Compute Moments of a Probability Distribution

Description

Calculates mean, variance, skewness, kurtosis, median, and mode for a given probability distribution defined by PDF and CDF.

Usage

dist_moments(pdf, cdf, support = c(0, Inf), params = list())

Arguments

Value

A data frame with Mean, Variance, Skewness, Kurtosis, Median, and Mode.

Examples

# Generalized Exponential Distribution
pdf_ge <- function(x, alpha, lambda) {
  alpha * lambda * (1 - exp(-lambda * x))^(alpha - 1) * exp(-lambda * x)
}
cdf_ge <- function(x, alpha, lambda) {
  (1 - exp(-lambda * x))^alpha
}
dist_moments(pdf_ge, cdf_ge, support = c(0, Inf), params = list(alpha = 2, lambda = 3))

# Exponentiated Weibull Distribution
pdf_expweibull <- function(x, a, b, c){
  a * b * c * exp(-(b*x)^c) * (b*x)^(c-1) * (1 - exp(-(b*x)^c))^(a-1)
}
cdf_expweibull <- function(x, a, b, c){
  (1 - exp(-(b*x)^c))^a
}
dist_moments(pdf_expweibull, cdf_expweibull, support = c(0, Inf),
             params = list(a = 1.0, b = 1.4, c = 2.3))

Plot CDF for the Probability of a Distribution

Description

#' Plots the CDF of a probability distribution. Supports multiple sets of parameters with shaded areas under the curves.

Usage

plot_cdf(
  cdf,
  param_list,
  xlim = c(0, 10),
  ylim = c(0, 1),
  n = 500,
  main = "Custom CDF Plot",
  xlab = "x",
  ylab = "CDF",
  colors = NULL,
  shade_colors = NULL,
  lwd = 3,
  lty = 2,
  grid = TRUE,
  grid_lty = 3,
  grid_col = "gray80",
  grid_lwd = 1
)

Arguments

Value

A CDF plot is displayed. The function invisibly returns NULL.

Examples

# Example 1: Generalized Exponential Distribution
ge_cdf <- function(x, alpha, lambda) {
  (1 - exp(-lambda * x))^alpha
}
param_values <- list(
  list(alpha = 1, lambda = 1),
  list(alpha = 2, lambda = 1),
  list(alpha = 3, lambda = 0.5),
  list(alpha = 4, lambda = 1.5),
  list(alpha = 5, lambda = 2.5)
)
plot_cdf(cdf = ge_cdf, param_list = param_values, main = "CDF GE Distribution")

# Example 2: Exponentiated Weibull Distribution
cdf_expweibull <- function(x, a, b, c){
  (1 - exp(-(b*x)^c))^a
}
param_values <- list(
  list(a = 0.3, b = 1.2, c = 1.0),
  list(a = 1.3, b = 0.4, c = 2.3),
  list(a = 1.5, b = 0.9, c = 3.0),
  list(a = 2.0, b = 1.8, c = 2.8),
  list(a = 3.7, b = 2.0, c = 1.5)
)
colors <- c("green", "purple", "yellow", "orange", "darkblue")
plot_cdf(cdf = cdf_expweibull, param_list = param_values,
         main = "CDF of EW Distribution", colors = colors, xlim = c(0, 5))

Plot Data Overview

Description

Generates a 2x2 plot layout showing: Histogram, Boxplot, Kernel Density, and TTT Plot.

Usage

plot_data(
  x,
  col = "steelblue",
  border = "black",
  transparency = 0.5,
  lwd = 2,
  breaks = "Sturges"
)

Arguments

Value

NULL (plots are drawn)

Examples

set.seed(123)
mydata <- rexp(100, 1)
plot_data(mydata, col = "darkblue", transparency = 0.35, lwd = 2)

Plot Fitted Distribution with Diagnostic Plots

Description

This function produces a 2x2 panel of plots for a fitted distribution: 1. Fitted PDF over histogram of data 2. Fitted CDF vs empirical CDF 3. QQ-plot (Theoretical vs Sample Quantiles) 4. PP-plot (Fitted CDF vs Empirical CDF)

Usage

plot_fitted(
  data,
  pdf_fun,
  cdf_fun,
  par,
  q_fun = NULL,
  xlim = NULL,
  ylim_pdf = NULL,
  ylim_cdf = NULL,
  lwd = 3,
  lty = 2,
  col_pdf = "yellow",
  col_cdf = "red",
  col_qq = "purple",
  col_pp = "darkgreen"
)

Arguments

Value

NULL (plots are generated as a side effect)

Plot Hazard Function of a Probability Distribution

Description

#' Plots the PDF of a probability distribution. Supports multiple sets of parameters with shaded areas under the curves.

