R: Multivariate t Distribution: Precision-Cholesky...

dist.Multivariate.t.Precision.Cholesky {LaplacesDemon} | R Documentation

## Multivariate t Distribution: Precision-Cholesky Parameterization

### Description

These functions provide the density and random number generation for the multivariate t distribution, otherwise called the multivariate Student distribution. These functions use the precision and Cholesky parameterization.

### Usage

    dmvtpc(x, mu, U, nu=Inf, log=FALSE)
    rmvtpc(n=1, mu, U, nu=Inf)

### 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 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 (equal to the expected value when `df > 1`, otherwise represented as nu > 1). It must be of length k, as defined above.
`U` | This is a k x k upper-triangular of the precision matrix that is Cholesky fator U of precision matrix Omega.
`nu` | This is the degrees of freedom nu, which must be positive.
`log` | Logical. If `log=TRUE`, then the logarithm of the density is returned.

### Details

- Application: Continuous Multivariate

- Density:

  p(theta) = (Gamma((nu+k)/2) / (Gamma(nu/2)*nu^(k/2)*pi^(k/2))) * |Omega|^(1/2) * (1 + (1/nu) (theta-mu)^T Omega (theta-mu))^(-(nu+k)/2)

- Inventor: Unknown (to me, anyway)

- Notation 1: theta ~ t[k](mu, Omega^(-1), nu)

- Notation 2: p(theta) = t[k](theta | mu, Omega^(-1), ν)

- Parameter 1: location vector mu

- Parameter 2: positive-definite k x k precision matrix Omega

- Parameter 3: degrees of freedom nu > 0

- Mean: E(theta) = mu, for nu > 1, otherwise undefined

- Variance: var(theta) = (nu / (nu - 2))*Omega^(-1), for nu> 2

- Mode: mode(theta) = mu

The multivariate t distribution, also called the multivariate Student or multivariate Student t distribution, is a multidimensional extension of the one-dimensional or univariate Student t distribution. A random vector is considered to be multivariate t-distributed if every linear combination of its components has a univariate Student t-distribution.

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 t density with the precision parameterization, because a matrix inversion can be avoided. The precision matrix is replaced with an upper-triangular k x k matrix that is Cholesky factor U, as per the `chol` function for Cholesky decomposition.

This distribution has a mean parameter vector mu of length k, and a k x k precision matrix Omega, which must be positive-definite. When degrees of freedom nu=1, this is the multivariate Cauchy distribution.

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 `dmvtp`, `dmvtpc` must additionally matrix-multiply the Cholesky back to the precision matrix, but it does not have to check for or correct the precision matrix to positive-definiteness, which overall is slower. Compared with `rmvtp`, `rmvtpc` is faster because the Cholesky decomposition has already been performed.

### Value

`dmvtpc` gives the density and `rmvtpc` generates random deviates.

### Author(s)

Statisticat, LLC. software@bayesian-inference.com

### See Also

`chol`, `dwishartc`, `dmvc`, `dmvcp`, `dmvtc`, `dst`, `dstp`, and `dt`.

### Examples

    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)
    nu <- 4
    f <- dmvtpc(cbind(x,y,z), mu, U, nu)
    X <- rmvtpc(1000, c(0,1,2), U, 5)
    joint.density.plot(X[,1], X[,2], color=TRUE)

[Package LaplacesDemon version 16.1.1 Index]