These functions provide the density and random number generation for the multivariate Cauchy distribution. These functions use the precision and Cholesky parameterization.
dmvcpc(x, mu, U, log=FALSE)
rmvcpc(n=1, mu, U)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 \(\Omega\).
This is the number of random draws.
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.
This is the \(k \times k\) upper-triangular matrix that is Cholesky factor \(\textbf{U}\) of the positive-definite precision matrix \(\Omega\).
Logical. If log=TRUE, then the logarithm of the
density is returned.
dmvcpc gives the density and
rmvcpc generates random deviates.
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.
# NOT RUN {
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)
# }
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