distributional (version 0.6.0)

dist_multivariate_normal: The multivariate normal distribution

Description

The multivariate normal distribution is a generalization of the univariate normal distribution to higher dimensions. It is widely used in multivariate statistics and describes the joint distribution of multiple correlated continuous random variables.

Usage

dist_multivariate_normal(mu = 0, sigma = diag(1))

Arguments

mu

A list of numeric vectors for the distribution's mean.

sigma

A list of matrices for the distribution's variance-covariance matrix.

Details

We recommend reading this documentation on pkgdown which renders math nicely. https://pkg.mitchelloharawild.com/distributional/reference/dist_multivariate_normal.html

In the following, let \(\mathbf{X}\) be a \(k\)-dimensional multivariate normal random variable with mean vector mu = \(\boldsymbol{\mu}\) and variance-covariance matrix sigma = \(\boldsymbol{\Sigma}\).

Support: \(\mathbf{x} \in \mathbb{R}^k\)

Mean: \(\boldsymbol{\mu}\)

Variance-covariance matrix: \(\boldsymbol{\Sigma}\)

Probability density function (p.d.f):

$$ f(\mathbf{x}) = \frac{1}{(2\pi)^{k/2} |\boldsymbol{\Sigma}|^{1/2}} \exp\left(-\frac{1}{2}(\mathbf{x} - \boldsymbol{\mu})^T \boldsymbol{\Sigma}^{-1}(\mathbf{x} - \boldsymbol{\mu})\right) $$

where \(|\boldsymbol{\Sigma}|\) is the determinant of \(\boldsymbol{\Sigma}\).

Cumulative distribution function (c.d.f):

$$ P(\mathbf{X} \le \mathbf{q}) = P(X_1 \le q_1, \ldots, X_k \le q_k) $$

The c.d.f. does not have a closed-form expression and is computed numerically.

Moment generating function (m.g.f):

$$ M(\mathbf{t}) = E(e^{\mathbf{t}^T \mathbf{X}}) = \exp\left(\mathbf{t}^T \boldsymbol{\mu} + \frac{1}{2}\mathbf{t}^T \boldsymbol{\Sigma} \mathbf{t}\right) $$

See Also

mvtnorm::dmvnorm(), mvtnorm::pmvnorm(), mvtnorm::qmvnorm(), mvtnorm::rmvnorm()

Examples

dist <- dist_multivariate_normal(mu = list(c(1,2)), sigma = list(matrix(c(4,2,2,3), ncol=2)))
dimnames(dist) <- c("x", "y")
dist

mean(dist)
variance(dist)
support(dist)
generate(dist, 10)

density(dist, cbind(2, 1))
density(dist, cbind(2, 1), log = TRUE)

cdf(dist, 4)

quantile(dist, 0.7, kind = "equicoordinate")
quantile(dist, 0.7, kind = "marginal")