R: Multivariate Cauchy Distribution: Cholesky Parameterization

dist.Multivariate.Cauchy.Cholesky {LaplacesDemon} | R Documentation

## Multivariate Cauchy Distribution: Cholesky Parameterization

### Description

These functions provide the density and random number generation for the multivariate Cauchy distribution, given the Cholesky parameterization.

### Usage

    dmvcc(x, mu, U, log=FALSE)
    rmvcc(n=1, mu, U)

### Arguments

`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.
`U` | This is the k x k upper-triangular matrix that is Cholesky factor U of the positive-definite scale matrix S.
`log` | Logical. If `log=TRUE`, then the logarithm of the density is returned.

### Details

- 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) =

- 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.

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, the Cholesky parameterization is faster than the traditional parameterization. Compared with `dmvc`, `dmvcc` must additionally matrix-multiply the Cholesky back to the scake matrix, but it does not have to check for or correct the scale matrix to positive-definiteness, which overall is slower. Compared with `rmvc`, `rmvcc` is faster because the Cholesky decomposition has already been performed.

### Value

`dmvcc` gives the density and `rmvcc` generates random deviates.

### Author(s)

Statisticat, LLC. software@bayesian-inference.com

### See Also

`chol`, `dcauchy`, `dinvwishartc`, `dmvcpc`, `dmvtc`, and `dmvtpc`.

### Examples

    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)
    U <- chol(Sigma)
    f <- dmvcc(cbind(x,y,z), mu, U)

    X <- rmvcc(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)

[Package LaplacesDemon version 16.1.1 Index]