Optimal Binning for Small-Angle Neutron Scattering Data Using the Freedman-Diaconis Rule

TL;DR

The paper addresses inefficiencies in fixed-bin SANS data reduction by introducing a statistically grounded binning strategy based on the Freedman-Diaconis rule $h = 2\, \mathrm{IQR}\, n^{-1/3}$. It derives the competing error terms for counting noise and binning distortion, showing that the optimal bin width scales as $h_{opt} \propto N_{total}^{-1/3}$ and that the classical FD formula provides a practical adaptive binning estimator. Using synthetic Debye-fragment data for a Gaussian polymer, the authors demonstrate faithful reproduction of $I(Q)$ curvature while suppressing random fluctuations, with an optimal bin count $K_{opt}$ around 38 in the tested scenario. The approach offers a physics-informed, adaptive histogramming framework that connects statistics, instrument resolution, and data representation, with clear pathways to higher-dimensional extensions and correlation considerations.

Abstract

Small-Angle Neutron Scattering (SANS) data analysis often relies on fixed-width binning schemes that overlook variations in signal strength and structural complexity. We introduce a statistically grounded approach based on the Freedman-Diaconis (FD) rule, which minimizes the mean integrated squared error between the histogram estimate and the true intensity distribution. By deriving the competing scaling relations for counting noise ($\propto h^{-1}$) and binning distortion ($\propto h^{2}$), we establish an optimal bin width that balances statistical precision and structural resolution. Application to synthetic data from the Debye scattering function of a Gaussian polymer chain demonstrates that the FD criterion quantitatively determines the most efficient binning, faithfully reproducing the curvature of $I(Q)$ while minimizing random error. The optimal width follows the expected scaling $h_{\mathrm{opt}} \propto N_{\mathrm{total}}^{-1/3}$, delineating the transition between noise- and resolution-limited regimes. This framework provides a unified, physics-informed basis for adaptive, statistically efficient binning in neutron scattering experiments.

Figures (2)

• Figure 1: Reconstruction of a synthetic SANS intensity profile generated from the Debye scattering function of a Gaussian polymer chain Debye1947. The analytical $I(Q)$ (gray line) serves as the reference against which histograms obtained using different bin numbers are compared. (a) Coarse binning ($K = 10$) yields a smooth intensity profile with low statistical noise but significant distortion due to averaging over wide intervals. (b) Optimal binning ($K_{\mathrm{opt}} = 38$) minimizes the total mean-squared deviation in Eq. \ref{['eq:DeltaI_total']}, accurately reproducing the curvature and overall decay of the Debye function. (c) Over-binning ($K = 120$) produces strong statistical fluctuations dominated by counting noise. The three panels collectively demonstrate how the balance between counting variance ($\propto h^{-1}$) and binning distortion ($\propto h^{2}$) determines the optimal bin width, which preserves the intrinsic structural features of the scattering profile while minimizing random error.
• Figure 2: Evolution of the Freedman--Diaconis (FD) optimal binning with total detector counts. (a) Mean-squared error (MSE) versus bin width $h$ for increasing total counts, shown from black to blue. The red dashed line marks the detector-pixel width, and black crosses indicate FD-optimal bin sizes. The minima shift from larger to smaller $h$ as counts increase, marking a transition from the noise-limited to the resolution-limited regime. (b) Minimum MSE values as a function of total counts. The solid line shows the FD-predicted scaling $\langle (\Delta I)^2 \rangle_{\mathrm{opt}} \propto N_{\mathrm{total}}^{-1}$, while open squares correspond to pixel-sized bins. (c) FD-optimal bin width $h_{\mathrm{opt}}$ versus total counts. The black crosses follow the expected scaling $h_{\mathrm{opt}} \propto N_{\mathrm{total}}^{-1/3}$, with the red dashed line denoting the pixel width. The crossover point marks where detector resolution begins to limit statistical improvement.