R: Multivariate Normal Distribution

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 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 `log=TRUE`, then the logarithm of the density is returned.

### Details

- 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.

### Value

`dmvn` gives the density and `rmvn` generates random deviates.

### Author(s)

Statisticat, LLC. software@bayesian-inference.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)

[Package LaplacesDemon version 16.1.1 Index]