dist.Multivariate.Cauchy.Precision.Cholesky {LaplacesDemon}  R Documentation 
Multivariate Cauchy Distribution: PrecisionCholesky 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 
n 
This is the number of random draws. 
mu 
This is a numeric vector representing the location parameter,

U 
This is the 
log 
Logical. If 
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: positivedefinite
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 onedimensional or univariate Cauchy distribution. A random vector is considered to be multivariate Cauchydistributed 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 positivedefinite. The precision
matrix is replaced with the uppertriangular Cholesky factor, as in
chol
.
In practice, \textbf{U}
is fully unconstrained for proposals
when its diagonal is logtransformed. 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
matrixmultiply the Cholesky back to the covariance matrix, but it
does not have to check for or correct the precision matrix to
positivedefiniteness, 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@bayesianinference.com
See Also
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)