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Kernel_Triangular.R
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#' @title Triangular Kernel
#'
#' @description Mathematical and statistical functions for the Triangular kernel defined by the pdf,
#' \deqn{f(x) = 1 - |x|}
#' over the support \eqn{x \in (-1,1)}{x \epsilon (-1,1)}.
#'
#' @name TriangularKernel
#' @template class_distribution
#' @template class_kernel
#' @template method_pdfsquared2Norm
#'
#'
#' @export
TriangularKernel <- R6Class("TriangularKernel",
inherit = Kernel, lock_objects = F,
public = list(
name = "TriangularKernel",
short_name = "Tri",
description = "Triangular 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 (xi >= 0 & xi <= 1) {
if (ui >= (xi - 1) & ui <= 0) {
ret[i] <- (xi^3 + (- 3 * ui^2 - 6 * ui - 3) *
xi + 2 * ui^3 + 6 * ui^2 + 6 * ui + 2) / 6
} else if (ui >= 0 & ui <= xi) {
ret[i] <- (- 2 * ui^3 + 3 * ui^2 * xi -
6 * ui * xi + 6 * ui + xi^3 - 3 * xi + 2) / 6
} else if (ui >= xi & ui <= 1) {
ret[i] <- (2 * ui^3 - 3 * ui^2 * xi -
6 * ui^2 + 6 * ui * xi + 6 * ui +
3 * xi^3 - 6 * xi^2 - 3 * xi + 2) / 6
} else if (ui == Inf | ui > 1) {
ret[i] <- (3 * xi^3 - 6 * xi^2 + 4) / 6
} else {
ret[i] <- 0
}
} else if (xi >= 1 & xi <= 2) {
if (ui == Inf | ui >= 1) {
ret[i] <- (- xi^3 + 6 * xi^2 - 12 * xi + 8) / 6
} else if (ui >= (xi - 1) & ui <= 1) {
ret[i] <- (- xi^3 + 6 * xi^2 - (- 3 * ui^2 + 6 * ui + 9) *
xi - 2 * ui^3 + 6 * ui + 4) / 6
} else {
ret[i] <- 0
}
} else if (xi >= -1 & xi <= 0) {
if (ui == Inf | ui >= xi + 1) {
ret[i] <- (- xi^3 + 6 * xi^2 + 4) / 6
} else if (ui >= - 1 & ui <= xi) {
ret[i] <- (- (3 * ui ^2 + 6 * ui + 3) * xi +
2 * ui^3 + 6 * ui ^2 + 6 * ui + 2) / 6
} else if (ui >= xi & ui <= 0) {
ret[i] <- (- (xi * (2 * xi^2 + 6 * xi -
3 * ui * (ui + 2) + 3) +
2 * (ui^3 - 3 * ui - 1))) / 6
} else if (ui >= 0 & ui <= xi + 1) {
ret[i] <- (2 * ui^3 - 3 * ui^2 * xi -
6 * ui^2 + 6 * ui * xi + 6 * ui - 2 * xi^3 -
6 * xi^2 - 3 * xi + 2) / 6
} else {
ret[i] <- 0
}
} else if (xi >= - 2 & xi <= - 1) {
if (ui == Inf | ui >= xi + 1) {
ret[i] <- (xi^3 + 6 * xi^2 + 12 * xi + 8) / 6
} else if (ui >= -1 & ui <= xi + 1) {
ret[i] <- ((3 * ui^2 + 6 * ui) * xi -
2 * ui^3 + 6 * ui + 3 * xi + 4) / 6
}
} else {
ret[i] <- 0
}
}
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) {
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 <= 1) {
if (ui <= -1) {
ret[i] <- 0
} else if (ui >= -1 & ui <= xi - 1) {
ret[i] <- 0
