A Linear Iterative Unfolding Method

TL;DR

This paper introduces a linear iterative unfolding method for reconstructing true probability distributions from smeared measurements, avoiding assumptions on the initial distribution and balancing bias and errors through iteration stopping.

Contribution

The proposed method is a novel linear iterative approach that does not require initial distribution assumptions and includes explicit error propagation, with proven convergence under general conditions.

Findings

Proves binwise convergence to the true distribution without measurement errors.

Provides explicit formulas for propagating bias, statistical, and systematic errors.

Includes a numerical C library with automatic error propagation features.

Abstract

A frequently faced task in experimental physics is to measure the probability distribution of some quantity. Often this quantity to be measured is smeared by a non-ideal detector response or by some physical process. The procedure of removing this smearing effect from the measured distribution is called unfolding, and is a delicate problem in signal processing, due to the well-known numerical ill behavior of this task. Various methods were invented which, given some assumptions on the initial probability distribution, try to regularize the unfolding problem. Most of these methods definitely introduce bias into the estimate of the initial probability distribution. We propose a linear iterative method, which has the advantage that no assumptions on the initial probability distribution is needed, and the only regularization parameter is the stopping order of the iteration, which can be used to choose the best compromise between the introduced bias and the propagated statistical and systematic errors. The method is consistent: "binwise" convergence to the initial probability distribution is proved in absence of measurement errors under a quite general condition on the response function. This condition holds for practical applications such as convolutions, calorimeter response functions, momentum reconstruction response functions based on tracking in magnetic field etc. In presence of measurement errors, explicit formulae for the propagation of the three important error terms is provided: bias error, statistical error, and systematic error. A trade-off between these three error terms can be used to define an optimal iteration stopping criterion, and the errors can be estimated there. We provide a numerical C library for the implementation of the method, which incorporates automatic statistical error propagation as well.