MatrixTDistribution[Σrow,Σcol,ν]
represents zero mean matrix 
distribution with row covariance matrix Σrow, column covariance matrix Σcol, and degrees of freedom parameter ν.
MatrixTDistribution[μ,Σrow,Σcol,ν]
represents matrix 
distribution with mean matrix μ.
MatrixTDistribution
MatrixTDistribution[Σrow,Σcol,ν]
represents zero mean matrix 
distribution with row covariance matrix Σrow, column covariance matrix Σcol, and degrees of freedom parameter ν.
MatrixTDistribution[μ,Σrow,Σcol,ν]
represents matrix 
distribution with mean matrix μ.
Details
- The probability density for a matrix
of dimensions 
in a matrix 
distribution is proportional to 
with 
an identity matrix of length 
. - MatrixTDistribution[Σrow,Σcol,ν] is the distribution of MatrixNormalDistribution[Σ,Σcol] with
sampled from InverseWishartMatrixDistribution[ν+n-1,Σrow]. - MatrixTDistribution[μ,c Σrow,c-1 Σcol,ν] has the same distribution as MatrixTDistribution[μ,Σrow,Σcol,ν] for any positive real constant c.
- The covariance matrices Σrow and Σcol can be any symmetric positive definite matrices of real numbers of dimensions {n,n} and {m,m}, respectively. The degrees of freedom parameter ν can be any positive number, and the mean matrix μ can be any matrix of real numbers of dimensions {n,m}.
- MatrixTDistribution can be used with such functions as MatrixPropertyDistribution, EstimatedDistribution, and RandomVariate.
Examples
open all close allBasic Examples (2)
Sample from matrix 
distribution:
sigR = {{3, 1}, {1, 4}};
sigC = {{1, -1 / 2}, {-1 / 2, 2}};RandomVariate[MatrixTDistribution[sigR, sigC, 3]]Mean[MatrixTDistribution[{{1, 2}, {3, 4}, {5, 6}}, DiagonalMatrix[{2, 1, 3}], {{2, 1}, {1, 3}}, 4]]//MatrixFormVariance[MatrixTDistribution[{{1, 2}, {3, 4}, {5, 6}}, DiagonalMatrix[{2, 1, 3}], {{2, 1}, {1, 3}}, 4]]//MatrixFormScope (6)
Generate a single pseudorandom matrix:
RandomVariate[MatrixTDistribution[{{1, 1 / 3}, {1 / 3, 1}}, IdentityMatrix[3], 2]]Generate a single pseudorandom matrix with nonzero mean:
RandomVariate[MatrixTDistribution[{{1, 2, 3}, {4, 5, 6}}, {{1, 1 / 3}, {1 / 3, 1}}, IdentityMatrix[3], 2]]Generate a set of pseudorandom matrices:
RandomVariate[MatrixTDistribution[{{1, 1 / 3}, {1 / 3, 1}}, IdentityMatrix[3], 2], 3]RandomVariate[MatrixTDistribution[{{1, 1 / 3}, {1 / 3, 1}}, IdentityMatrix[3], 2], WorkingPrecision -> 20]RandomVariate[MatrixTDistribution[{{1, 2, 3}, {4, 5, 6}}, {{1, 1 / 3}, {1 / 3, 1}}, IdentityMatrix[3], 2], WorkingPrecision -> 20]Distribution parameters estimation:
dist = MatrixTDistribution[{{3, 1}, {1, 2}}, ToeplitzMatrix[{4, 2, 1}], 4];sample = RandomVariate[dist, 10 ^ 3];Estimate the distribution parameters from sample data:
edist = EstimatedDistribution[sample, MatrixTDistribution[Array[row, {2, 2}], Array[col, {3, 3}], ν]]Compare LogLikelihood for both distributions:
LogLikelihood[#, sample]& /@ {edist, dist}Skewness[MatrixTDistribution[{{2, 1}, {1, 3}}, IdentityMatrix[3], 5]]//MatrixFormKurtosis[MatrixTDistribution[{{2, 1}, {1, 3}}, IdentityMatrix[3], 5]]//MatrixFormdist = MatrixTDistribution[IdentityMatrix[2], DiagonalMatrix[{2, 1}], 3];PDF[dist, {{1., 2.}, {3, 4.}}]Plot PDF for a diagonal matrices:
Plot3D[PDF[dist, DiagonalMatrix[{a, b}]], {a, -2, 2}, {b, -1, 1}]Properties & Relations (4)
Matrix t distribution is defined up to a positive multiplicative constant:
sigr = {{2, 1}, {1, 4}};
sigc = ToeplitzMatrix[{9, 3, 1}];
𝒟1 = MatrixTDistribution[sigr, sigc, 3];Equivalent distribution with row and column scale matrices multiplied and divided by a positive constant:
c = 2;
𝒟2 = MatrixTDistribution[c * sigr, sigc / c, 3];Compute the PDF of the distributions at a random point:
mat = RandomReal[{-10, 10}, {2, 3}]PDF[𝒟1, mat] == PDF[𝒟2, mat]MatrixTDistribution[Σrow,Σcol,ν] is a parameter mixture of MatrixNormalDistribution[Σ,Σcol] with 