} else if (ui >= xi - 1 & ui <= 0) {
ret[i] <- (- xi^5 + 10 * xi^2 - 15 * xi + 6) / 120 +
(ui * (30 * xi^2 - 60 * xi + 30)) / 120 +
(ui^2 * (30 * xi^2 - 90 * xi + 60)) / 120 +
(ui^3 * (10 * xi^2 - 60 * xi + 60)) / 120 +
(ui^4 * (30 - 15 * xi)) / 120 +
ui^5 / 20
} else if (ui >= 0 & ui <= xi) {
ret[i] <- (-xi^5 / 120) + (ui * (30 * xi^2 - 60 * xi + 30)) / 120 +
(ui^2 * (30 * xi^2 - 90 * xi + 60)) / 120 + xi^2 / 12 +
(ui^3 * (-10 * xi^2 - 20 * xi + 40)) / 120 +
(ui^4 * xi) / 8 - xi / 8 - ui^5 / 20 + 1 / 20
} else if (ui >= xi & ui <= 1) {
ret[i] <- (-xi^5 / 60) + (- xi^5 + 40 * xi^3 - 30 * xi) / 120 +
xi^4 / 12 - xi^3 / 6 + (ui^3 * (10 * xi^2 + 60 * xi + 20)) / 120 +
xi^2 / 12 + (ui^2 * (- 30 * xi^2 - 30 * xi + 60)) / 120 +
(ui * (-30 * xi^2 - 60 * xi + 30)) / 120 +
xi / 8 + (ui^4 * (-15 * xi - 30)) / 120 +
ui^5 / 20 + 1 / 20
} else if (ui >= xi & ui <= xi + 1) {
ret[i] <- (-3 * xi^5 + 10 * xi^4 + 20 * xi^3 + 20 * xi^2 - 8) / 120 +
(ui * (-60 * xi^2 - 120 * xi + 60)) / 120 +
(ui^2 * (60 * xi + 60)) / 120 - ui^3 / 6
} else if (ui >= xi + 1) {
ret[i] <- (1 / 120) * (- 28 + 120 * ui - 60 * xi -
40 * xi^2 + 10 * xi^4 - 3 * xi^5)
}
} else if (xi >= 1 & xi <= 2) {
if (ui <= -1) {
ret[i] <- 0
} else if (ui >= - 1 & ui <= xi - 1) {
ret[i] <- 0
} else if (ui >= xi - 1 & ui <= 0) {
ret[i] <- (- (ui^5 / 20)) + (ui^4 * xi) / 8 +
(1 / 120) * ui^3 * (40 - 20 * xi - 10 * xi^2) +
(1 / 120) * ui^2 * (60 - 90 * xi + 30 * xi^2) +
(1 / 120) * ui * (30 - 60 * xi + 30 * xi^2) +
(1 / 120) * (4 - 5 * xi - 10 * xi^2 + 20 * xi^3 - 10 * xi^4 + xi^5)
} else if (ui >= 0 & ui <= xi) {
ret[i] <- (1 / 120) * (20 * ui^3 - 60 * ui^2 * (xi - 1) +
60 * ui * (xi - 1)^2 + xi^5 -
10 * xi^4 + 20 * xi^3 - 20 * xi^2 + 20 * xi - 12)
} else if (ui >= xi & ui <= xi + 1) {
ret[i] <- (xi^5 - 10 * xi^4 + 60 * xi^3 + (- 60 * ui - 20) * xi^2 +
(60 * ui^2 - 120 * ui + 20) * xi -
20 * ui^3 + 60 * ui^2 + 60 * ui - 12) / 120
} else if (ui >= xi + 1) {
ret[i] <- (xi^5 - 10 * xi^4 + 40 * xi^3 - 80 * xi^2 -
40 * xi + 120 * ui - 32) / 120
}
} else if (xi >= - 1 & 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) {
ret[i] <- ui^5 / 20 + (1 / 120) * ui^4 * (30 - 15 * xi) +
(1 / 120) * ui^3 * (60 - 60 * xi + 10 * xi^2) +
(1 / 120) * (6 - 15 * xi + 10 * xi^2) +
(1 / 120) * ui^2 * (60 - 90 * xi + 30 * xi^2) +
(1 / 120) * ui * (30 - 60 * xi + 30 * xi^2)
} else if (ui >= xi & ui <= 0) {
ret[i] <- (xi^5 + 10 * xi^4 + 20 * xi^3 - 30 * xi) / 120 +
