LaplacesDemon (version 16.1.0)

dist.Multivariate.t.Precision: Multivariate t Distribution: Precision Parameterization

Description

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 parameterization.

Usage

dmvtp(x, mu, Omega, nu=Inf, log=FALSE)
rmvtp(n=1, mu, Omega, nu=Inf)

Arguments

x

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 $\mathrm{\Omega }$.

n

This is the number of random draws.

mu

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.

Omega

This is a $k\times k$ positive-definite precision matrix $\mathrm{\Omega }$.

nu

This is the degrees of freedom $\nu$, which must be positive.

log

Logical. If log=TRUE, then the logarithm of the density is returned.

Value

dmvtp gives the density and rmvtp generates random deviates.

Details

• Application: Continuous Multivariate

• Density: $$p(\theta )=\frac{\mathrm{\Gamma }((\nu +k)/2)}{\mathrm{\Gamma }(\nu /2){\nu }^{k/2}{\pi }^{k/2}}|\mathrm{\Omega }{|}^{1/2}(1+\frac{1}{\nu }(\theta -\mu {)}^T\mathrm{\Omega }(\theta -\mu ){)}^{-(\nu +k)/2}$$

• Inventor: Unknown (to me, anyway)

• Notation 1: $\theta \sim {\mathrm{t}}_k(\mu ,{\mathrm{\Omega }}^{-1},\nu )$

• Notation 2: $p(\theta )={\mathrm{t}}_k(\theta |\mu ,{\mathrm{\Omega }}^{-1},\nu )$

• Parameter 1: location vector $\mu$

• Parameter 2: positive-definite $k\times k$ precision matrix $\mathrm{\Omega }$

• Parameter 3: degrees of freedom $\nu >0$

• Mean: $E(\theta )=\mu$, for $\nu >1$, otherwise undefined

• Variance: $var(\theta )=\frac{\nu }{\nu -2}{\mathrm{\Omega }}^{-1}$, 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.

This distribution has a mean parameter vector $\mu$ of length $k$, and a $k\times k$ precision matrix $\mathrm{\Omega }$, which must be positive-definite. When degrees of freedom $\nu =1$, this is the multivariate Cauchy distribution.

See Also

dwishart, dmvc, dmvcp, dmvt, dst, dstp, and dt.

Examples

# 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)
nu <- 4
f <- dmvtp(cbind(x,y,z), mu, Omega, nu)
X <- rmvtp(1000, c(0,1,2), diag(3), 5)
joint.density.plot(X[,1], X[,2], color=TRUE)
# }