How to transform data to uniform distribution (uniform percentiles)?

Question

Given the list of data points (normalized in [0,1] range), I plot the histogram of values and compute percentiles (shown as x ticks).

How to find a transformation of data values so the histogram is approximately uniform. Which would, in turn, make percentile values also uniformly distributed.

Answer 1

Hi: You can calculate the empirical cumulative distribution of the data. By this, I mean, given some observation in the sample, $x_i$, calculate $P(X < x_{i})$ by calculating the proportion of observations that are less than $x_{i}$ (i.e. the percentiles ). Then, do this for all the $x_{i}$ so that you have the cumulative distribution of the $x_{i}$.

Then, $P(X < x)$ is uniform for a given value of $x$.

In fact, it seems like you already did this but the percentile values should be on the vertical axis and the values of the data should be on the x-axis.

Note that page 14 of this PDF explains the concept more clearly than I have.

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Example Implementation

Below is a quick-and-dirty attempt to illustrate this answer. The image below shows the original histogram of the Gaussian, the empirical cumulative distribution function of that data, and then the histogram of the converted data.

R Code Below

par(mfrow=c(3,1))
# First, generate some Gaussian numbers.
gaussian <- rnorm(1000,0.0,0.05)
gh <- hist(gaussian, breaks=1000)

empirical_cumulative_distribution <- cumsum(gh$counts)/1000

plot(gh$mids, empirical_cumulative_distribution)


uniformize <- function(x) {
  ans_x <- x
  for (idx in seq(1,length(x))){
    max_idx <- max(which(gh$mids < x[idx]))
    ans_x[idx] <- empirical_cumulative_distribution[max_idx]
  }
  return(ans_x)
}

uniform2 <- uniformize(gaussian )
hist(uniform2, breaks=100)
par(mfrow=c(1,1))

Answer 2

Python Version:

import matplotlib.pyplot as plt 
import numpy as np

def uniformize(x,nbins=1000):
    which = lambda lst:list(np.where(lst)[0])

    gh = np.histogram(x,bins=nbins)
    
    empirical_cumulative_distribution = np.cumsum(gh[0])/nbins
    
    ans_x = x
    for idx in range(len(x)):
        max_idx = max(which(gh[1]<x[idx])+[0])
        ans_x[idx] = empirical_cumulative_distribution[max_idx]
    
    return ans_x

if __name__ == '__main__':
    #number of bins to use
    numb = 1000
    
    # Distribution you want to transform
    dist_transform = np.random.normal(3,5,numb)
    
    # Plotting original distribution and CDF
    fig, (ax1,ax2,x3) = plt.subplots(3,1)
    n,bins,patches = ax1.hist(dist_transform,bins=numb)
    ax2.plot(bins[1:],np.cumsum(n)/numb)

    uniform_dist =  uniformize(dist_transform)   
    x3.hist(uniform_dist,bins = 100,alpha=0.5)