From Data to Probability Densities without Histograms

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.

💡 Research Summary

The paper addresses a common problem in data analysis: estimating and visualizing the probability density (PD) of a continuous variable without the drawbacks of traditional histograms. Histograms require the arbitrary choice of bin size, which creates a trade‑off between resolution and statistical noise. Moreover, the resulting PD estimate can be noisy, especially for limited data sets. The authors propose a method that bypasses histograms entirely by exploiting the empirical cumulative distribution function (ECDF), which is parameter‑free but a step function. By differentiating a step function one would obtain a sum of Dirac deltas, not a smooth density, so the authors introduce a two‑stage procedure that yields a smooth PD estimate together with reliable error bars.

Step 1 – Linear baseline and remainder.
Given a sorted data set {x₁,…,xₙ}, the ECDF F(x) jumps by 1/n at each data point. The authors define a simple linear baseline F₀(x) that goes from 0 at the lower bound a to 1 at the upper bound b (typically a = smallest data point, b = largest, but can be narrowed for heavy‑tailed distributions). The remainder R(x) = F(x) – F₀(x) satisfies R(a)=R(b)=0, which makes it amenable to a pure sine Fourier series: R(x) = Σ_{i=1}^{m} d_i sin