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Multivariate Normal Density and Random Deviates

Mvnorm.Rd

These functions provide the density function and a random number generator for the multivariate normal distribution with mean equal to mean and covariance matrix sigma.

Usage

dmvnorm(x, mean = rep(0, p), sigma = diag(p), log = FALSE, checkSymmetry = TRUE)
rmvnorm(n, mean = rep(0, nrow(sigma)), sigma = diag(length(mean)),
           method=c("eigen", "svd", "chol"), pre0.9_9994 = FALSE, 
           checkSymmetry = TRUE, rnorm = stats::rnorm)

Arguments

x

vector or matrix of quantiles. When x is a matrix, each row is taken to be a quantile and columns correspond to the number of dimensions, p.

n

number of observations.

mean

mean vector, default is rep(0, length = ncol(x)). In ldmvnorm or sldmvnorm, mean is a matrix with observation-specific means arranged in columns.

sigma

covariance matrix, default is diag(ncol(x)).

log

logical; if TRUE, densities d are given as log(d).

method

string specifying the matrix decomposition used to determine the matrix root of sigma. Possible methods are eigenvalue decomposition ("eigen", default), singular value decomposition ("svd"), and Cholesky decomposition ("chol"). The Cholesky is typically fastest, not by much though.

pre0.9_9994

logical; if FALSE, the output produced in mvtnorm versions up to 0.9-9993 is reproduced. In 0.9-9994, the output is organized such that rmvnorm(10,...) has the same first ten rows as rmvnorm(100, ...) when called with the same seed.

checkSymmetry

logical; if FALSE, skip checking whether the covariance matrix is symmetric or not. This will speed up the computation but may cause unexpected outputs when ill-behaved sigma is provided. The default value is TRUE.

rnorm

a function with the same interface as rnorm. This allows switching to other generators of standard normal variables.

Details

dmvnorm computes the density function of the multivariate normal specified by mean and the covariance matrix sigma.

rmvnorm generates multivariate normal variables.

See also

pmvnorm, rnorm, qmvnorm, vignette("lmvnorm_src", package = "mvtnorm")

Examples

dmvnorm(x=c(0,0))
#> [1] 0.1591549
dmvnorm(x=c(0,0), mean=c(1,1))
#> [1] 0.05854983

sigma <- matrix(c(4,2,2,3), ncol=2)
x <- rmvnorm(n=500, mean=c(1,2), sigma=sigma)
colMeans(x)
#> [1] 1.017563 2.004162
var(x)
#>          [,1]     [,2]
#> [1,] 4.052903 2.044061
#> [2,] 2.044061 3.008296
dS <- dmvnorm(x, sigma = sigma)

### alternative interface
C <- t(chol(sigma))
(C <- ltMatrices(C[lower.tri(C, diag = TRUE)], diag = TRUE))
#> , , 1
#> 
#>          1        2
#> 1 2.000000        .
#> 2 1.000000 1.414214
#> 
dC <- exp(ldmvnorm(obs = t(x), chol = C, logLik = FALSE))
all.equal(dS, dC)
#> [1] TRUE

x <- rmvnorm(n=500, mean=c(1,2), sigma=sigma, method="chol")
colMeans(x)
#> [1] 0.9191267 1.9707497
var(x)
#>          [,1]     [,2]
#> [1,] 3.813205 1.900035
#> [2,] 1.900035 2.784946

plot(x)