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Matrix t-distribution

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

f(\mathbf {X

where the constant of integration K is given by

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

Properties

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

Expected values

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

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

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

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

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

where $\operatorname {tr}$ denotes trace.

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

{\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:

\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:

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

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

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

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

\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:

\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)=

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

\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]

\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

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

Notes

1 2 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.
1 2 3 4 5 Gupta, Arjun K and Nagar, Daya K (1999). Matrix variate distributions. CRC Press. pp. Chapter 4.{{cite book}}: CS1 maint: multiple names: authors list (link)
1 2 3 4 5 Iranmanesh, Anis, M. Arashi and S. M. M. Tabatabaey (2010). "On Conditional Applications of Matrix Variate Normal Distribution" Archived 2016-03-04 at the Wayback Machine. Iranian Journal of Mathematical Sciences and Informatics, 5:2, pp. 33–43.

External links

A C++ library for random matrix generator

Probability distributions (list)

Discrete
univariate

with finite support	Benford Bernoulli Beta-binomial Binomial Categorical Hypergeometric Negative Poisson binomial Rademacher Soliton Discrete uniform Zipf Zipf–Mandelbrot
with infinite support	Beta negative binomial Borel Conway–Maxwell–Poisson Discrete phase-type Delaporte Extended negative binomial Flory–Schulz Gauss–Kuzmin Geometric Logarithmic Mixed Poisson Negative binomial Panjer Parabolic fractal Poisson Skellam Yule–Simon Zeta

Continuous
univariate

supported on a bounded interval	Arcsine ARGUS Balding–Nichols Bates Beta Generalized Beta rectangular Continuous Bernoulli Continuous binomial Irwin–Hall Kumaraswamy Logit-normal Noncentral beta PERT Power function Raised cosine Reciprocal Triangular U-quadratic Uniform Wigner semicircle
supported on a semi-infinite interval	Benini Benktander 1st kind Benktander 2nd kind Beta prime Burr Chi Chi-squared Noncentral Inverse Scaled Dagum Davis Erlang Hyper Exponential Hyperexponential Hypoexponential Logarithmic F Noncentral Folded normal Fréchet Gamma Generalized Inverse gamma/Gompertz Gompertz Shifted Half-logistic Half-normal Hotelling's T-squared Hartman–Watson Inverse Gaussian Generalized Kolmogorov Lévy Log-Cauchy Log-Laplace Log-logistic Log-normal Log-t Lomax Matrix-exponential Maxwell–Boltzmann Maxwell–Jüttner Mittag-Leffler Nakagami Pareto Phase-type Poly-Weibull Rayleigh Relativistic Breit–Wigner Rice Truncated normal type-2 Gumbel Weibull Discrete Wilks's lambda
supported on the whole real line	Cauchy Exponential power Fisher's z Kaniadakis κ-Gaussian Gaussian q Generalized hyperbolic Generalized logistic (logistic-beta) Generalized normal Geometric stable Gumbel Holtsmark Hyperbolic secant Johnson's S_U Landau Laplace Asymmetric Logistic Noncentral t Normal (Gaussian) Normal-inverse Gaussian Skew normal Slash Stable Student's t Tracy–Widom Variance-gamma Voigt
with support whose type varies	Generalized chi-squared Generalized extreme value Generalized Pareto Marchenko–Pastur Kaniadakis κ-exponential Kaniadakis κ-Gamma Kaniadakis κ-Weibull Kaniadakis κ-Logistic Kaniadakis κ-Erlang q-exponential q-Gaussian q-Weibull Shifted log-logistic Tukey lambda

Mixed
univariate

continuous- discrete	Rectified Gaussian

Multivariate
(joint)

Discrete:
Ewens
Multinomial
- Dirichlet
- Negative
Continuous:
Dirichlet
- Generalized
Multivariate Laplace
Multivariate normal
Multivariate stable
Multivariate t
Normal-gamma
- Inverse
Matrix-valued:
LKJ
Matrix beta
Matrix F
Matrix normal
Matrix t
Matrix gamma
- Inverse
Wishart
- Normal
- Inverse
- Normal-inverse
- Complex
Uniform distribution on a Stiefel manifold

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

v t e Random matrix theory
Concepts	Ensemble Spectrum Universality Resolvent Level repulsion Integrability Free probability Noncrossing partition Coulomb gas Dyson Brownian motion Riemann–Hilbert problem Determinantal point process
Ensembles	Gaussian ensemble Wishart ensemble Jacobi ensemble Ginibre ensemble Beta ensemble Circular ensemble Deformed ensemble Matrix t-distribution Random band ensemble Heavy-tailed
Laws	Wigner semicircle distribution Marchenko–Pastur law Circular law Tracy–Widom distribution BBP transition Wigner surmise
Techniques	Stieltjes transformation Isserlis's theorem Fredholm determinant Orthogonal polynomials Skew-orthogonal polynomials Christoffel–Darboux formula Cavity method Weingarten function Selberg integral Mean-field theory Airy process Bessel process sine process Painlevé transcendents KPZ equation Green's function

A carregar…

Matrix t-distribution

Definition

Properties

Expected values

Transformation

Re-parameterized matrix t-distribution

Properties

See also

Notes

External links