Usage

plot_hf(
  pdf,
  cdf,
  param_list,
  xlim = c(0, 10),
  ylim = c(0, 6),
  n = 500,
  main = "Hazard Function Plot",
  xlab = "x",
  ylab = "h(x)",
  colors = NULL,
  shade_colors = NULL,
  lwd = 3,
  lty = 2,
  grid = TRUE,
  grid_lty = 3,
  grid_col = "gray80",
  grid_lwd = 1
)

Arguments

Value

A hazard function plot is displayed. The function invisibly returns NULL.

Examples

# Example 1: Generalized Exponential Distribution
pdf_ge <- function(x, alpha, lambda) {
  alpha * lambda * exp(-lambda*x) * (1 - exp(-lambda*x))^(alpha - 1)
}
cdf_ge <- function(x, alpha, lambda) {
  1 - (1 - exp(-lambda*x))^alpha
}
param_values <- list(
  list(alpha = 1, lambda = 1),
  list(alpha = 2, lambda = 1),
  list(alpha = 3, lambda = 0.5),
  list(alpha = 4, lambda = 1.5),
  list(alpha = 5, lambda = 2.5)
)
plot_hf(pdf_ge, cdf_ge, param_values, xlim=c(0,5), ylim=c(0,4), main="HF GE Distribution")

# Example 2: Exponentiated Weibull Distribution
pdf_expweibull <- function(x, a, b, c){
  a * b * c * exp(-(b*x)^c) * (b*x)^(c-1) * (1 - exp(-(b*x)^c))^(a-1)
}
cdf_expweibull <- function(x, a, b, c){
  1 - (1 - exp(-(b*x)^c))^a
}
param_values <- list(
  list(a = 0.3, b = 1.2, c = 1.0),
  list(a = 1.3, b = 0.4, c = 2.3),
  list(a = 1.5, b = 0.9, c = 3.0),
  list(a = 2.0, b = 1.8, c = 2.8),
  list(a = 3.7, b = 2.0, c = 1.5)
)
plot_hf(pdf_expweibull, cdf_expweibull, param_values)

Plot Multiple Fitted Distributions

Description

Creates 2x2 plots for multiple fitted distributions: Fitted PDFs, Fitted CDFs vs Empirical CDF, QQ-Plots, and PP-Plots.

Usage

plot_multi_fitted(
  data,
  pdf_list,
  cdf_list,
  qf_list,
  params_list,
  dist_names = NULL,
  col_list = NULL,
  lty_list = NULL,
  lwd_list = NULL,
  main_pdf = "Fitted PDFs",
  main_cdf = "Fitted CDFs",
  main_qq = "QQ Plots",
  main_pp = "PP Plots",
  xlab = "x"
)

Arguments

Value

NULL (plots are generated as a side effect)

Examples

# Example Multiple Distributions
set.seed(1)
data <- rexp(200, 1.1)

# Exponential
pdf_exp <- function(x, par) par[1] * exp(-par[1] * x)
cdf_exp <- function(x, par) 1 - exp(-par[1] * x)
qf_exp  <- function(p, par) -log(1 - p) / par[1]

# Generalized Exponential
pdf_gexp <- function(x, par) {
  a <- par[1]; l <- par[2]
  a * l * exp(-l*x) * (1-exp(-l*x))^(a-1)
}
cdf_gexp <- function(x, par) {
  a <- par[1]; l <- par[2]
  (1-exp(-l*x))^a
}
qf_gexp <- function(p, par) {
  a <- par[1]; l <- par[2]
  -log(1 - p^(1/a)) / l
}

# Weibull
pdf_weibull <- function(x, par) {
  k <- par[1]; l <- par[2]
  (k/l) * (x/l)^(k-1) * exp(-(x/l)^k)
}
cdf_weibull <- function(x, par) {
  k <- par[1]; l <- par[2]
  1 - exp(-(x/l)^k)
}
qf_weibull <- function(p, par) {
  k <- par[1]; l <- par[2]
  l * (-log(1 - p))^(1/k)
}

# Normal
pdf_norm <- function(x, par) dnorm(x, par[1], par[2])
cdf_norm <- function(x, par) pnorm(x, par[1], par[2])
qf_norm <- function(p, par) qnorm(p, par[1], par[2])

data <- rexp(200, 1)
# Call the plot function
plot_multi_fitted(
  data = data,
  pdf_list = list(pdf_exp, pdf_gexp, pdf_weibull, pdf_norm),
  cdf_list = list(cdf_exp, cdf_gexp, cdf_weibull, cdf_norm),
  qf_list  = list(qf_exp, qf_gexp, qf_weibull, qf_norm),
  params_list = list(
    c(1.1),
    c(2, 1.3),
    c(1.5, 2),
    c(0, 1)
  ),
  dist_names = c("Exp", "GExp", "Weibull", "Normal"),
  col_list = c("blue", "red", "darkgreen", "purple"),
  lty_list = c(1, 2, 3, 4),
  lwd_list = c(3, 3, 3, 3)
)

Plot for the PDF of a Probability distribution.

