MultivariateNormal: Multivariate Normal Distribution Class In distr6: The Complete R6 Probability Distributions Interface

 MultivariateNormal R Documentation

Multivariate Normal Distribution Class

Description

Mathematical and statistical functions for the Multivariate Normal distribution, which is commonly used to generalise the Normal distribution to higher dimensions, and is commonly associated with Gaussian Processes.

Details

The Multivariate Normal distribution parameterised with mean, μ, and covariance matrix, Σ, is defined by the pdf,

f(x_1,...,x_k) = (2 * π)^{-k/2}det(Σ)^{-1/2}exp(-1/2(x-μ)^TΣ^{-1}(x-μ))

for μ ε R^{k} and Σ ε R^{k x k}.

Sampling is performed via the Cholesky decomposition using chol.

Number of variables cannot be changed after construction.

Value

Returns an R6 object inheriting from class SDistribution.

Distribution support

The distribution is supported on the Reals and only when the covariance matrix is positive-definite.

Default Parameterisation

MultiNorm(mean = rep(0, 2), cov = c(1, 0, 0, 1))

Omitted Methods

`cdf` and `quantile` are omitted as no closed form analytic expression could be found, decorate with `FunctionImputation` for a numerical imputation.

N/A

Super classes

`distr6::Distribution` -> `distr6::SDistribution` -> `MultivariateNormal`

Public fields

`name`

Full name of distribution.

`short_name`

Short name of distribution for printing.

`description`

Brief description of the distribution.

Active bindings

`properties`

Returns distribution properties, including skewness type and symmetry.

Methods

Public methods

Inherited methods

Method `new()`

Creates a new instance of this R6 class. Number of variables cannot be changed after construction.

Usage
```MultivariateNormal\$new(
mean = rep(0, 2),
cov = c(1, 0, 0, 1),
prec = NULL,
decorators = NULL
)```
Arguments
`mean`

`(numeric())`
Vector of means, defined on the Reals.

`cov`

`(matrix()|vector())`
Covariance of the distribution, either given as a matrix or vector coerced to a matrix via `matrix(cov, nrow = K, byrow = FALSE)`. Must be semi-definite.

`prec`

`(matrix()|vector())`
Precision of the distribution, inverse of the covariance matrix. If supplied then `cov` is ignored. Given as a matrix or vector coerced to a matrix via `matrix(cov, nrow = K, byrow = FALSE)`. Must be semi-definite.

`decorators`

`(character())`
Decorators to add to the distribution during construction.

Method `mean()`

The arithmetic mean of a (discrete) probability distribution X is the expectation

E_X(X) = ∑ p_X(x)*x

with an integration analogue for continuous distributions.

Usage
`MultivariateNormal\$mean(...)`
`...`

Unused.

Method `mode()`

The mode of a probability distribution is the point at which the pdf is a local maximum, a distribution can be unimodal (one maximum) or multimodal (several maxima).

Usage
`MultivariateNormal\$mode(which = "all")`
Arguments
`which`

`(character(1) | numeric(1)`
Ignored if distribution is unimodal. Otherwise `"all"` returns all modes, otherwise specifies which mode to return.

Method `variance()`

The variance of a distribution is defined by the formula

var_X = E[X^2] - E[X]^2

where E_X is the expectation of distribution X. If the distribution is multivariate the covariance matrix is returned.

Usage
`MultivariateNormal\$variance(...)`
`...`

Unused.

Method `entropy()`

The entropy of a (discrete) distribution is defined by

- ∑ (f_X)log(f_X)

where f_X is the pdf of distribution X, with an integration analogue for continuous distributions.

Usage
`MultivariateNormal\$entropy(base = 2, ...)`
Arguments
`base`

`(integer(1))`
Base of the entropy logarithm, default = 2 (Shannon entropy)

`...`

Unused.

Method `mgf()`

The moment generating function is defined by

mgf_X(t) = E_X[exp(xt)]

where X is the distribution and E_X is the expectation of the distribution X.

Usage
`MultivariateNormal\$mgf(t, ...)`
Arguments
`t`

`(integer(1))`
t integer to evaluate function at.

`...`

Unused.

Method `cf()`

The characteristic function is defined by

cf_X(t) = E_X[exp(xti)]

where X is the distribution and E_X is the expectation of the distribution X.

Usage
`MultivariateNormal\$cf(t, ...)`
Arguments
`t`

`(integer(1))`
t integer to evaluate function at.

`...`

Unused.

Method `pgf()`

The probability generating function is defined by

pgf_X(z) = E_X[exp(z^x)]

where X is the distribution and E_X is the expectation of the distribution X.

Usage
`MultivariateNormal\$pgf(z, ...)`
Arguments
`z`

`(integer(1))`
z integer to evaluate probability generating function at.

`...`

Unused.

Method `getParameterValue()`

Returns the value of the supplied parameter.

Usage
`MultivariateNormal\$getParameterValue(id, error = "warn")`
Arguments
`id`

`character()`
id of parameter support to return.

`error`

`(character(1))`
If `"warn"` then returns a warning on error, otherwise breaks if `"stop"`.

Method `setParameterValue()`

Sets the value(s) of the given parameter(s).

Usage
```MultivariateNormal\$setParameterValue(
...,
lst = list(...),
error = "warn",
resolveConflicts = FALSE
)```
Arguments
`...`

`ANY`
Named arguments of parameters to set values for. See examples.

`lst`

`(list(1))`
Alternative argument for passing parameters. List names should be parameter names and list values are the new values to set.

`error`

`(character(1))`
If `"warn"` then returns a warning on error, otherwise breaks if `"stop"`.

`resolveConflicts`

`(logical(1))`
If `FALSE` (default) throws error if conflicting parameterisations are provided, otherwise automatically resolves them by removing all conflicting parameters.

Method `clone()`

The objects of this class are cloneable with this method.

Usage
`MultivariateNormal\$clone(deep = FALSE)`
Arguments
`deep`

Whether to make a deep clone.

References

McLaughlin, M. P. (2001). A compendium of common probability distributions (pp. 2014-01). Michael P. McLaughlin.

Gentle, J.E. (2009). Computational Statistics. Statistics and Computing. New York: Springer. pp. 315–316. doi:10.1007/978-0-387-98144-4. ISBN 978-0-387-98143-7.

Other continuous distributions: `Arcsine`, `BetaNoncentral`, `Beta`, `Cauchy`, `ChiSquaredNoncentral`, `ChiSquared`, `Dirichlet`, `Erlang`, `Exponential`, `FDistributionNoncentral`, `FDistribution`, `Frechet`, `Gamma`, `Gompertz`, `Gumbel`, `InverseGamma`, `Laplace`, `Logistic`, `Loglogistic`, `Lognormal`, `Normal`, `Pareto`, `Poisson`, `Rayleigh`, `ShiftedLoglogistic`, `StudentTNoncentral`, `StudentT`, `Triangular`, `Uniform`, `Wald`, `Weibull`
Other multivariate distributions: `Dirichlet`, `EmpiricalMV`, `Multinomial`