These functions provide the density and random number generation for the multivariate t distribution, otherwise called the multivariate Student distribution, given the Cholesky parameterization.
dmvtc(x, mu, U, df=Inf, log=FALSE)
rmvtc(n=1, mu, U, df=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 scale matrix \(\textbf{S}\).
This is the number of random draws.
This is a numeric vector or matrix representing the location
parameter,\(\mu\) (the mean vector), of the multivariate
distribution (equal to the expected value when df > 1,
otherwise represented as \(\nu > 1\)). When a vector, it
must be of length \(k\), or must have \(k\) columns as a matrix,
as defined above.
This is the \(k \times k\) upper-triangular matrix
that is Cholesky factor \(\textbf{U}\) of scale matrix
\(\textbf{S}\), such that S*df/(df-2) is the
variance-covariance matrix when df > 2.
This is the degrees of freedom, and is often represented with \(\nu\).
Logical. If log=TRUE, then the logarithm of the
density is returned.
dmvtc gives the density and
rmvtc generates random deviates.
Application: Continuous Multivariate
Density: $$p(\theta) = \frac{\Gamma[(\nu+k)/2]}{\Gamma(\nu/2)\nu^{k/2}\pi^{k/2}|\Sigma|^{1/2}[1 + (1/\nu)(\theta-\mu)^{\mathrm{T}} \Sigma^{-1} (\theta-\mu)]^{(\nu+k)/2}}$$
Inventor: Unknown (to me, anyway)
Notation 1: \(\theta \sim \mathrm{t}_k(\mu, \Sigma, \nu)\)
Notation 2: \(p(\theta) = \mathrm{t}_k(\theta | \mu, \Sigma, \nu)\)
Parameter 1: location vector \(\mu\)
Parameter 2: positive-definite \(k \times k\) scale matrix \(\Sigma\)
Parameter 3: degrees of freedom \(\nu > 0\) (df in the functions)
Mean: \(E(\theta) = \mu\), for \(\nu > 1\), otherwise undefined
Variance: \(var(\theta) = \frac{\nu}{\nu - 2} \Sigma\), 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.
This distribution has a mean parameter vector \(\mu\) of length
\(k\), and an upper-triangular \(k \times k\) matrix that is
Cholesky factor \(\textbf{U}\), as per the chol
function for Cholesky decomposition. 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 dmvt, dmvtc must additionally
matrix-multiply the Cholesky back to the scale matrix, but it
does not have to check for or correct the scale matrix to
positive-definiteness, which overall is slower. The same is true when
comparing rmvt and rmvtc.
# NOT RUN {
library(LaplacesDemon)
x <- seq(-2,4,length=21)
y <- 2*x+10
z <- x+cos(y)
mu <- c(1,12,2)
S <- matrix(c(1,2,0,2,5,0.5,0,0.5,3), 3, 3)
U <- chol(S)
df <- 4
f <- dmvtc(cbind(x,y,z), mu, U, df)
X <- rmvtc(1000, c(0,1,2), U, 5)
joint.density.plot(X[,1], X[,2], color=TRUE)
# }
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