Data Unfolding Methods in High Energy Physics

TL;DR

The paper surveys common unfolding methods for binned high-energy physics data, focusing on how to solve the ill-posed problem of recovering a truth distribution from detector-level measurements. It compares bin-by-bin, matrix inversion, template fits with and without regularisation, and two iterative approaches (EM and IDS), and it introduces practical criteria (L-curve and average global correlations) to select regularisation strength using a toy example. Closure and data tests reveal a trade-off: regularisation reduces oscillations and correlations but biases the result, while unregularised methods can be unbiased but unstable; consequently, careful validation through closure tests is essential. The findings guide practitioners in choosing and tuning unfolding methods for reliable, testable results in high-energy physics analyses.

Abstract

A selection of unfolding methods commonly used in High Energy Physics is compared. The methods discussed here are: bin-by-bin correction factors, matrix inversion, template fit, Tikhonov regularisation and two examples of iterative methods. Two procedures to choose the strength of the regularisation are tested, namely the L-curve scan and a scan of global correlation coefficients. The advantages and disadvantages of the unfolding methods and choices of the regularisation strength are discussed using a toy example.

Figures (7)

• Figure 1: Toy example, distribution on truth level (left), on reconstructed level (middle), response matrix (right).
• Figure 2: Unfolding result using the matrix inversion method. Shown are the unfolded distribution compared to data and Monte Carlo truth (left), the correlation coefficients (middle) and the unfolded result folded back using the response matrix, compared to Data and Monte Carlo (right).
• Figure 3: Unfolding result using the constrained template fit with Tikhonov regularisation and parameter $\tau=0.0068$. Further details are given in Fig. \ref{['fig:inversion3']} caption.
• Figure 4: Parametric plot of $X=\log\chi^2_A$ and $Y=\log\chi^2_L$ (L-curve).
• Figure 5: Results of two scans of the parameter $\tau$: L-curve curvature scan (left panels) and scan of average global correlation coefficients (right panels). The scans and the corresponding unfolding result at the final choice of $\tau$ are shown.