Description

Plots the PDF of a probability distribution. Supports multiple sets of parameters with shaded areas under the curves.

Usage

plot_pdf(
  pdf,
  param_list,
  xlim = c(0, 10),
  ylim = c(0, 6),
  n = 500,
  main = "PDF Plot",
  xlab = "x",
  ylab = "Density",
  colors = NULL,
  shade_colors = NULL,
  lwd = 3,
  lty = 2,
  grid = TRUE,
  grid_lty = 3,
  grid_col = "gray80",
  grid_lwd = 1
)

Arguments

Value

A PDF plot is displayed. The function invisibly returns NULL.

Examples

# Example 1 with Generalized Exponential Distribution
ge_pdf <- function(x, alpha, lambda) {
alpha * lambda * exp(-lambda * x) * (1 - exp(-lambda * x))^(alpha - 1)
}
param_values <- list(list(alpha = 1, lambda = 1),
                    list(alpha = 2, lambda = 1),
                    list(alpha = 3, lambda = 0.5),
                    list(alpha = 4, lambda = 1.5),
                    list(alpha = 5, lambda = 2.5))
plot_pdf(pdf = ge_pdf, param_list = param_values, ylim = c(0, 1),
        main = "Generalized Exponential Distribution")

# Example 2 with Exponentiated Weibull Distribution
pdf_expweibull <- function(x, a, b, c){
a * b * c * exp(-(b*x)^c) *
 (b*x)^(c-1) * (1 - exp(-(b*x)^c))^(a-1)
}
param_values <- list(list(a = 0.3, b = 1.2, c = 1.0),
                    list(a = 1.3, b = 0.4, c = 2.3),
                    list(a = 1.5, b = 0.9, c = 3.0),
                    list(a = 2.0, b = 1.8, c = 2.8),
                    list(a = 3.7, b = 2.0, c = 1.5))
colors <- c("green", "purple", "yellow", "orange", "darkblue")
plot_pdf(pdf = pdf_expweibull, param_list = param_values,
        main = "PDF of EW Distribution",
        colors = colors, xlim = c(0, 5), ylim = c(0, 3))

Plot for the Survival Function of a Probability Distribution

Description

#' Plots the SF of a probability distribution. Supports multiple sets of parameters with shaded areas under the curves.

Usage

plot_sf(
  sf,
  param_list,
  xlim = c(0, 10),
  ylim = c(0, 1),
  n = 500,
  main = "SF Plot",
  xlab = "x",
  ylab = "SF",
  colors = NULL,
  shade_colors = NULL,
  lwd = 3,
  lty = 2,
  grid = TRUE,
  grid_lty = 3,
  grid_col = "gray80",
  grid_lwd = 1
)

Arguments

Value

A SF plot is displayed. The function invisibly returns NULL.

Examples

# Example 1: Generalized Exponential Distribution
ge_sf <- function(x, alpha, lambda) {
  1 - (1 - exp(-lambda * x))^alpha
}
param_values <- list(
  list(alpha = 1, lambda = 1),
  list(alpha = 2, lambda = 1),
  list(alpha = 3, lambda = 0.5),
  list(alpha = 4, lambda = 1.5),
  list(alpha = 5, lambda = 2.5)
)
plot_sf(sf = ge_sf, param_list = param_values, main = "SF GE Distribution")

# Example 2: Exponentiated Weibull Distribution
sf_expweibull <- function(x, a, b, c) {
  1 - (1 - exp(-(b*x)^c))^a
}
param_values <- list(
  list(a = 0.3, b = 1.2, c = 1.0),
  list(a = 1.3, b = 0.4, c = 2.3),
  list(a = 1.5, b = 0.9, c = 3.0),
  list(a = 2.0, b = 1.8, c = 2.8),
  list(a = 3.7, b = 2.0, c = 1.5)
)
colors <- c("green", "purple", "yellow", "orange", "darkblue")
plot_sf(sf = sf_expweibull, param_list = param_values,
        main = "SF of EW Distribution", colors = colors, xlim = c(0, 5))