dist.Multivariate.t.Cholesky: Multivariate t Distribution: Cholesky Parameterization
In LaplacesDemonR/LaplacesDemonCpp: C++ Extension for LaplacesDemon

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

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

1 2	dmvtc(x, mu, U, df=Inf, log=FALSE) rmvtc(n=1, mu, U, df=Inf)

`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.

Application: Continuous Multivariate
Density:

p(theta) = Gamma[(nu+k)/2] / {Gamma(nu/2)nu^(k/2)pi^(k/2)|Sigma|^(1/2)[1 + (1/nu)(theta-mu)^T*Sigma^(-1)(theta-mu)]^[(nu+k)/2]}
Inventor: Unknown (to me, anyway)
Notation 1: theta ~ t[k](mu, Sigma, nu)
Notation 2: p(theta) = t[k](theta | mu, Sigma, nu)
Parameter 1: location vector mu
Parameter 2: positive-definite k x k scale matrix Sigma
Parameter 3: degrees of freedom nu > 0 (df in the functions)
Mean: E(theta) = mu, for nu > 1, otherwise undefined
Variance: var(theta) = (nu / (nu - 2))*Sigma, 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. 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.

dmvtc gives the density and rmvtc generates random deviates.

Statisticat, LLC. software@bayesian-inference.com

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

library(LaplacesDemonCpp)
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)