Multivariate Cauchy Distribution
These functions provide the density and random number generation for the multivariate Cauchy distribution.
dmvc(x, mu, S, log=FALSE) rmvc(n=1, mu, S)
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 scale matrix S. |
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. |
S |
This is a k x k positive-definite scale matrix S. |
log |
Logical. If |
Application: Continuous Multivariate
Density:
p(theta) = Gamma[(1+k)/2] / {Gamma(1/2)1^(k/2)pi^(k/2)|Sigma|^(1/2)[1+(theta-mu)^T*Sigma^(-1)(theta-mu)]^[(1+k)/2]}
Inventor: Unknown (to me, anyway)
Notation 1: theta ~ MC[k](mu, Sigma)
Notation 2: p(theta) = MC[k](theta | mu, Sigma)
Parameter 1: location vector mu
Parameter 2: positive-definite k x k scale matrix Sigma
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. The multivariate Cauchy distribution is equivalent to a multivariate t distribution with 1 degree of freedom. A random vector is considered to be multivariate Cauchy-distributed if every linear combination of its components has a univariate Cauchy distribution.
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.
dmvc gives the density and
rmvc generates random deviates.
Statisticat, LLC. software@bayesian-inference.com
dcauchy,
dinvwishart,
dmvcp,
dmvt, and
dmvtp.
library(LaplacesDemon)
x <- seq(-2,4,length=21)
y <- 2*x+10
z <- x+cos(y)
mu <- c(1,12,2)
Sigma <- matrix(c(1,2,0,2,5,0.5,0,0.5,3), 3, 3)
f <- dmvc(cbind(x,y,z), mu, Sigma)
X <- rmvc(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.