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

## Multivariate Cauchy Distribution: Precision-Cholesky Parameterization

### Description

These functions provide the density and random number generation for the multivariate Cauchy distribution. These functions use the precision and Cholesky parameterization.

### Usage

dmvcpc(x, mu, U, log=FALSE)
rmvcpc(n=1, mu, U)


### 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. It must be of length k, as defined above. U This is the k \times k upper-triangular matrix that is Cholesky factor \textbf{U} of the positive-definite precision matrix \Omega. log Logical. If log=TRUE, then the logarithm of the density is returned.

### Details

• Application: Continuous Multivariate

• Density:

p(\theta) = \frac{\Gamma((1+k)/2)}{\Gamma(1/2)1^{k/2}\pi^{k/2}} |\Omega|^{1/2} (1 + (\theta-\mu)^T \Omega (\theta-\mu))^{-(1+k)/2}

• Inventor: Unknown (to me, anyway)

• Notation 1: \theta \sim \mathcal{MC}_k(\mu, \Omega^{-1})

• Notation 2: p(\theta) = \mathcal{MC}_k(\theta | \mu, \Omega^{-1})

• Parameter 1: location vector \mu

• Parameter 2: positive-definite k \times k precision matrix \Omega

• Mean: E(\theta) = \mu

• Variance: var(\theta) =

• Mode: mode(\theta) = \mu

The multivariate Cauchy distribution is a multidimensional extension of the one-dimensional or univariate Cauchy distribution. A random vector is considered to be multivariate Cauchy-distributed if every linear combination of its components has a univariate Cauchy distribution. The multivariate Cauchy distribution is equivalent to a multivariate t distribution with 1 degree of freedom.

The Cauchy distribution is known as a pathological distribution because its mean and variance are undefined, and it does not satisfy the central limit theorem.

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 Cauchy 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 \Omega, which must be positive-definite. The precision matrix is replaced with the upper-triangular Cholesky factor, as in chol.

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, Cholesky parameterization is faster than the traditional parameterization. Compared with dmvcp, dmvcpc must additionally matrix-multiply the Cholesky back to the covariance matrix, but it does not have to check for or correct the precision matrix to positive-definiteness, which overall is slower. Compared with rmvcp, rmvcpc is faster because the Cholesky decomposition has already been performed.

### Value

dmvcpc gives the density and rmvcpc generates random deviates.

### Author(s)

Statisticat, LLC. software@bayesian-inference.com

chol, dcauchy, dmvcc, dmvtc, dmvtpc, and dwishartc.

### 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)
f <- dmvcpc(cbind(x,y,z), mu, U)

X <- rmvcpc(1000, rep(0,2), diag(2))
X <- X[rowSums((X >= quantile(X, probs=0.025)) &
(X <= quantile(X, probs=0.975)))==2,]
joint.density.plot(X[,1], X[,2], color=TRUE)


[Package LaplacesDemon version 16.1.6 Index]