Multivariate Cauchy Distribution: Precision Parameterization
These functions provide the density and random number generation for the multivariate Cauchy distribution. These functions use the precision parameterization.
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 a k x k positive-definite precision matrix Omega.
Application: Continuous Multivariate
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.
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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) 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)