dist.Multivariate.Normal {LaplacesDemon}  R Documentation 
Multivariate Normal Distribution
Description
These functions provide the density and random number generation for the multivariate normal distribution.
Usage
dmvn(x, mu, Sigma, log=FALSE)
rmvn(n=1, mu, Sigma)
Arguments
x 
This is data or parameters in the form of a vector of length

n 
This is the number of random draws. 
mu 
This is mean vector 
Sigma 
This is the 
log 
Logical. If 
Details
Application: Continuous Multivariate
Density:
p(\theta) = \frac{1}{(2\pi)^{k/2}\Sigma^{1/2}} \exp(\frac{1}{2}(\theta  \mu)'\Sigma^{1}(\theta  \mu))
Inventors: Robert Adrain (1808), PierreSimon Laplace (1812), and Francis Galton (1885)
Notation 1:
\theta \sim \mathcal{MVN}(\mu, \Sigma)
Notation 2:
\theta \sim \mathcal{N}_k(\mu, \Sigma)
Notation 3:
p(\theta) = \mathcal{MVN}(\theta  \mu, \Sigma)
Notation 4:
p(\theta) = \mathcal{N}_k(\theta  \mu, \Sigma)
Parameter 1: location vector
\mu
Parameter 2: positivedefinite
k \times k
covariance matrix\Sigma
Mean:
E(\theta) = \mu
Variance:
var(\theta) = \Sigma
Mode:
mode(\theta) = \mu
The multivariate normal distribution, or multivariate Gaussian
distribution, is a multidimensional extension of the onedimensional
or univariate normal (or Gaussian) distribution. A random vector is
considered to be multivariate normally distributed if every linear
combination of its components has a univariate normal distribution.
This distribution has a mean parameter vector \mu
of length
k
and a k \times k
covariance matrix
\Sigma
, which must be positivedefinite.
The conjugate prior of the mean vector is another multivariate normal
distribution. The conjugate prior of the covariance matrix is the
inverse Wishart distribution (see dinvwishart
).
When applicable, the alternative Cholesky parameterization should be
preferred. For more information, see dmvnc
.
For models where the dependent variable, Y, is specified to be
distributed multivariate normal given the model, the Mardia test (see
plot.demonoid.ppc
, plot.laplace.ppc
, or
plot.pmc.ppc
) may be used to test the residuals.
Value
dmvn
gives the density and
rmvn
generates random deviates.
Author(s)
Statisticat, LLC. software@bayesianinference.com
See Also
dinvwishart
,
dmatrixnorm
,
dmvnc
,
dmvnp
,
dnorm
,
dnormp
,
dnormv
,
plot.demonoid.ppc
,
plot.laplace.ppc
, and
plot.pmc.ppc
.
Examples
library(LaplacesDemon)
x < dmvn(c(1,2,3), c(0,1,2), diag(3))
X < rmvn(1000, c(0,1,2), diag(3))
joint.density.plot(X[,1], X[,2], color=TRUE)