R/Kernel_Quartic.R

In distr6: The Complete R6 Probability Distributions Interface

#' @title Quartic Kernel
#'
#' @description Mathematical and statistical functions for the Quartic kernel defined by the pdf,
#' \deqn{f(x) = 15/16(1 - x^2)^2}
#' over the support \eqn{x \in (-1,1)}{x \epsilon (-1,1)}.
#'
#' @details Quantile is omitted as no closed form analytic expression could be found, decorate with
#' FunctionImputation for numeric results.
#'
#' @name Quartic
#' @template class_distribution
#' @template class_kernel
#' @template method_pdfsquared2Norm
#'
#' @export
Quartic <- R6Class("Quartic",
  inherit = Kernel, lock_objects = F,
  public = list(
    name = "Quartic",
    short_name = "Quart",
    description = "Quartic Kernel",

    #' @description
    #' The squared 2-norm of the pdf is defined by
    #' \deqn{\int_a^b (f_X(u))^2 du}
    #' where X is the Distribution, \eqn{f_X} is its pdf and \eqn{a, b}
    #' are the distribution support limits.
    pdfSquared2Norm = function(x = 0, upper = Inf) {
      xl <- length(x)
      ul <- length(upper)
      len <- max(xl, ul)

      ret <- numeric(len)
      for (i in seq(len)) {
        xi <- x[ifelse(i %% xl == 0, xl, i %% xl)]
        ui <- upper[ifelse(i %% ul == 0, ul, i %% ul)]
        if (abs(xi) >= 2) {
          ret[i] <- 0
        } else if (xi >= 0) {
          if (ui <= xi - 1) {
            ret[i] <- 0
          } else if (ui >= xi - 1 & ui <= 1) {
            ret[i] <- (15 / 16)^2 * (- xi^9 / 630 + (4 * xi^7) / 105 -
              (8 * xi^5) / 15 + ui * (xi^4 - 2 * xi^2 + 1) +
              (8 * xi^4) / 15 + ui^5 * (xi^4 / 5 - (14 * xi^2) / 5 + 6 / 5) +
              ui^3 * (- (2 * xi^4) / 3 + (10 * xi^2) / 3 - 4 / 3) +
              ui^4 * (2 * xi^3 - 3 * xi) + (2 * xi^3) / 3 +
              ui^6 * (2 * xi - (2 * xi^3) / 3) +
              ui^2 * (2 * xi - 2 * xi^3) - (64 * xi^2) / 105 +
              ui^7 * ((6 * xi^2) / 7 - 4 / 7) - (ui^8 * xi) / 2 - xi / 2 +
              ui^9 / 9 + 128 / 315)
          } else if (ui >= 1 | ui == Inf) {
            ret[i] <- (15 / 16)^2 * (1 / 630) *
              (- xi^9 + 24 * xi^7 - 336 * xi^5 + 672 * xi^4 - 768 * xi^2 + 512)
          }
        } else if (xi <= 0) {
          if (ui <= -1) {
            ret[i] <- 0
          } else if (ui <= xi + 1 & ui >= -1) {
            ret[i] <- (15 / 16)^2 * (ui * (xi^4 - 2 * xi^2 + 1) + (8 * xi^4) / 15 +
                                        ui^5 * (xi^4 / 5 - (14 * xi^2) / 5 + 6 / 5) +
                                        ui^3 * (- (2 * xi^4) / 3 +
                                                        (10 * xi^2) / 3 - 4 / 3) +
                                        ui^4 * (2 * xi^3 - 3 * xi) + (2 * xi^3) / 3 +
                                        ui^6 * (2 * xi - (2 * xi^3) / 3) +
                                        ui^2 * (2 * xi - 2 * xi^3) - (64 * xi^2) / 105 +
                                        ui^7 * ((6 * xi^2) / 7 - 4 / 7) -
                                         (ui^8 * xi) / 2 -
                                         xi / 2 + ui^9 / 9 + 128 / 315)
          } else if (ui == Inf | ui >= xi + 1) {
            ret[i] <- (15 / 16)^2 * (1 / 630) *
                       (xi^9 - 24 * xi^7 + 336 * xi^5 + 672 * xi^4 - 768 * xi^2 + 512)
          }
        }
      }
      return(ret)
    },

    #' @description
    #' The squared 2-norm of the cdf is defined by
    #' \deqn{\int_a^b (F_X(u))^2 du}
    #' where X is the Distribution, \eqn{F_X} is its pdf and \eqn{a, b}
    #' are the distribution support limits.
    cdfSquared2Norm = function(x = 0, upper = 0) {

      #when c \in [ 0, 2]
      #i. when a \in [c-1, 1]
      f1 <- function(x, upper) {
        ret <- (1 / 236544) * ((1 + upper - x)^4 *
                                 (11360 + 756 * upper^7 + 15872 * x +
                                  8528 * x^2 + 1645 * x^3 - 380 * x^4 -
              80 * x^5 + 12 * x^6 + 3 * x^7 - 378 * upper^6 * (8 + 3 * x) +
              56 * upper^5 * (25 + 27 * x + 3 * x^2) +
              35 * upper^4 * (272 + 196 * x + 24 * x^2 + 3 * x^3) +
              20 * upper^3 * (-503 - 518 * x - 26 * x^2 + 21 * x^3 + 3 * x^4) +
              2 * upper^2 * (-6032 - 6855 * x - 2460 * x^2 - 235 * x^3 + 90 * x^4 + 15 * x^5) +
              4 * upper * (3424 + 4184 * x + 1200 * x^2 - 445 * x^3 - 65 * x^4 + 15 * x^5 +
              3 * x^6)))
        return(ret)
      }

