The matrixNormal package in the Comprehensive R Archive
Network (CRAN) [1]
consists of two different types of functions: distribution and matrix
ones. First, one can compute densities, probabilities and quantiles of
the Matrix Normal Distribution. Second, one can perform useful but
simple matrix operations like creating identity matrices and matrices of
ones, calculating the trace of a matrix, and implementing the matrix
operator vec().
You can install the released version of matrixNormal
from CRAN[1]
with install.packages("matrixNormal") and can load the
package by:
The matrix normal distribution is a generalization of the
multivariate normal distribution to matrix-valued random values. The
parameters consist of a n x p mean matrix M
and two positive-definite covariance matrices, one for rows
U and another for columns V. Suppose
that any n x p matrix A ∼ MatNormn, p(M, U, V).
The mean of A is M, and the variance
of vec(A) is the Kronecker product of V
and U.
The matrix normal distribution is a conjugate prior of the coefficients used in multivariate regression. Suppose there are p predictors for k dependent variables. In univariate regression, only one dependent variable exists, so the conjugate prior for the k regression coefficients is the multivariate normal distribution. For multivariate regression, the extension of the conjugate prior of the coefficient matrix is the matrix normal distribution. This package also contains the PDF, CDF, and random number generation for the matrix normal distribution (*matnorm). See matrixNormal_Distribution file for more information.
For instance, the USArrests dataset in the dataset package examines statistics in arrests per 100,000 residents for assault, murder, and rape in each of 50 states since 1973. Suppose that a researcher wants to determine whether the outcomes of assault and murder rates are associated with urban population and rape. The researcher then decides to use a Bayesian multiple linear regression and makes the assumption that the states are independent. However, the covariance between assault and murder is nonzero and needs to be taken into account. In fact, it has a correlation of 0.411 as given below.
> library(datasets)
> data(USArrests)
> X <- cbind(USArrests$Assault, USArrests$Murder)
> Y <- cbind(USArrests$UrbanPop, USArrests$Rape)
> cor(Y)
[,1] [,2]
[1,] 1.0000000 0.4113412
[2,] 0.4113412 1.0000000We can assume that the outcome Y follows a matrix normal distribution with mean matrix M, which is the matrix product of the coefficient matrix, Ψ, times X. The covariance across states is assumed to be independent, and the covariance across the predictors be Σ. Or succinctly, with k = 2 outcomes, Y ∼ MatNormnx2(M = XΨ, U = I(n), V = Σ2)
of coefficient matrix times X, and covariance across
the predictors Φ. The overall
covariance matrix of Y, Σ, is a block diagonal matrix of
Φ. For instance, suppose that
Y has the following distribution, and if we know the parameters, we can
calculate its density using dmatnorm().
> # Y is n = 50 x p = 2 that follows a matrix normal with mean matrix M, which is product
> # of
> M <- (100 * toeplitz(50:1))[, 1:2]
> dim(M)
[1] 50 2
> head(M)
[,1] [,2]
[1,] 5000 4900
[2,] 4900 5000
[3,] 4800 4900
[4,] 4700 4800
[5,] 4600 4700
[6,] 4500 4600
> U <- I(50) # Covariance across states: Assumed to be independent
> U[1:5, 1:5]
1 2 3 4 5
1 1 0 0 0 0
2 0 1 0 0 0
3 0 0 1 0 0
4 0 0 0 1 0
5 0 0 0 0 1
> V <- cov(X) # Covariance across predictors
> V
[,1] [,2]
[1,] 6945.1657 291.06237
[2,] 291.0624 18.97047
> # Find the density if Y has the density with these arguments.
> matrixNormal::dmatnorm(Y, M, U, V)
[1] -30446783The coefficient matrix, Psi, for urban population and rape has dimensions 2 x 3 for the k = 2 outcomes and the p = 3 predictors. A semi-conjugate prior can be constructed on Ψ to be a Matrix Normal distribution with mean Ψ0 as a matrix of ones, covariance across the predictors as X′X, and with no covariance across states (due to independence). The conjugate prior for the covariance between the outcomes Σ is the inverse-Wishart distribution with mean covariance Σ0 and degrees of freedom ν. After defining the parameters, one random matrix is generated from this prior.
> # Generate a random matrix from this prior. The prior mean of regression matrix
> J(2, 3)
1 2 3
1 1 1 1
2 1 1 1
> # The prior variance between rape and population
> t(X) %*% X
[,1] [,2]
[1,] 1798262 80756.0
[2,] 80756 3962.2
> # The prior variance between regression parameters
> I(3)
1 2 3
1 1 0 0
2 0 1 0
3 0 0 1
> # Random draw for prior would have these values
> A <- matrixNormal::rmatnorm(M = J(2, 3), U = t(X) %*% X, V = I(3))
Warning in matrixNormal::rmatnorm(M = J(2, 3), U = t(X) %*% X, V = I(3)): The
construction of sigma has been found to be incorrect. Please upgrade to new
version.
