Multivariate Cauchy Distribution: Precision-Cholesky Parameterization
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)
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. |
U |
This is the k x k upper-triangular matrix that is Cholesky factor U of the positive-definite precision matrix Omega. |
log |
Logical. If |
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) =
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. The precision
matrix is replaced with the upper-triangular Cholesky factor, as in
chol.
In practice, 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.
dmvcpc gives the density and
rmvcpc generates random deviates.
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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)Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.