MultivariateNormal: 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.

Value

Returns an R6 object inheriting from class SDistribution.

Constructor

MultivariateNormal$new(mean = rep(0,2), cov = c(1,0,0,1), prec = NULL, decorators = NULL, verbose = FALSE)

Constructor Arguments

Argument	Type	Details
`mean`	numeric	vector of means.
`cov`	numeric	vector or matrix. See details.
`prec`	numeric	vector or matrix. See details.

decorators Decorator decorators to add functionality. See details.

Constructor Details

The Multivariate Normal distribution is parameterised with mean as a vector of numerics and either cov or prec as positive semi-definite matrices. These are related via, $$prec = cov^{-1}$$ If prec is given then cov is ignored. The covariance matrix can either be supplied as a matrix or as a vector that can be coerced via matrix(cov, nrow = K, byrow = FALSE).

Public Variables

Variable	Return
`name`	Name of distribution.
`short_name`	Id of distribution.
`description`	Brief description of distribution.

Public Methods

Accessor Methods	Link
`decorators()`	`decorators`
`traits()`	`traits`
`valueSupport()`	`valueSupport`
`variateForm()`	`variateForm`
`type()`	`type`
`properties()`	`properties`
`support()`	`support`
`symmetry()`	`symmetry`
`sup()`	`sup`
`inf()`	`inf`
`dmax()`	`dmax`
`dmin()`	`dmin`
`skewnessType()`	`skewnessType`
`kurtosisType()`	`kurtosisType`

Statistical Methods Link pdf(x1, ..., log = FALSE, simplify = TRUE) pdf cdf(x1, ..., lower.tail = TRUE, log.p = FALSE, simplify = TRUE) cdf quantile(p, ..., lower.tail = TRUE, log.p = FALSE, simplify = TRUE) quantile.Distribution rand(n, simplify = TRUE) rand mean() mean.Distribution variance() variance stdev() stdev prec() prec cor() cor skewness() skewness kurtosis(excess = TRUE) kurtosis entropy(base = 2) entropy mgf(t) mgf cf(t) cf pgf(z) pgf median() median.Distribution iqr() iqr mode(which = "all") mode

Parameter Methods Link parameters(id) parameters getParameterValue(id, error = "warn") getParameterValue setParameterValue(..., lst = NULL, error = "warn") setParameterValue

Validation Methods Link liesInSupport(x, all = TRUE, bound = FALSE) liesInSupport liesInType(x, all = TRUE, bound = FALSE) liesInType

Representation Methods Link strprint(n = 2) strprint print(n = 2) print summary(full = T) summary.Distribution

Details

The Multivariate Normal distribution parameterised with mean, $\mu$, and covariance matrix, $\Sigma$, is defined by the pdf, $$f(x_1,...,x_k) = (2 * \pi)^{-k/2}det(\Sigma)^{-1/2}exp(-1/2(x-\mu)^T\Sigma^{-1}(x-\mu))$$ for $\mu \epsilon R^{k}$ and $\Sigma \epsilon R^{k x k}$.

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

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

The parameter K is automatically updated by counting the length of the mean vector and once constructed this cannot be changed. If a mean vector of length greater than K is given then this is truncated to the correct length. If a mean vector of length less than K is given then this replicated and truncated to the correct length. Similarly cov and prec are internally coerced with matrix(cov, nrow = K, byrow = FALSE). Sampling is performed via the Cholesky decomposition using chol.

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<U+2013>316. doi:10.1007/978-0-387-98144-4. ISBN 978-0-387-98143-7.

Examples

Run this code

# NOT RUN {
# Different parameterisations
MultivariateNormal$new(mean = c(0,0,0), cov = matrix(c(3,-1,-1,-1,1,0,-1,0,1), byrow=TRUE,nrow=3))
MultivariateNormal$new(mean = c(0,0,0), cov = c(3,-1,-1,-1,1,0,-1,0,1)) # Equivalently
MultivariateNormal$new(mean = c(0,0,0), prec = c(3,-1,-1,-1,1,0,-1,0,1))

# Default is bivariate standard normal
x <- MultivariateNormal$new()

# Update parameters
x$setParameterValue(mean = c(1, 2))
# When any parameter is updated, all others are too!
x$setParameterValue(prec = c(1,0,0,1))
x$parameters()

# d/p/q/r
# Note the difference from R stats
x$pdf(1, 2)
# This allows vectorisation:
x$pdf(1:3, 2:4)
x$rand(4)

# Statistics
x$mean()
x$variance()

summary(x)

# }

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