}|^{-{\frac {n}{2}}}|{\boldsymbol {\Sigma }}|^{-{\frac {p}{2}}}.}$

Here ${\displaystyle \Gamma _{p}}$ is the multivariate gamma function.

Properties

If ${\displaystyle \mathbf {X} \sim {\mathcal {T}}_{n\times p}(\nu ,\mathbf {M} ,\mathbf {\Sigma } ,\mathbf {\Omega } )}$, then we have the following properties ^[2] :

Expected values

The mean, or expected value is, if ${\displaystyle \nu >1}$:

${\displaystyle E[\mathbf {X} ]=\mathbf {M} }$

and we have the following second-order expectations, if ${\displaystyle \nu >2}$:

${\displaystyle E[(\mathbf {X} -\mathbf {M} )(\mathbf {X} -\mathbf {M} )^{T}]={\frac {\mathbf {\Sigma } \operatorname {tr} (\mathbf {\Omega } )}{\nu -2}}}$

${\displaystyle E[(\mathbf {X} -\mathbf {M} )^{T}(\mathbf {X} -\mathbf {M} )]={\frac {\mathbf {\Omega } \operatorname {tr} (\mathbf {\Sigma } )}{\nu -2}}}$

where ${\displaystyle \operatorname {tr} }$ denotes trace.

More generally, for appropriately dimensioned matrices A,B,C:

${\displaystyle {\begin{aligned}E[(\mathbf {X} -\mathbf {M} )\mathbf {A} (\mathbf {X} -\mathbf {M} )^{T}]&={\frac {\mathbf {\Sigma } \operatorname {tr} (\mathbf {A} ^{T}\mathbf {\Omega } )}{\nu -2}}\\E[(\mathbf {X} -\mathbf {M} )^{T}\mathbf {B} (\mathbf {X} -\mathbf {M} )]&={\frac {\mathbf {\Omega } \operatorname {tr} (\mathbf {B} ^{T}\mathbf {\Sigma } )}{\nu -2}}\\E[(\mathbf {X} -\mathbf {M} )\mathbf {C} (\mathbf {X} -\mathbf {M} )]&={\frac {\mathbf {\Sigma } \mathbf {C} ^{T}\mathbf {\Omega } }{\nu -2}}\end{aligned}}}$

Transformation

Transpose transform:

${\displaystyle \mathbf {X} ^{T}\sim {\mathcal {T}}_{p\times n}(\nu ,\mathbf {M} ^{T},\mathbf {\Omega } ,\mathbf {\Sigma } )}$

Linear transform: let A (r-by-n), be of full rank r ≤ n and B (p-by-s), be of full rank s ≤ p, then:

${\displaystyle \mathbf {AXB} \sim {\mathcal {T}}_{r\times s}(\nu ,\mathbf {AMB} ,\mathbf {A\Sigma A} ^{T},\mathbf {B} ^{T}\mathbf {\Omega B} )}$

The characteristic function and various other properties can be derived from the re-parameterised formulation (see below).

Re-parameterized matrix t-distribution

Re-parameterized matrix t

Notation	${\displaystyle {\rm {T}}_{n,p}(\alpha ,\beta ,\mathbf {M} ,{\boldsymbol {\Sigma }},{\boldsymbol {\Omega }})}$
Parameters	${\displaystyle \mathbf {M} }$ location (real ${\displaystyle n\times p}$ matrix) ${\displaystyle {\boldsymbol {\Omega }}}$ scale (positive-definite real ${\displaystyle p\times p}$ matrix) ${\displaystyle {\boldsymbol {\Sigma }}}$ scale (positive-definite real ${\displaystyle n\times n}$ matrix) ${\displaystyle \alpha >(p-1)/2}$ shape parameter ${\displaystyle \beta >0}$ scale parameter
Support	${\displaystyle \mathbf {X} \in \mathbb {R} ^{n\times p}}$
PDF	${\displaystyle {\frac {\Gamma _{p}(\alpha +n/2)}{(2\pi /\beta )^{\frac {np}{2}}\Gamma _{p}(\alpha )}}|{\boldsymbol {\Omega }}|^{-{\frac {n}{2}}}|{\boldsymbol {\Sigma }}|^{-{\frac {p}{2}}}}$ ${\displaystyle \times \left|\mathbf {I} _{n}+{\frac {\beta }{2}}{\boldsymbol {\Sigma }}^{-1}(\mathbf {X} -\mathbf {M} ){\boldsymbol {\Omega }}^{-1}(\mathbf {X} -\mathbf {M} )^{\rm {T}}\right|^{-(\alpha +n/2)}}$ ${\displaystyle \Gamma _{p}}$ is the multivariate gamma function.
CDF	No analytic expression
Mean	${\displaystyle \mathbf {M} }$ if ${\displaystyle \alpha >p/2}$, else undefined
Variance	${\displaystyle {\frac {2({\boldsymbol {\Sigma }}\otimes {\boldsymbol {\Omega }})}{\beta (2\alpha -p-1)}}}$ if ${\displaystyle \alpha >(p+1)/2}$, else undefined
CF	see below

