LaplacesDemon (version 16.1.6)

dist.Multivariate.Cauchy.Cholesky: Multivariate Cauchy Distribution: Cholesky Parameterization

Description

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

Usage

dmvcc(x, mu, U, log=FALSE)
rmvcc(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 scale matrix \(\textbf{S}\).

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 scale matrix \(\textbf{S}\).

log

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

Value

dmvcc gives the density and rmvcc generates random deviates.

Details

• Application: Continuous Multivariate

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

• Inventor: Unknown (to me, anyway)

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

• Notation 2: \(p(\theta) = \mathcal{MC}_k(\theta | \mu, \Sigma)\)

• Parameter 1: location vector \(\mu\)

• Parameter 2: positive-definite \(k \times k\) scale matrix \(\Sigma\)

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

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.

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 dmvc, dmvcc must additionally matrix-multiply the Cholesky back to the scake matrix, but it does not have to check for or correct the scale matrix to positive-definiteness, which overall is slower. Compared with rmvc, rmvcc is faster because the Cholesky decomposition has already been performed.

See Also

chol, dcauchy, dinvwishartc, dmvcpc, dmvtc, and dmvtpc.

Examples

# NOT RUN {
library(LaplacesDemon)
x <- seq(-2,4,length=21)
y <- 2*x+10
z <- x+cos(y) 
mu <- c(1,12,2)
Sigma <- matrix(c(1,2,0,2,5,0.5,0,0.5,3), 3, 3)
U <- chol(Sigma)
f <- dmvcc(cbind(x,y,z), mu, U)

X <- rmvcc(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)
# }