dist.Multivariate.t | R Documentation |
These functions provide the density and random number generation for the multivariate t distribution, otherwise called the multivariate Student distribution.
dmvt(x, mu, S, df=Inf, log=FALSE)
rmvt(n=1, mu, S, df=Inf)
x |
This is either a vector of length |
n |
This is the number of random draws. |
mu |
This is a numeric vector or matrix representing the location
parameter, |
S |
This is a |
df |
This is the degrees of freedom, and is often represented
with |
log |
Logical. If |
Application: Continuous Multivariate
Density:
p(\theta) =
\frac{\Gamma[(\nu+k)/2]}{\Gamma(\nu/2)\nu^{k/2}\pi^{k/2}|\Sigma|^{1/2}[1
+ (1/\nu)(\theta-\mu)^{\mathrm{T}} \Sigma^{-1}
(\theta-\mu)]^{(\nu+k)/2}}
Inventor: Unknown (to me, anyway)
Notation 1: \theta \sim \mathrm{t}_k(\mu,
\Sigma, \nu)
Notation 2: p(\theta) = \mathrm{t}_k(\theta | \mu,
\Sigma, \nu)
Parameter 1: location vector \mu
Parameter 2: positive-definite k \times 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) = \frac{\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 a k \times k
scale matrix \textbf{S}
,
which must be positive-definite. When degrees of freedom
\nu=1
, this is the multivariate Cauchy distribution.
dmvt
gives the density and
rmvt
generates random deviates.
Statisticat, LLC. software@bayesian-inference.com
dinvwishart
,
dmvc
,
dmvcp
,
dmvtp
,
dst
,
dstp
, and
dt
.
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)
df <- 4
f <- dmvt(cbind(x,y,z), mu, S, df)
X <- rmvt(1000, c(0,1,2), S, 5)
joint.density.plot(X[,1], X[,2], color=TRUE)
Add the following code to your website.
For more information on customizing the embed code, read Embedding Snippets.