General methodology to calculate the inverse function via a numeric approach.

inverse_function.R

# INVERSE FUNCTION
# by: Rodrigo Hernández Mota
# created in:  29/04/2016
# last modify: 02/05/2016

source("generate_dataset.R")
library(ggplot2)
library(tibble)
library(tidyr)


# Inverse Function Estimated ----------------------------------------------
# This is the numerical estimation of the inverse function.
# The following variables area needed:
# - u: the variable [0, 1] to calculate the inverse into (runif)
# - x: the x-axis
# - z: the accumulated values (cdf function)

inverse_func <- function(u, x, z){
  
  # determine the "position" of u in z
  i <- z <= u
  i <- sum(i)
  
  # use linear interpolation to map
  # the variable
  m <- (x[i+1]-x[i])/(z[i+1]-z[i])
  b <- x[i] - m * z[i]
  
  # return the result
  m * u + b
  }

# numeric estimation and generation of rv

n <- 5000
uniform_rv <- runif(n)
sub_data <- dataset[dataset$dens == "cdf", ]
random_variables <- sapply(uniform_rv, inverse_func, 
                           x = sub_data$x, z = sub_data$freq)


# ggplot(data_frame(random_variables), aes(random_variables, y = ..density..)) +
#   geom_density(fill = "blue", alpha = 0.2) + theme_light()
aux <- data_frame(x = 1:n, Desired = random_variables, Uniform = uniform_rv)
aux1 <- gather(aux[1:50, ], Distribution, Value, -x)


ggplot(aux, aes(Uniform, Desired)) + theme_light() + geom_point(size = 0.5) +
  ggtitle("Relation Among the Random Numbers") + xlab("Uniform distr. rv") +
  ylab("Desired distr. rv")

ggplot(aux1, aes(x,Value, group = Distribution)) + theme_light() +
  geom_point(aes(color = Distribution)) + 
  geom_line(aes(color = Distribution), alpha = 0.3) +
  scale_color_brewer(palette = 3, direction = 1) +
  ggtitle("Sample: first 50 transformations") +
  xlab("Number") + ylab("Value")

# Estimated inverse function ----------------------------------------------


ggplot(data_frame(x = (1:n)/n, y = random_variables[order(random_variables)]),
       aes(x,y)) + theme_light() + 
  geom_ribbon(aes(ymin = 0, ymax = y), fill = "blue", alpha = 0.2) +
  geom_line(size = 0.75) + geom_hline(yintercept = 0) + 
  ggtitle("Estimated Inverse Function") + 
  xlab("Freq.") + ylab("Value")

# Frequency histogram -----------------------------------------------------


ggplot(data_frame(random_variables), aes(random_variables, y = ..density..)) +
  geom_histogram(fill = "blue", alpha = 0.3, bins = floor(sqrt(n)), color = "black") + 
  theme_light() +
  ggtitle("Frequency histogram of generated random variables")