R/SDistribution_Pareto.R

In distr6: The Complete R6 Probability Distributions Interface

#' @name Pareto
#' @template SDist
#' @templateVar ClassName Pareto
#' @templateVar DistName Pareto
#' @templateVar uses in Economics to model the distribution of wealth and the 80-20 rule
#' @templateVar params shape, \eqn{\alpha}, and scale, \eqn{\beta},
#' @templateVar pdfpmf pdf
#' @templateVar pdfpmfeq \deqn{f(x) = (\alpha\beta^\alpha)/(x^{\alpha+1})}
#' @templateVar paramsupport \eqn{\alpha, \beta > 0}
#' @templateVar distsupport \eqn{[\beta, \infty)}
#' @templateVar default shape = 1, scale = 1
#' @details
#' Currently this is implemented as the Type I Pareto distribution, other types
#' will be added in the future. Characteristic function is omitted as no suitable incomplete
#' gamma function with complex inputs implementation could be found.
#'
#' @template class_distribution
#' @template method_mode
#' @template method_entropy
#' @template method_kurtosis
#' @template method_pgf
#' @template method_mgfcf
#' @template method_setParameterValue
#' @template param_decorators
#' @template field_packages
#' @template param_shape
#' @template param_scale
#'
#' @family continuous distributions
#' @family univariate distributions
#'
#' @export
Pareto <- R6Class("Pareto",
  inherit = SDistribution, lock_objects = F,
  public = list(
    # Public fields
    name = "Pareto",
    short_name = "Pare",
    description = "Pareto (Type I) Probability Distribution.",
    packages = c("extraDistr", "pracma"),

    # Public methods
    # initialize

    #' @description
    #' Creates a new instance of this [R6][R6::R6Class] class.
    initialize = function(shape = NULL, scale = NULL, decorators = NULL) {
      super$initialize(
        decorators = decorators,
        support = Interval$new(1, Inf, type = "[)"),
        type = PosReals$new(zero = T)
      )
    },

    # 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(...) {
      shape <- unlist(self$getParameterValue("shape"))
      scale <- unlist(self$getParameterValue("scale"))

      mean <- rep(Inf, length(shape))
      mean[shape > 1] <- (shape[shape > 1] * scale[shape > 1]) / (shape[shape > 1] - 1)
      return(mean)
    },

    #' @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") {
      unlist(self$getParameterValue("scale"))
    },

    #' @description
    #' Returns the median of the distribution. If an analytical expression is available
    #' returns distribution median, otherwise if symmetric returns `self$mean`, otherwise
    #' returns `self$quantile(0.5)`.
    median = function() {
      unlist(self$getParameterValue("scale")) * 2^(1 / unlist(self$getParameterValue("shape"))) # nolint
    },

    #' @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(...) {
      shape <- unlist(self$getParameterValue("shape"))
      scale <- unlist(self$getParameterValue("scale"))

      var <- rep(Inf, length(shape))
      var[shape > 2] <- (shape[shape > 2] * scale[shape > 2]^2) /
        ((shape[shape > 2] - 1)^2 * (shape[shape > 2] - 2))
      return(var)
    },

    #' @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(...) {
      shape <- unlist(self$getParameterValue("shape"))

      skew <- rep(NaN, length(shape))
      skew[shape > 3] <- ((2 * (1 + shape[shape > 3])) / (shape[shape > 3] - 3)) *
        sqrt((shape[shape > 3] - 2) / shape[shape > 3])
      return(skew)
    },

    #' @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, ...) {
      shape <- unlist(self$getParameterValue("shape"))

      kur <- rep(NaN, length(shape))
      kur[shape > 4] <- (6 * (shape[shape > 4]^3 + shape[shape > 4]^2 - 6 * shape[shape > 4] - 2)) /
        (shape[shape > 4] * (shape[shape > 4] - 3) * (shape[shape > 4] - 4))

      if (excess) {
        return(kur)
      } else {
        return(kur + 3)
      }
    },

    #' @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, ...) {
      shape <- unlist(self$getParameterValue("shape"))
      scale <- unlist(self$getParameterValue("scale"))

      return(log((scale / shape) * exp(1 + 1 / shape), base))
    },

    #' @description The moment generating function is defined by
    #' \deqn{mgf_X(t) = E_X[exp(xt)]}
    #' where X is the distribution and \eqn{E_X} is the expectation of the distribution X.
    #' @param ... Unused.
    mgf = function(t, ...) {
      if (t < 0) {
        shape <- self$getParameterValue("shape")
        scale <- self$getParameterValue("scale")
        return(shape * (-scale * t)^shape * pracma::incgam(-scale * t, -shape))
      } else {
        return(NaN)
      }
    },

    #' @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, ...) {
      return(NaN)
    }
  ),

  active = list(
    #' @field properties
    #' Returns distribution properties, including skewness type and symmetry.
    properties = function() {
      prop <- super$properties
      prop$support <- Interval$new(
        self$getParameterValue("scale"),
        Inf,
        type = "[)"
      )
      prop
    }
  ),

  private = list(
    # dpqr
    .pdf = function(x, log = FALSE) {
      if (checkmate::testList(self$getParameterValue("shape"))) {
        mapply(
          extraDistr::dpareto,
          a = self$getParameterValue("shape"),
          b = self$getParameterValue("scale"),
          MoreArgs = list(x = x, log = log)
        )
      } else {
        extraDistr::dpareto(
          x,
          a = self$getParameterValue("shape"),
          b = self$getParameterValue("scale"),
          log = log
        )
      }
    },
    .cdf = function(x, lower.tail = TRUE, log.p = FALSE) {
      if (checkmate::testList(self$getParameterValue("shape"))) {
        mapply(
          extraDistr::ppareto,
          a = self$getParameterValue("shape"),
          b = self$getParameterValue("scale"),
          MoreArgs = list(
            q = x,
            lower.tail = lower.tail,
            log.p = log.p
          )
        )
      } else {
        extraDistr::ppareto(
          x,
          a = self$getParameterValue("shape"),
          b = self$getParameterValue("scale"),
          lower.tail = lower.tail,
          log.p = log.p
        )
      }
    },
    .quantile = function(p, lower.tail = TRUE, log.p = FALSE) {
      if (checkmate::testList(self$getParameterValue("shape"))) {
        mapply(
          extraDistr::qpareto,
          a = self$getParameterValue("shape"),
          b = self$getParameterValue("scale"),
          MoreArgs = list(
            p = p,
            lower.tail = lower.tail,
            log.p = log.p
          )
        )
      } else {
        extraDistr::qpareto(
          p,
          a = self$getParameterValue("shape"),
          b = self$getParameterValue("scale"),
          lower.tail = lower.tail,
          log.p = log.p
        )
      }
    },
    .rand = function(n) {
      if (checkmate::testList(self$getParameterValue("shape"))) {
        mapply(
          extraDistr::rpareto,
          a = self$getParameterValue("shape"),
          b = self$getParameterValue("scale"),
          MoreArgs = list(n = n)
        )
      } else {
        extraDistr::rpareto(
          n,
          a = self$getParameterValue("shape"),
          b = self$getParameterValue("scale")
        )
      }
    },

    # traits
    .traits = list(valueSupport = "continuous", variateForm = "univariate")
  )
)

.distr6$distributions <- rbind(
  .distr6$distributions,
  data.table::data.table(
    ShortName = "Pare", ClassName = "Pareto",
    Type = "\u211D+", ValueSupport = "continuous",
    VariateForm = "univariate",
    Package = "extraDistr", Tags = ""
  )
)