Comparative Analysis of Richardson-Lucy Deconvolution and Data Unfolding with Mean Integrated Square Error Optimization

TL;DR

The paper addresses the challenge of comparing unfolding methods when external validation is unavailable by employing internal quality criteria. It introduces Mean Integrated Squared Error (MISE) and the Minimum Condition Number (MCN) of the unfolded-bin correlation matrix as robust metrics, and contrasts Richardson-Lucy with a new MISE-optimized Data Unfolding method (NG) that uses entropy regularization and system identification to construct the response matrix. Across simulated data with 10000 and 1000 events, NG consistently achieves lower optimal MISE and MCN, indicating better accuracy and stability, particularly in the presence of multicollinearity. The work highlights the practical value of internal criteria for method selection in unfolding tasks and suggests that the NG approach offers a more reliable alternative when external references are scarce or unavailable.

Abstract

Two maximum likelihood-based algorithms for unfolding or deconvolution are considered: the Richardson-Lucy method and the Data Unfolding method with Mean Integrated Square Error (MISE) optimization [10]. Unfolding is viewed as a procedure for estimating an unknown probability density function. Both external and internal quality assessment methods can be applied for this purpose. In some cases, external criteria exist to evaluate deconvolution quality. A typical example is the deconvolution of a blurred image, where the sharpness of the restored image serves as an indicator of quality. However, defining such external criteria can be challenging, particularly when a measurement has not been performed previously. In such instances, internal criteria are necessary to assess the quality of the result independently of external information. The article discusses two internal criteria: MISE for the unfolded distribution and the condition number of the correlation matrix of the unfolded distribution. These internal quality criteria are applied to a comparative analysis of the two methods using identical numerical data. The results of the analysis demonstrate the superiority of the Data Unfolding method with MISE optimization over the Richardson-Lucy method.

Figures (18)

• Figure 1: Acceptance $A(x)$ and resolution function $R(x,y)$ , for $x=10$ .
• Figure 2: True PDF $\phi (x)$ (left) , PDF used for simulation $\phi^s(x)$ (right).
• Figure 3: Matrix $R$ calculated for RL method (left) and NG method (right).
• Figure 4: Measured distribution histogram for the RL method (left) and measured distribution histogram for NG method (right). The true distribution is shown by the curve.
• Figure 5: MISE for RL method (left) and MISE for the NG method(right). The standard deviations of the estimated MISE are presented as error bars.