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#' @name Arcsine
#' @template SDist
#' @templateVar ClassName Arcsine
#' @templateVar DistName Arcsine
#' @templateVar uses in the study of random walks and as a special case of the Beta distribution
#' @templateVar params lower, \eqn{a}, and upper, \eqn{b}, limits
#' @templateVar pdfpmf pdf
#' @templateVar pdfpmfeq \deqn{f(x) = 1/(\pi\sqrt{(x-a)(b-x))}}
#' @templateVar paramsupport \eqn{-\infty < a \le b < \infty}
#' @templateVar distsupport \eqn{[a, b]}
#' @templateVar default lower = 0, upper = 1
#'
#' @template param_lower
#' @template param_upper
#' @template class_distribution
#' @template method_mode
#' @template method_entropy
#' @template method_kurtosis
#' @template method_pgf
#' @template method_mgfcf
#' @template method_setParameterValue
#' @template param_decorators
#'
#' @family continuous distributions
#' @family univariate distributions
#'
#' @export
Arcsine <- R6Class("Arcsine",
inherit = SDistribution, lock_objects = F,
public = list(
# Public fields
name = "Arcsine",
short_name = "Arc",
description = "Arcsine Probability Distribution.",
# Public methods
# initialize
#' @description
#' Creates a new instance of this [R6][R6::R6Class] class.
initialize = function(lower = NULL, upper = NULL, decorators = NULL) {
super$initialize(
decorators = decorators,
support = Interval$new(0, 1),
symmetry = "sym",
type = Reals$new()
)
},
# stats
#' @description
#' The arithmetic mean of a (discrete) probability distribution X is the expectation
#' \deqn{E_X(X) = \sum p_X(x)*x}
#' with an integration analogue for continuous distributions.
#' @param ... Unused.
mean = function(...) {
return((unlist(self$getParameterValue("upper")) + unlist(self$getParameterValue("lower"))) /
2)
},
#' @description
#' The mode of a probability distribution is the point at which the pdf is
#' a local maximum, a distribution can be unimodal (one maximum) or multimodal (several
#' maxima).
mode = function(which = "all") {
lower <- self$getParameterValue("lower")
upper <- self$getParameterValue("upper")
if (checkmate::testList(lower)) {
modes <- data.table(lower, upper)
if (which == "all") {
return(modes)
} else {
return(unlist(modes[, ..which]))
}
} else {
if (which == "all") {
return(c(lower, upper))
} else {
return(c(lower, upper)[which])
}
}
},
#' @description
#' The variance of a distribution is defined by the formula
#' \deqn{var_X = E[X^2] - E[X]^2}
#' where \eqn{E_X} is the expectation of distribution X. If the distribution is multivariate the
#' covariance matrix is returned.
#' @param ... Unused.
variance = function(...) {
((unlist(self$getParameterValue("upper")) - unlist(self$getParameterValue("lower")))^2) / 8
},
#' @description
#' The skewness of a distribution is defined by the third standardised moment,
#' \deqn{sk_X = E_X[\frac{x - \mu}{\sigma}^3]}{sk_X = E_X[((x - \mu)/\sigma)^3]}
#' where \eqn{E_X} is the expectation of distribution X, \eqn{\mu} is the mean of the
#' distribution and \eqn{\sigma} is the standard deviation of the distribution.
#' @param ... Unused.
skewness = function(...) {
rep(0, length(self$getParameterValue("lower")))
},
#' @description
#' The kurtosis of a distribution is defined by the fourth standardised moment,
#' \deqn{k_X = E_X[\frac{x - \mu}{\sigma}^4]}{k_X = E_X[((x - \mu)/\sigma)^4]}
#' where \eqn{E_X} is the expectation of distribution X, \eqn{\mu} is the mean of the
#' distribution and \eqn{\sigma} is the standard deviation of the distribution.
#' Excess Kurtosis is Kurtosis - 3.
#' @param ... Unused.
kurtosis = function(excess = TRUE, ...) {
if (excess) {
return(rep(-1.5, length(self$getParameterValue("lower"))))
} else {
return(rep(1.5, length(self$getParameterValue("lower"))))
}
},
#' @description
#' The entropy of a (discrete) distribution is defined by
#' \deqn{- \sum (f_X)log(f_X)}
#' where \eqn{f_X} is the pdf of distribution X, with an integration analogue for
#' continuous distributions.
#' @param ... Unused.
entropy = function(base = 2, ...) {
rep(log(pi / 4, base), length(self$getParameterValue("lower")))
},
#' @description The probability generating function is defined by
#' \deqn{pgf_X(z) = E_X[exp(z^x)]}
#' where X is the distribution and \eqn{E_X} is the expectation of the distribution X.
#' @param ... Unused.
pgf = function(z, ...) {
NaN
}
),
active = list(
#' @field properties
#' Returns distribution properties, including skewness type and symmetry.
properties = function() {
prop <- super$properties
prop$support <- Interval$new(
self$getParameterValue("lower"),
self$getParameterValue("upper")
)
prop
}
),
private = list(
# dpqr
.pdf = function(x, log = FALSE) {
lower <- self$getParameterValue("lower")
upper <- self$getParameterValue("upper")
if (checkmate::testList(lower)) {
return(C_ArcsinePdf(x, unlist(lower), unlist(upper), log))
} else {
return(as.numeric(C_ArcsinePdf(x, lower, upper, log)))
}
},
.cdf = function(x, lower.tail = TRUE, log.p = FALSE) {
lower <- self$getParameterValue("lower")
upper <- self$getParameterValue("upper")
if (checkmate::testList(lower)) {
return(C_ArcsineCdf(x, unlist(lower), unlist(upper), lower.tail, log.p))
} else {
return(as.numeric(C_ArcsineCdf(x, lower, upper, lower.tail, log.p)))
}
},
.quantile = function(p, lower.tail = TRUE, log.p = FALSE) {
lower <- self$getParameterValue("lower")
upper <- self$getParameterValue("upper")
if (checkmate::testList(lower)) {
return(C_ArcsineQuantile(p, unlist(lower), unlist(upper), lower.tail, log.p))
} else {
return(as.numeric(C_ArcsineQuantile(p, lower, upper, lower.tail, log.p)))
}
},
.rand = function(n) {
self$quantile(runif(n))
},
# traits
.traits = list(valueSupport = "continuous", variateForm = "univariate")
)
)
.distr6$distributions <- rbind(
.distr6$distributions,
data.table::data.table(
ShortName = "Arc", ClassName = "Arcsine",
Type = "\u211D", ValueSupport = "continuous", VariateForm = "univariate",
Package = "-", Tags = "limits"
)
)
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