dist.Multivariate.Cauchy.Precision: Multivariate Cauchy Distribution: Precision Parameterization

In LaplacesDemon: Complete Environment for Bayesian Inference

Description

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

Usage

dmvcp(x, mu, Omega, log=FALSE)
rmvcp(n=1, mu, Omega)

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.
Omega	This is a k x k 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) = (Gamma((nu+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 ~ MC[k](mu, Omega^(-1))

• Notation 2: p(theta) = MC[k](theta | mu, Omega^(-1))

• Parameter 1: location vector mu

• Parameter 2: positive-definite k x k precision matrix Omega

• Mean: E(theta) = mu

• Variance: var(theta) = undefined

• 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 x k precision matrix Omega, which must be positive-definite.

Value

dmvcp gives the density and rmvcp generates random deviates.

Author(s)

Statisticat, LLC. [e-mail]

See Also

dcauchy, dmvc, dmvt, dmvtp, and dwishart.

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)
f <- dmvcp(cbind(x,y,z), mu, Omega)

X <- rmvcp(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)

Example output

LaplacesDemon documentation built on July 9, 2021, 5:07 p.m.