MatrixNormalDistribution

MatrixNormalDistribution[Σrow,Σcol]

represents zero mean matrix normal distribution with row covariance matrix Σrow and column covariance matrix Σcol.

MatrixNormalDistribution[μ,Σrow,Σcol]

represents matrix normal distribution with mean matrix μ.

Details

Examples

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Basic Examples  (2)

Sample from matrix normal distribution:

Mean and variance:

Scope  (7)

Generate a single pseudorandom matrix:

Generate a single pseudorandom matrix with nonzero mean:

Generate a set of pseudorandom matrices:

Sample at extended precision:

Distribution parameters estimation:

Estimate the distribution parameters from sample data:

Compare LogLikelihood for both distributions:

Find the probability that smallest eigenvalue TemplateBox[{{lambda, _, {(, min, )}}}, Abs]>0.1`:

Probability density function:

Plot PDF for a diagonal matrices:

Skewness and kurtosis:

Applications  (2)

Visualize sample matrices from matrix normal distributions:

Use matrix normal distribution to simulate a vector autoregressive process:

Construct TemporalData from sampled values:

Estimate diagonal vector autoregressive process:

Compare to the original values:

Properties & Relations  (6)

Matrix normal distribution is defined up to a positive multiplicative constant:

Equivalent distribution with row and column scale matrices multiplied and divided by a positive constant:

Compute the PDF of the distributions at a random point:

  • MatrixTDistribution[Σrow,Σcol,ν] is a parameter mixture of MatrixNormalDistribution[Σ,Σcol] with following InverseWishartMatrixDistribution[ν+n-1,Σrow]:
  • Create a sample following the parameter mixture of MatrixNormalDistribution with InverseWishartMatrixDistribution:

    Fit the sample data to MatrixTDistribution:

    Compute log-likelihood ratio statistic against the appropriate MatrixTDistribution:

    Log-likelihood ratio follows ChiSquareDistribution with the parameter equal to the number of degrees of freedom:

    Compute the -value of log-likelihood ratio test:

    Sample from matrix normal distribution with independent rows:

    Test the hypothesis that rows follow multinormal distribution with the column covariance matrix:

    Sample from matrix normal distribution with independent rows:

    Test the hypothesis that rows follow multinormal distribution with the column covariance matrix:

    Sample from matrix normal distribution with independent rows:

    Computing sample inter-row covariances shows different rows are pairwise independent:

    Computing sample inter-column covariances shows different columns are dependent:

    By joining the rows of the matrix-valued random variable together, a matrix normal distribution can be regarded as a multivariate normal distribution:

    The covariance matrix of the vectorized random matrix is the Kronecker product of and :

    Possible Issues  (1)

    Matrix normal distribution is defined up to a multiplicative scaling constant. The estimated parameters may not be close to the ones that specify the underlying distribution:

    Sample from the matrix normal distribution:

    Estimate the distribution:

    Compare the estimated scale parameters with the ones of the underlying distribution:

    Kronecker products of the scale matrices are close to each other:

    The LogLikelihood of the distribution indicates that the estimate is good:

    Wolfram Research (2015), MatrixNormalDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/MatrixNormalDistribution.html (updated 2017).

    Text

    Wolfram Research (2015), MatrixNormalDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/MatrixNormalDistribution.html (updated 2017).

    BibTeX

    @misc{reference.wolfram_2021_matrixnormaldistribution, author="Wolfram Research", title="{MatrixNormalDistribution}", year="2017", howpublished="\url{https://reference.wolfram.com/language/ref/MatrixNormalDistribution.html}", note=[Accessed: 25-September-2021 ]}

    BibLaTeX

    @online{reference.wolfram_2021_matrixnormaldistribution, organization={Wolfram Research}, title={MatrixNormalDistribution}, year={2017}, url={https://reference.wolfram.com/language/ref/MatrixNormalDistribution.html}, note=[Accessed: 25-September-2021 ]}

    CMS

    Wolfram Language. 2015. "MatrixNormalDistribution." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2017. https://reference.wolfram.com/language/ref/MatrixNormalDistribution.html.

    APA

    Wolfram Language. (2015). MatrixNormalDistribution. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/MatrixNormalDistribution.html