In statistics , the matrix t -distribution (or matrix variate t -distribution ) is the generalization of the multivariate t -distribution from vectors to matrices .[1] t -distribution shares the same relationship with the multivariate t -distribution that the matrix normal distribution shares with the multivariate normal distribution .[clarification needed 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 .[citation needed [2]
Matrix t Notation
T
n
,
p
(
ν
,
M
,
Σ
,
Ω
)
{\displaystyle {\rm {T}}_{n,p}(\nu ,\mathbf {M} ,{\boldsymbol {\Sigma }},{\boldsymbol {\Omega }})}
Parameters
M
{\displaystyle \mathbf {M} }
location (real
n
×
p
{\displaystyle n\times p}
matrix )
Ω
{\displaystyle {\boldsymbol {\Omega }}}
scale (positive-definite real
p
×
p
{\displaystyle p\times p}
matrix )
Σ
{\displaystyle {\boldsymbol {\Sigma }}}
scale (positive-definite real
n
×
n
{\displaystyle n\times n}
matrix )
ν
{\displaystyle \nu }
degrees of freedom Support
X
∈
R
n
×
p
{\displaystyle \mathbf {X} \in \mathbb {R} ^{n\times p}}
PDF
Γ
p
(
ν
+
n
+
p
−
1
2
)
(
π
)
n
p
2
Γ
p
(
ν
+
p
−
1
2
)
|
Ω
|
−
n
2
|
Σ
|
−
p
2
{\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}}}}
×
|
I
n
+
Σ
−
1
(
X
−
M
)
Ω
−
1
(
X
−
M
)
T
|
−
ν
+
n
+
p
−
1
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
M
{\displaystyle \mathbf {M} }
ν
+
p
−
n
>
1
{\displaystyle \nu +p-n>1}
Mode
M
{\displaystyle \mathbf {M} }
Variance
Σ
⊗
Ω
ν
−
2
{\displaystyle {\frac {{\boldsymbol {\Sigma }}\otimes {\boldsymbol {\Omega }}}{\nu -2}}}
ν
>
2
{\displaystyle \nu >2}
CF
see below
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 .
Definition
edit
For a matrix t -distribution, the probability density function at the point
X
{\displaystyle \mathbf {X} }
n
×
p
{\displaystyle n\times p}
f
(
X
;
ν
,
M
,
Σ
,
Ω
)
=
K
×
|
I
n
+
Σ
−
1
(
X
−
M
)
Ω
−
1
(
X
−
M
)
T
|
−
ν
+
n
+
p
−
1
2
,
{\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
K
=
Γ
p
(
ν
+
n
+
p
−
1
2
)
(
π
)
n
p
2
Γ
p
(
ν
+
p
−
1
2
)
|
Ω
|
−
n
2
|
Σ
|
−
p
2
.
{\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
Γ
p
{\displaystyle \Gamma _{p}}
multivariate gamma function .
The characteristic function and various other properties can be derived from the generalized matrix t -distribution (see below).
Generalized matrix t -distribution
edit
Generalized matrix t Notation
T
n
,
p
(
α
,
β
,
M
,
Σ
,
Ω
)
{\displaystyle {\rm {T}}_{n,p}(\alpha ,\beta ,\mathbf {M} ,{\boldsymbol {\Sigma }},{\boldsymbol {\Omega }})}
Parameters
M
{\displaystyle \mathbf {M} }
location (real
n
×
p
{\displaystyle n\times p}
matrix )
Ω
{\displaystyle {\boldsymbol {\Omega }}}
scale (positive-definite real
p
×
p
{\displaystyle p\times p}
matrix )
Σ
{\displaystyle {\boldsymbol {\Sigma }}}
scale (positive-definite real
n
×
n
{\displaystyle n\times n}
matrix )
α
>
(
p
−
1
)
/
2
{\displaystyle \alpha >(p-1)/2}
shape parameter
β
>
0
{\displaystyle \beta >0}
scale parameter Support
X
∈
R
n
×
p
{\displaystyle \mathbf {X} \in \mathbb {R} ^{n\times p}}
PDF
Γ
p
(
α
+
n
/
2
)
(
2
π
/
β
)
n
p
2
Γ
p
(
α
)
|
Ω
|
−
n
2
|
Σ
|
−
p
2
{\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}}}}
×
|
I
n
+
β
2
Σ
−
1
(
X
−
M
)
Ω
−
1
(
X
−
M
)
T
|
−
(
α
+
n
/
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)}}
CDF
No analytic expression Mean
M
{\displaystyle \mathbf {M} }
Variance
2
(
Σ
⊗
Ω
)
β
(
2
α
−
p
−
1
)
{\displaystyle {\frac {2({\boldsymbol {\Sigma }}\otimes {\boldsymbol {\Omega }})}{\beta (2\alpha -p-1)}}}
CF
see below
The generalized matrix t -distribution is a generalization of the matrix t -distribution with two parameters α and β in place of ν .[3]
This reduces to the standard matrix t -distribution with
β
=
2
,
α
=
ν
+
p
−
1
2
.
{\displaystyle \beta =2,\alpha ={\frac {\nu +p-1}{2}}.}
The generalized matrix t -distribution is 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
edit
If
X
∼
T
n
,
p
(
α
,
β
,
M
,
Σ
,
Ω
)
{\displaystyle \mathbf {X} \sim {\rm {T}}_{n,p}(\alpha ,\beta ,\mathbf {M} ,{\boldsymbol {\Sigma }},{\boldsymbol {\Omega }})}
[citation needed
X
T
∼
T
p
,
n
(
α
,
β
,
M
T
,
Ω
,
Σ
)
.
{\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 :
det
(
I
n
+
β
2
Σ
−
1
(
X
−
M
)
Ω
−
1
(
X
−
M
)
T
)
=
{\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)=}
det
(
I
p
+
β
2
Ω
−
1
(
X
T
−
M
T
)
Σ
−
1
(
X
T
−
M
T
)
T
)
.
{\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
X
∼
T
n
,
p
(
α
,
β
,
M
,
Σ
,
Ω
)
{\displaystyle \mathbf {X} \sim {\rm {T}}_{n,p}(\alpha ,\beta ,\mathbf {M} ,{\boldsymbol {\Sigma }},{\boldsymbol {\Omega }})}
A
(
n
×
n
)
{\displaystyle \mathbf {A} (n\times n)}
B
(
p
×
p
)
{\displaystyle \mathbf {B} (p\times p)}
nonsingular matrices then[citation needed
A
X
B
∼
T
n
,
p
(
α
,
β
,
A
M
B
,
A
Σ
A
T
,
B
T
Ω
B
)
.
{\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]
ϕ
T
(
Z
)
=
exp
(
t
r
(
i
Z
′
M
)
)
|
Ω
|
α
Γ
p
(
α
)
(
2
β
)
α
p
|
Z
′
Σ
Z
|
α
B
α
(
1
2
β
Z
′
Σ
Z
Ω
)
,
{\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
B
δ
(
W
Z
)
=
|
W
|
−
δ
∫
S
>
0
exp
(
t
r
(
−
S
W
−
S
−
1
Z
)
)
|
S
|
−
δ
−
1
2
(
p
+
1
)
d
S
,
{\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
B
δ
{\displaystyle B_{\delta }}
Bessel function of Herz[clarification needed of a matrix argument.
See also
edit
Notes
edit
^ 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.
^ 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)^ a b 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
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A C++ library for random matrix generator
