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 |
n |
This is the number of random draws. |
mu |
This is a numeric vector representing the location parameter,
|
U |
This is a |
nu |
This is the degrees of freedom |
log |
Logical. If |
Details
Application: Continuous Multivariate
Density:
Inventor: Unknown (to me, anyway)
Notation 1:
Notation 2:
Parameter 1: location vector
Parameter 2: positive-definite
precision matrix
Parameter 3: degrees of freedom
Mean:
, for
, otherwise undefined
Variance:
, for
Mode:
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
matrix that is Cholesky factor
, as per the
chol
function for Cholesky
decomposition.
This distribution has a mean parameter vector of length
, and a
precision matrix
, which must be positive-definite. When degrees of
freedom
, this is the multivariate Cauchy distribution.
In practice, 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)