dist.Multivariate.t.Precision.Cholesky {LaplacesDemon} R Documentation

## Multivariate t Distribution: Precision-Cholesky 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 and Cholesky parameterization.

### Usage

dmvtpc(x, mu, U, nu=Inf, log=FALSE)
rmvtpc(n=1, mu, U, 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 \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. U This is a k \times k upper-triangular of the precision matrix that is Cholesky fator \textbf{U} of precision matrix \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.

### Details

• 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.

### Value

dmvtpc gives the density and rmvtpc generates random deviates.

### Author(s)

Statisticat, LLC. software@bayesian-inference.com

chol, dwishartc, dmvc, dmvcp, dmvtc, dst, dstp, and dt.

### Examples

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)


[Package LaplacesDemon version 16.1.6 Index]