Dirichlet: Dirichlet Distribution Class

Description

Mathematical and statistical functions for the Dirichlet distribution, which is commonly used as a prior in Bayesian modelling and is multivariate generalisation of the Beta distribution.

Value

Returns an R6 object inheriting from class SDistribution.

Constructor

Dirichlet$new(params = c(1, 1), decorators = NULL, verbose = FALSE)

Constructor Arguments

Argument	Type	Details
`params`	numeric	vector of concentration parameters.

decorators Decorator decorators to add functionality. See details.

Constructor Details

The Dirichlet distribution is parameterised with params as a vector of positive numerics. The parameter K is automatically calculated by counting the length of the params vector, once constructed this cannot be changed.

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`



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 Dirichlet distribution parameterised with concentration parameters, $\alpha_1,...,\alpha_k$, is defined by the pdf, $$f(x_1,...,x_k) = (\prod \Gamma(\alpha_i))/(\Gamma(\sum \alpha_i))\prod(x_i^{\alpha_i - 1})$$ for $\alpha = \alpha_1,...,\alpha_k; \alpha > 0$, where $\Gamma$ is the gamma function.

The distribution is supported on $x_i \ \epsilon \ (0,1), \sum x_i = 1$.

mgf and cf 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.

Sampling is performed via sampling independent Gamma distributions and normalising the samples (Devroye, 1986).

References

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

Devroye, Luc (1986). Non-Uniform Random Variate Generation. Springer-Verlag. ISBN 0-387-96305-7.

Examples

Run this code

# NOT RUN {
# Different parameterisations
x <- Dirichlet$new(params = c(2,5,6))

# Update parameters
x$setParameterValue(params = c(3, 2, 3))
# 'K' parameter is automatically calculated
x$parameters()
# }
# NOT RUN {
# This errors as less than three parameters supplied
x$setParameterValue(params = c(1, 2))
# }
# NOT RUN {
# d/p/q/r
# Note the difference from R stats
x$pdf(0.1, 0.4, 0.5)
# This allows vectorisation:
x$pdf(c(0.3, 0.2), c(0.6, 0.9), c(0.9,0.1))
x$rand(4)

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

summary(x)

# }

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