These functions provide the density and random number generation for the multivariate t distribution, otherwise called the multivariate Student distribution. These functions use the precision and Cholesky parameterization.
dmvtpc(x, mu, U, nu=Inf, log=FALSE)
rmvtpc(n=1, mu, U, nu=Inf)This is either a vector of length \(k\) or a matrix with a number of columns, \(k\), equal to the number of columns in precision matrix \(\Omega\).
This is the number of random draws.
This is a numeric vector representing the location parameter,
\(\mu\) (the mean vector), of the multivariate distribution
(equal to the expected value when df > 1, otherwise
represented as \(\nu > 1\)). It must be of length
\(k\), as defined above.
This is a \(k \times k\) upper-triangular of the precision matrix that is Cholesky fator \(\textbf{U}\) of precision matrix \(\Omega\).
This is the degrees of freedom \(\nu\), which must be positive.
Logical. If log=TRUE, then the logarithm of the
density is returned.
dmvtpc gives the density and
rmvtpc generates random deviates.
Application: Continuous Multivariate
Density: $$p(\theta) = \frac{\Gamma((\nu+k)/2)}{\Gamma(\nu/2)\nu^{k/2}\pi^{k/2}} |\Omega|^{1/2} (1 + \frac{1}{\nu} (\theta-\mu)^T \Omega (\theta-\mu))^{-(\nu+k)/2}$$
Inventor: Unknown (to me, anyway)
Notation 1: \(\theta \sim \mathrm{t}_k(\mu, \Omega^{-1}, \nu)\)
Notation 2: \(p(\theta) = \mathrm{t}_k(\theta | \mu, \Omega^{-1}, \nu)\)
Parameter 1: location vector \(\mu\)
Parameter 2: positive-definite \(k \times k\) precision matrix \(\Omega\)
Parameter 3: degrees of freedom \(\nu > 0\)
Mean: \(E(\theta) = \mu\), for \(\nu > 1\), otherwise undefined
Variance: \(var(\theta) = \frac{\nu}{\nu - 2} \Omega^{-1}\), for \(\nu > 2\)
Mode: \(mode(\theta) = \mu\)
The multivariate t distribution, also called the multivariate Student or multivariate Student t distribution, is a multidimensional extension of the one-dimensional or univariate Student t distribution. A random vector is considered to be multivariate t-distributed if every linear combination of its components has a univariate Student t-distribution.
It is usually parameterized with mean and a covariance matrix, or in
Bayesian inference, with mean and a precision matrix, where the
precision matrix is the matrix inverse of the covariance matrix. These
functions provide the precision parameterization for convenience and
familiarity. It is easier to calculate a multivariate t density
with the precision parameterization, because a matrix inversion can be
avoided. The precision matrix is replaced with an upper-triangular
\(k \times k\) matrix that is Cholesky factor
\(\textbf{U}\), as per the chol function for Cholesky
decomposition.
This distribution has a mean parameter vector \(\mu\) of length \(k\), and a \(k \times k\) precision matrix \(\Omega\), which must be positive-definite. When degrees of freedom \(\nu=1\), this is the multivariate Cauchy distribution.
In practice, \(\textbf{U}\) is fully unconstrained for proposals
when its diagonal is log-transformed. The diagonal is exponentiated
after a proposal and before other calculations. Overall, the Cholesky
parameterization is faster than the traditional parameterization.
Compared with dmvtp, dmvtpc must additionally
matrix-multiply the Cholesky back to the precision matrix, but it
does not have to check for or correct the precision matrix to
positive-definiteness, which overall is slower. Compared with
rmvtp, rmvtpc is faster because the Cholesky decomposition
has already been performed.
# NOT RUN {
library(LaplacesDemon)
x <- seq(-2,4,length=21)
y <- 2*x+10
z <- x+cos(y)
mu <- c(1,12,2)
Omega <- matrix(c(1,2,0,2,5,0.5,0,0.5,3), 3, 3)
U <- chol(Omega)
nu <- 4
f <- dmvtpc(cbind(x,y,z), mu, U, nu)
X <- rmvtpc(1000, c(0,1,2), U, 5)
joint.density.plot(X[,1], X[,2], color=TRUE)
# }
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