dist.Multivariate.Cauchy.Precision: Multivariate Cauchy Distribution: Precision Parameterization
In LaplacesDemonR/LaplacesDemonCpp: C++ Extension for LaplacesDemon

Description Usage Arguments Details Value Author(s) See Also Examples

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

1 2	dmvcp(x, mu, Omega, log=FALSE) rmvcp(n=1, mu, Omega)

`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.

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.

dmvcp gives the density and rmvcp generates random deviates.

Statisticat, LLC. software@bayesian-inference.com

dcauchy, dmvc, dmvt, dmvtp, and dwishart.

library(LaplacesDemonCpp)
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)