xi^5 / 120 + xi^2 / 12 + (ui^3 * (- 10 * xi^2 + 20 * xi + 40)) / 120 +
(ui^2 * (- 30 * xi^2 - 30 * xi + 60)) / 120 +
(ui * (- 30 * xi^2 - 60 * xi + 30)) / 120 +
(ui^4 * xi) / 8 + xi / 8 - ui^5 / 20 + 1 / 20
} else if (ui >= 0 & ui <= xi + 1) {
ret[i] <- xi^5 / 60 + xi^4 / 12 + (xi^3) / 6 +
(ui * (- 30 * ui * (xi^2 + xi - 2) +
10 * ui^2 * (xi * (xi + 6) + 2) -
30 * xi * (xi + 2) - 15 * ui^3 * (xi + 2) +
6 * ui^4 + 30)) / 120 + xi^2 / 12 - xi / 8 + 1 / 20
} else if (ui >= xi + 1 & ui <= 1) {
ret[i] <- (3 * xi^5 + 10 * xi^4 - 40 * xi^2 -
60 * xi - 20 * ui *
(ui^2 - 3 * ui - 3) - 8) / 120
} else if (ui >= 1) {
ret[i] <- (1 / 120) * (- 28 + 120 * ui - 60 * xi -
40 * xi^2 + 10 * xi^4 + 3 * xi^5)
}
} else if (xi >= -2 & xi <= -1) {
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] <- (ui^3 * (- 10 * xi^2 + 20 * xi + 40)) / 120 +
(- 10 * xi^2 - 25 * xi + 4) / 120 +
(ui^2 * (- 30 * xi^2 - 30 * xi + 60)) / 120 +
(ui * (- 30 * xi^2 - 60 * xi + 30)) / 120 +
(ui^4 * xi) / 8 - ui^5 / 20
} else if (ui >= xi + 1 & ui <= 0) {
ret[i] <- (- xi^5 - 10 * xi^4 - 40 * x [i]^3 - 80 * xi^2 -
80 * xi + 20 * ui^3 + 60 * ui^2 + 60 * ui - 12) / 120
} else if (ui >= 0 & ui <= 1) {
ret[i] <- (- xi^5 - 10 * xi^4 - 40 * xi^3 - 80 * xi^2 -
80 * xi - 20 * ui * (ui^2 - 3 * ui - 3) - 12) / 120
} else if (ui >= 1) {
ret[i] <- (- xi^5 - 10 * xi^4 - 40 * xi^3 - 80 * xi^2 -
80 * xi + 120 * ui - 32) / 120
}
} else if (xi >= 2) {
if (ui <= xi - 1) {
ret[i] <- 0
} else if (ui >= xi - 1 & ui <= xi) {
ret[i] <- (1 / 6) * (1 + ui - xi)^3
} else if (ui >= xi & ui <= xi + 1) {
ret[i] <- (xi^3 + (3 - 3 * ui) * xi^2 +
(3 * ui^2 - 6 * ui - 3) *
xi - ui^3 + 3 * ui^2 + 3 * ui) / 6 + 1 / 6
} else if (ui >= xi + 1) {
ret[i] <- ui - xi
}
} else if (xi <= -2) {
if (ui <= -1) {
ret[i] <- 0
} else if (ui >= -1 & ui <= 0) {
ret[i] <- (ui^3 + 3 * ui^2 + 3 * ui + 1) / 6
} else if (ui >= 0 & ui <= 1) {
ret[i] <- (- ui^3 + 3 * ui^2 + 3 * ui + 1) / 6
} 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 / 6)
}
),
private = list(
.pdf = function(x, log = FALSE) {
C_TriangularKernelPdf(x, log)
},
.cdf = function(x, lower.tail = TRUE, log.p = FALSE) {
C_TriangularKernelCdf(x, lower.tail, log.p)
},
.quantile = function(p, lower.tail = TRUE, log.p = FALSE) {
C_TriangularKernelQuantile(p, lower.tail, log.p)
}
)
)
.distr6$kernels <- rbind(
.distr6$kernels,
data.table::data.table(
ShortName = "Tri", ClassName = "TriangularKernel",
Support = "[-1,1]", Packages = "-"
)
)