following InverseWishartMatrixDistribution[ν+n-1,Σrow]:
ν = 3;
n = 2;
Σr = IdentityMatrix[n];
Σc = IdentityMatrix[3];
invW = InverseWishartMatrixDistribution[ν + n - 1, Σr];Create a sample following the parameter mixture of MatrixNormalDistribution with InverseWishartMatrixDistribution:
sampleSize = 10 ^ 3;
sigmas = RandomVariate[invW, 10 ^ 3];
sample = Table[RandomVariate[MatrixNormalDistribution[s, Σc]], {s, sigmas}];Fit the sample data to MatrixTDistribution:
edist = EstimatedDistribution[sample, MatrixTDistribution[Array[s1, {2, 2}], Array[s2, {3, 3}], nu]]Compute log-likelihood ratio statistic against the appropriate MatrixTDistribution
ratio = LogLikelihood[edist, sample] - LogLikelihood[MatrixTDistribution[Σr, Σc, ν], sample]Log-likelihood ratio follows ChiSquareDistribution with the parameter equal to the number of degrees of freedom:
dof = (Length[Σr] + 1) * Length[Σr] / 2 + (Length[Σc] + 1) * Length[Σc] / 2 + 1Compute the 
-value of log-likelihood ratio test:
pval = SurvivalFunction[ChiSquareDistribution[dof], 2ratio]For matrix 
sampled from matrix 
distribution, the expression 
follows Student 
distribution for any nonzero vectors 
and 
with lengths that match with the dimension of 
:
v1 = RandomReal[1, 2];v2 = RandomReal[1, 3];Use MatrixPropertyDistribution to sample values of the expression 
:
Subscript[Σ, r] = {{1, 4 / 5}, {4 / 5, 1}};Subscript[Σ, c] = {{2, 1, 0}, {1, 3, 1}, {0, 1, 2}};df = 5;data = RandomVariate[MatrixPropertyDistribution[v1.𝓂.v2, 𝓂MatrixTDistribution[Subscript[Σ, r], Subscript[Σ, c], df]], 10 ^ 4];Check agreement with the expected 
distribution:
t𝒟 = StudentTDistribution[0, Sqrt[(v1.Subscript[Σ, r].v1)(v2.Subscript[Σ, c].v2) / df], df];DistributionFitTest[data, t𝒟]Show[Histogram[data, Automatic, PDF], Plot[PDF[t𝒟, x], {x, -4, 4}, PlotRange -> All]]For matrix 
sampled from matrix 
distribution, 
follows multivariate 
distribution for any nonzero vector 
with length that matches with the number of columns of 
:
v = RandomReal[1, 3];Use MatrixPropertyDistribution to sample values of 
:
Subscript[Σ, r] = {{1, 4 / 5}, {4 / 5, 1}};Subscript[Σ, c] = {{2, 1, 0}, {1, 3, 1}, {0, 1, 2}};df = 5;data = RandomVariate[MatrixPropertyDistribution[𝓂.v, 𝓂MatrixTDistribution[Subscript[Σ, r], Subscript[Σ, c], df]], 10 ^ 4];Verify goodness of fit with the expected distribution:
DistributionFitTest[data, MultivariateTDistribution[Subscript[Σ, r] (v.Subscript[Σ, c].v) / df, df]]Possible Issues (1)
Matrix 
distribution is defined up to a multiplicative scaling constant. The estimated parameters may not be close to the ones that specify the underlying distribution:
ν = 4;
sigr = ToeplitzMatrix[{9, 3, 1}];
sigc = DiagonalMatrix[{1, 2}];
dist = MatrixTDistribution[sigr, sigc, ν];Sample from the matrix 
distribution:
sample = RandomVariate[dist, 10 ^ 4];edist = EstimatedDistribution[sample, MatrixTDistribution[Array[s1, {3, 3}], Array[s2, {2, 2}], ν]];Compare the estimated scale parameters with the ones of the underlying distribution:
MatrixForm /@ {esigr = First[edist], sigr}MatrixForm /@ {esigc = edist[[2]], sigc}Kronecker products of the scale matrices are close to each other:
NumberForm[KroneckerProduct[sigr, sigc] - KroneckerProduct[esigr, esigc]//MatrixForm, 4]The LogLikelihood of the distributions indicate that the estimate is good:
LogLikelihood[#, sample]& /@ {dist, edist}Text
Wolfram Research (2015), MatrixTDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/MatrixTDistribution.html (updated 2017).
CMS
Wolfram Language. 2015. "MatrixTDistribution." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2017. https://reference.wolfram.com/language/ref/MatrixTDistribution.html.
APA
Wolfram Language. (2015). MatrixTDistribution. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/MatrixTDistribution.html