An alternative parameterisation of the matrix t-distribution uses two parameters ${\displaystyle \alpha }$ and ${\displaystyle \beta }$ in place of ${\displaystyle \nu }$. ^[3]

This formulation reduces to the standard matrix t-distribution with ${\displaystyle \beta =2,\alpha ={\frac {\nu +p-1}{2}}.}$

This formulation of the matrix t-distribution can be derived as the compound distribution that results from an infinite mixture of a matrix normal distribution with an inverse multivariate gamma distribution placed over either of its covariance matrices.

Properties

If ${\displaystyle \mathbf {X} \sim {\rm {T}}_{n,p}(\alpha ,\beta ,\mathbf {M} ,{\boldsymbol {\Sigma }},{\boldsymbol {\Omega }})}$ then ^{[2] [3]}

${\displaystyle \mathbf {X} ^{\rm {T}}\sim {\rm {T}}_{p,n}(\alpha ,\beta ,\mathbf {M} ^{\rm {T}},{\boldsymbol {\Omega }},{\boldsymbol {\Sigma }}).}$

The property above comes from Sylvester's determinant theorem:

${\displaystyle \det \left(\mathbf {I} _{n}+{\frac {\beta }{2}}{\boldsymbol {\Sigma }}^{-1}(\mathbf {X} -\mathbf {M} ){\boldsymbol {\Omega }}^{-1}(\mathbf {X} -\mathbf {M} )^{\rm {T}}\right)=}$

${\displaystyle \det \left(\mathbf {I} _{p}+{\frac {\beta }{2}}{\boldsymbol {\Omega }}^{-1}(\mathbf {X} ^{\rm {T}}-\mathbf {M} ^{\rm {T}}){\boldsymbol {\Sigma }}^{-1}(\mathbf {X} ^{\rm {T}}-\mathbf {M} ^{\rm {T}})^{\rm {T}}\right).}$

If ${\displaystyle \mathbf {X} \sim {\rm {T}}_{n,p}(\alpha ,\beta ,\mathbf {M} ,{\boldsymbol {\Sigma }},{\boldsymbol {\Omega }})}$ and ${\displaystyle \mathbf {A} (n\times n)}$ and ${\displaystyle \mathbf {B} (p\times p)}$ are nonsingular matrices then ^{[2] [3]}

${\displaystyle \mathbf {AXB} \sim {\rm {T}}_{n,p}(\alpha ,\beta ,\mathbf {AMB} ,\mathbf {A} {\boldsymbol {\Sigma }}\mathbf {A} ^{\rm {T}},\mathbf {B} ^{\rm {T}}{\boldsymbol {\Omega }}\mathbf {B} ).}$

The characteristic function is ^[3]

${\displaystyle \phi _{T}(\mathbf {Z} )={\frac {\exp({\rm {tr}}(i\mathbf {Z} '\mathbf {M} ))|{\boldsymbol {\Omega }}|^{\alpha }}{\Gamma _{p}(\alpha )(2\beta )^{\alpha p}}}|\mathbf {Z} '{\boldsymbol {\Sigma }}\mathbf {Z} |^{\alpha }B_{\alpha }\left({\frac {1}{2\beta }}\mathbf {Z} '{\boldsymbol {\Sigma }}\mathbf {Z} {\boldsymbol {\Omega }}\right),}$

where

${\displaystyle B_{\delta }(\mathbf {WZ} )=|\mathbf {W} |^{-\delta }\int _{\mathbf {S} >0}\exp \left({\rm {tr}}(-\mathbf {SW} -\mathbf {S^{-1}Z} )\right)|\mathbf {S} |^{-\delta -{\frac {1}{2}}(p+1)}d\mathbf {S} ,}$

and where ${\displaystyle B_{\delta }}$ is the type-two Bessel function of Herz^{[ clarification needed ]} of a matrix argument.

See also

• Multivariate t-distribution
• Matrix normal distribution

Notes

1. Zhu, Shenghuo and Kai Yu and Yihong Gong (2007). "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.
2. Gupta, Arjun K and Nagar, Daya K (1999). Matrix variate distributions. CRC Press. pp. Chapter 4.
3. Iranmanesh, Anis, M. Arashi and S. M. M. Tabatabaey (2010). "On Conditional Applications of Matrix Variate Normal Distribution". Iranian Journal of Mathematical Sciences and Informatics, 5:2, pp. 33–43.

External links

• A C++ library for random matrix generator