> A
1 2 3
[1,] 1356.09810 -1218.82242 -992.17417
[2,] 60.73219 -19.68277 -69.17957
> # Predicted Counts for y can be given as:
> ceiling(rowSums(X %*% A))
[1] -202127 -225119 -251568 -162678 -236205 -174621 -94131 -203631 -286824
[10] -180873 -39474 -102660 -213162 -96806 -47936 -98482 -93456 -213302
[19] -71015 -256787 -127503 -218339 -61628 -221871 -152425 -93352 -87320
[28] -215777 -48788 -136137 -243966 -217456 -288466 -38492 -102793 -129275
[37] -136066 -90796 -148847 -238921 -73628 -161092 -172191 -102677 -41097
[46] -133603 -124072 -69407 -45382 -137829However, the predicted counts do not have much meaning because the
prior is uninformed from the data. We should use the posterior
distribution to predict Y. Iranmanesh 2010 [2]
shows the posterior distributions to be $$
\Psi | \Sigma, Y, X \sim MatNorm_{kxp}( \frac{\hat{\Psi} +\Psi_0}{2},
(2X^TX)^{-1}, \Sigma)$$ and Σ|Y, X ∼ Wp−1(S*, 2ν + n)
where:
* Ψ̂ is the maximum likelihood
estimate (MLE) for the coefficient Ψ matrix. * S is the MLE
for the covariance matrix Σ:
(Y − X ⋅ Ψ̂)T * (Y − X ⋅ Ψ̂)
* S* is the
data-adjusted matrix of inverse Wishart, S* = 2Σ0 + S + (Ψ̂ − Ψ0)′(2XTX)−1(Ψ̂ − Ψ0)
* c is the dimension of Σ * n is the sample size,
the number of rows in Y and X.
Additional details can be found in Iranmanesh et al. 2010 [2]
and Pocuca et al. 2019 [3]. At any rate, the
matrixNormal package can be used in Bayesian Multivariate
Linear Regression.
In the matrixNormal package, useful but simple matrix
operations have been coded:
| Filename | Description |
|---|---|
| is.symmetric.matrix | Is a matrix square? symmetric? positive definite? or positive semi-definite? A tolerance is included here. |
| Special_matrices | Creates the identity matrix I and matrix of 1’s J. |
| tr | Calculates the usual trace of a matrix |
| vec | Stacks a matrix using matrix operator vec() and has option to keep names. |
| vech | Stacks elements of a numeric symmetric matrix A in lower triangular only (using half vectorization, vech()). |
For example, these functions can be applied to the following matrices.
> # Make a 3 x 3 Identity matrix
> I(3)
1 2 3
1 1 0 0
2 0 1 0
3 0 0 1
> # Make a 3 x 4 J matrix
> J(3, 4)
1 2 3 4
1 1 1 1 1
2 1 1 1 1
3 1 1 1 1
> # Make a 3 x 3 J matrix
> J(3, 3)
1 2 3
1 1 1 1
2 1 1 1
3 1 1 1
> # Calculate the trace of a J matrix
> tr(J(3, 3)) # Should be 3
[1] 3
> # Stack a matrix (used in distribution functions)
> A <- matrix(c(1:4), nrow = 2, dimnames = list(NULL, c("A", "B")))
> A
A B
[1,] 1 3
[2,] 2 4
> vec(A)
1:A 2:A 1:B 2:B
1 2 3 4
> # Test if matrix is symmetric (used in distribution function)
> is.symmetric.matrix(A)
[1] "A is not symmetric. Top of the matrix: "
A B
[1,] 1 3
[2,] 2 4
[1] FALSEAlthough other packages on CRAN have some matrixNormal functionality,
this package provides a general approach in randomly sampling a matrix
normal random variate. The MBSP::matrix.normal ,
matrixsampling::rmatrixnormal, and
LaplacesDemon::rmatrixnorm functions also randomly sample
from the matrix normal distribution [4–6].
The function in [MBSP] (https://CRAN.R-project.org/package=MBSP/) uses Cholesky
decomposition of individual matrices U and
V [4]. Similarly,
the function in [matrixsampling] (https://CRAN.R-project.org/package=matrixsampling/) uses
the Spectral decomposition of the individual matrices U
and V [5].
Comparatively, the new function, rmatnorm(), in
matrixNormal package is flexible in the decomposition of
the covariance matrix, which is Kronecker product of U
and V. While you can simulate many samples using the
rmatrixnormal() function from matrixsampling
package, the new rmatnorm() function only generates one
variate, but can generate many samples by placing
rmatnorm() function inside a for-loop. The
[LaplacesDemon] (https://cran.r-project.org/package=LaplacesDemon)
package also has a density function, dmatrixnorm(), which
calculates the log determinant of Cholesky decomposition of a positive
definite matrix [6].
However, the random matrix A that follows a matrix
normal distribution does not need to be positive definite [2].
There is no such restriction on the random matrix in
matrixNormal package.
In conclusion, the matrixNormal package collects all
forms of the Matrix Normal Distribution in one place: calculating the
PDF and CDF of the Matrix Normal distribution and simulating a random
variate from this distribution. The package allows the users to be
flexible in finding the random variate. Its main application in using
this package is the Bayesian multivariate regression.
This vignette is successfully processed using the following.
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date 2025-02-21
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-- Packages -------------------------------------------------------
package * version date (UTC) lib source
LaplacesDemon 16.1.6 2021-07-09 [2] RSPM (R 4.4.0)
matrixcalc 1.0-6 2022-09-14 [2] RSPM (R 4.4.0)
matrixsampling 2.0.0 2019-08-24 [2] RSPM (R 4.4.0)
MBSP 4.0 2023-05-17 [2] RSPM (R 4.4.0)
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