Multivariate Normal Distribution
These functions provide the density and random number generation for the multivariate normal distribution.
dmvn(x, mu, Sigma, log=FALSE) rmvn(n=1, mu, Sigma)
x |
This is data or parameters in the form of a vector of length k or a matrix with k columns. |
n |
This is the number of random draws. |
mu |
This is mean vector mu with length k or matrix with k columns. |
Sigma |
This is the k x k covariance matrix Sigma. |
log |
Logical. If |
Application: Continuous Multivariate
Density: p(theta) = (1/((2*pi)^(k/2)*|Sigma|^(1/2))) * exp(-(1/2)*(theta-mu)'*Sigma^(-1)*(theta-mu))
Inventors: Robert Adrain (1808), Pierre-Simon Laplace (1812), and Francis Galton (1885)
Notation 1: theta ~ MVN(mu, Sigma)
Notation 2: theta ~ N[k](mu, Sigma)
Notation 3: p(theta) = MVN(theta | mu, Sigma)
Notation 4: p(theta) = N[k](theta | mu, Sigma)
Parameter 1: location vector mu
Parameter 2: positive-definite k x 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 one-dimensional 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 x k covariance matrix Sigma, which must be positive-definite.
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.
dmvn gives the density and
rmvn generates random deviates.
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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)
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