      #ii. when upper \in [1, x+1]
      f2 <- function(x, upper) {
        ret <- (1 / 32) * (-31 + 16 * upper + 15 * upper^2 - 5 * (upper - x)^4 + (upper - x)^6 +
                        5 * (-1 + x)^4 - (-1 + x)^6 + 30 * x - 30 * upper * x)
        return(ret)
      }


      #2. c \ge 2
      f3 <- function(x, upper) {
        ret <- (1 / 32) * (1 + upper - x)^4 * (5 + upper^2 + 4 * x + x^2 - 2 * upper * (2 + x))
        return(ret)
      }

      #3. x \in [-2, 0]
      #i. upper \in [-1, x+1]

      f4 <- function(x, upper) {
        ret <- (1 / 236544) * ((1 + upper)^4 *
                                 (4 * (1 + upper)^3 *
                                    (2840 + 7 * upper *
                                       (- 728 + upper * (536 + 27 * (- 7 + upper) * upper))) -
              462 * (-8 + upper + 6 * upper^2 - 3 * upper^3)^2 * x +
              1320 * (1 + upper) * (10 + upper * (-50 + upper *
                                                    (66 + 7 * (-5 + upper) * upper))) * x^2 -
              1155 * (-1 + upper) * (15 + upper * (19 + 9 * (-3 + upper) * upper)) * x^3 +
              1980 * (-4 + upper * (16 + 3 * (-4 + upper) * upper)) *
                x^4 - 1386 * (5 + (-4 + upper) * upper) * x^5))
        return(ret)
      }


      #ii. upper \in [x+1, 1]

      f5 <- function(x, upper) {
        ret <- - (27 / 32) + upper / 2 + (15 * upper^2) / 32 - (5 * upper^4) / 32 +
          upper^6 / 32 - x - (5 * x^4) / 16 - (3 * x^5) / 16 - x^6 / 32
        return(ret)
      }

      #4. x \le -2
      f6 <- function(x, upper) {
        ret <- 5 / 32 + upper / 2 + (15 * upper^2) / 32 - (5 * upper^4) / 32 + upper^6 / 32
        return(ret)
      }

      xl <- length(x)
      ul <- length(upper)
      len <- max(xl, ul)

      ret <- numeric(len)
      for (i in seq(len)) {

        xi <- x[ifelse(i %% xl == 0, xl, i %% xl)]
        ui <- upper[ifelse(i %% ul == 0, ul, i %% ul)]

        if (xi >= 0 & xi <= 2) {
          if (ui <= -1) {
            ret[i] <- 0
          } else if (ui >= -1 & ui <= xi - 1) {
            ret[i] <- 0
          } else if (ui >= xi - 1 & ui <= 1) {
            ret[i] <-  f1(x = xi, upper = ui)
          } else if (ui >= 1 & ui <= xi + 1) {
            ret[i] <- f2(x = xi, upper = ui) + f1(x = xi, upper = 1)
          } else if (ui >= xi + 1) {
            ret[i] <- f2(x = xi, upper = xi + 1) + f1(x = xi, upper = 1) +
              (ui - xi - 1)
          }
        } else if (xi >= -2 & xi <= 0) {
          if (ui <= xi - 1) {
            ret[i] <- 0
          } else if (ui >= xi - 1 & ui <= - 1) {
            ret[i] <- 0
          } else if (ui >= - 1 & ui <= xi + 1) {
            ret[i] <-  f4(x = xi, upper = ui)
          } else if (ui >= xi + 1 & ui <= 1) {
            ret[i] <- f4(x = xi, upper = ui) + f5(x = xi, upper = xi + 1)
          } else if (ui >= 1) {
            ret[i] <- f5(x = xi, upper = 1) + f4(x = xi, upper = xi + 1) + (ui - 1)
          }
        } else if (xi >= 2) {
          if (ui <=  -1) {
            ret[i] <- 0
          } else if (ui >= -1 & ui <= 1) {
            ret[i] <- 0
          } else if (ui >= 1 & ui <= xi - 1) {
            ret[i] <- 0
          } else if (ui >= xi - 1 & ui <= xi + 1) {
            ret[i] <- f3(x = xi, upper = ui)
          } else if (ui >= xi + 1) {
            ret[i] <- ui - xi
          }
        } else if (xi <= -2) {
          if (ui <= xi - 1) {
            ret[i] <- 0
          } else if (ui >= xi - 1 & ui <= xi + 1) {
            ret[i] <- 0
          } else if (ui >= xi + 1 & ui <= -1) {
            ret[i] <- 0
          } else if (ui >= -1 & ui <= 1) {
            ret[i] <- f6(x = xi, upper = ui)
          } else if (ui >= 1) {
            ret[i] <- ui
          }
        }
      }
      return(ret)

    },

    #' @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(...) {
      return(1 / 7)
    }
  ),

  private = list(
    .isQuantile = 0L,
    .pdf = function(x, log = FALSE) {
      C_QuarticKernelPdf(x, log)
    },
    .cdf = function(x, lower.tail = TRUE, log.p = FALSE) {
      C_QuarticKernelCdf(x, lower.tail, log.p)
    }
  )
)

.distr6$kernels <- rbind(
  .distr6$kernels,
  data.table::data.table(
    ShortName = "Quart", ClassName = "Quartic",
    Support = "[-1,1]", Packages = "-"
  )
)