Matrix-variate Distributions
Matrix-variate distributions are the distributions whose variate forms are Matrixvariate (i.e each sample is a matrix). Abstract types for matrix-variate distributions:
const MatrixDistribution{S<:ValueSupport} = Distribution{Matrixvariate,S}
const DiscreteMatrixDistribution = Distribution{Matrixvariate, Discrete}
const ContinuousMatrixDistribution = Distribution{Matrixvariate, Continuous}More advanced functionalities related to random matrices can be found in the RandomMatrices.jl package.
Common Interface
All distributions implement the same set of methods:
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Base.size — Method
size(d::MatrixDistribution)Return the size of each sample from distribution d.
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Base.length — Method
length(d::MatrixDistribution)The length (i.e number of elements) of each sample from the distribution d.
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LinearAlgebra.rank — Method
rank(d::MatrixDistribution)The rank of each sample from the distribution d.
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Statistics.mean — Method
mean(d::MatrixDistribution)Return the mean matrix of d.
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Statistics.var — Method
var(d::MatrixDistribution)Compute the matrix of element-wise variances for distribution d.
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Statistics.cov — Method
cov(d::MatrixDistribution)Compute the covariance matrix for vec(X), where X is a random matrix with distribution d.
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Distributions.pdf — Method
pdf(d::Distribution{ArrayLikeVariate{N}}, x::AbstractArray{<:Real,N}) where {N}Evaluate the probability density function of d at x.
This function checks if the size of x is compatible with distribution d. This check can be disabled by using @inbounds.
Implementation
Instead of pdf one should implement _pdf(d, x) which does not have to check the size of x. However, since the default definition of pdf(d, x) falls back to logpdf(d, x) usually it is sufficient to implement logpdf.
See also: logpdf.
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Distributions.logpdf — Method
logpdf(d::Distribution{ArrayLikeVariate{N}}, x::AbstractArray{<:Real,N}) where {N}Evaluate the logarithm of the probability density function of d at x.
This function checks if the size of x is compatible with distribution d. This check can be disabled by using @inbounds.
Implementation
Instead of logpdf one should implement _logpdf(d, x) which does not have to check the size of x.
See also: pdf.
Distributions
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Distributions.MatrixNormal — Type
MatrixNormal(M, U, V)M::AbstractMatrix n x p mean
U::AbstractPDMat n x n row covariance
V::AbstractPDMat p x p column covarianceThe matrix normal distribution generalizes the multivariate normal distribution to real matrices . If , then its probability density function is
if and only if .
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Distributions.Wishart — Type
Wishart(ν, S)ν::Real degrees of freedom (whole number or a real number greater than p - 1)
S::AbstractPDMat p x p scale matrixThe Wishart distribution generalizes the gamma distribution to real, positive semidefinite matrices .
If , then has rank and its probability density function is
If , then is rank and it has a density with respect to a suitably chosen volume element on the space of positive semidefinite matrices. See here.
For integer , a random matrix given by
has . For non-integer , Wishart matrices can be generated via the Bartlett decomposition.
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Distributions.InverseWishart — Type
InverseWishart(ν, Ψ)ν::Real degrees of freedom (greater than p - 1)
Ψ::AbstractPDMat p x p scale matrixThe inverse Wishart distribution generalizes the inverse gamma distribution to real, positive definite matrices . If , then its probability density function is
if and only if .
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Distributions.MatrixTDist — Type
MatrixTDist(ν, M, Σ, Ω)ν::Real positive degrees of freedom
M::AbstractMatrix n x p location
Σ::AbstractPDMat n x n scale
Ω::AbstractPDMat p x p scaleThe matrix t-distribution generalizes the multivariate t-distribution to real matrices . If , then its probability density function is
where
If the joint distribution is given by
then the marginal distribution of is .
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Distributions.MatrixBeta — Type
MatrixBeta(p, n1, n2)p::Int dimension
n1::Real degrees of freedom (greater than p - 1)
n2::Real degrees of freedom (greater than p - 1)The matrix beta distribution generalizes the beta distribution to real matrices for which and are both positive definite. If , then its probability density function is
If and are independent, and we use to denote the lower Cholesky factor, then
has .
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Distributions.MatrixFDist — Type
MatrixFDist(n1, n2, B)n1::Real degrees of freedom (greater than p - 1)
n2::Real degrees of freedom (greater than p - 1)
B::AbstractPDMat p x p scaleThe matrix F-distribution (sometimes called the matrix beta type II distribution) generalizes the F-Distribution to real, positive definite matrices . If , then its probability density function is
If the joint distribution $p(\boldsymbol{\Psi},\boldsymbol{\Sigma})=p(\boldsymbol{\Psi})p(\boldsymbol{\Sigma} |
\boldsymbol{\Psi})$ is given by |
then the marginal distribution of is .
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Distributions.LKJ — Type
LKJ(d, η)d::Int dimension
η::Real positive shapeThe LKJ distribution is a distribution over real correlation matrices (positive-definite matrices with ones on the diagonal). If , then its probability density function is
If , then the LKJ distribution is uniform over the space of correlation matrices.
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if a Cholesky factor of the correlation matrix is desired, it is more efficient to use |