dist.Multivariate.t.Cholesky: Multivariate t Distribution: Cholesky Parameterization

Description Usage Arguments Details Value Author(s) See Also Examples

Description

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

Usage

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dmvtc(x, mu, U, df=Inf, log=FALSE)
rmvtc(n=1, mu, U, df=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 scale matrix S.

n

This is the number of random draws.

mu

This is a numeric vector or matrix 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). When a vector, it must be of length k, or must have k columns as a matrix, as defined above.

U

This is the k x k upper-triangular matrix that is Cholesky factor U of scale matrix S, such that S*df/(df-2) is the variance-covariance matrix when df > 2.

df

This is the degrees of freedom, and is often represented with nu.

log

Logical. If log=TRUE, then the logarithm of the density is returned.

Details

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. This distribution has a mean parameter vector mu of length k, and an upper-triangular k x k matrix that is Cholesky factor U, as per the chol function for Cholesky decomposition. 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 dmvt, dmvtc must additionally matrix-multiply the Cholesky back to the scale matrix, but it does not have to check for or correct the scale matrix to positive-definiteness, which overall is slower. The same is true when comparing rmvt and rmvtc.

Value

dmvtc gives the density and rmvtc generates random deviates.

Author(s)

Statisticat, LLC. software@bayesian-inference.com

See Also

chol, dinvwishartc, dmvc, dmvcp, dmvtp, dst, dstp, and dt.

Examples

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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)
S <- matrix(c(1,2,0,2,5,0.5,0,0.5,3), 3, 3)
U <- chol(S)
df <- 4
f <- dmvtc(cbind(x,y,z), mu, U, df)
X <- rmvtc(1000, c(0,1,2), U, 5)
joint.density.plot(X[,1], X[,2], color=TRUE)

LaplacesDemon documentation built on July 9, 2021, 5:07 p.m.