From Data to Probability Densities without Histograms

TL;DR

This paper introduces a novel method to estimate smooth probability densities from continuous data without histograms, using Fourier series and statistical tests, with error estimation via jackknife, demonstrated through examples.

Contribution

It presents a new histogram-free approach for density estimation that overcomes noise and bin size issues, employing Fourier series and Kolmogorov tests.

Findings

Provides a smooth probability density estimate from data

Includes error bars using jackknife method

Demonstrates effectiveness with multiple examples

Abstract

When one deals with data drawn from continuous variables, a histogram is often inadequate to display their probability density. It deals inefficiently with statistical noise, and binsizes are free parameters. In contrast to that, the empirical cumulative distribution function (obtained after sorting the data) is parameter free. But it is a step function, so that its differentiation does not give a smooth probability density. Based on Fourier series expansion and Kolmogorov tests, we introduce a simple method, which overcomes this problem. Error bars on the estimated probability density are calculated using a jackknife method. We give several examples and provide computer code reproducing them. You may want to look at the corresponding figures 4 to 9 first.