A Linear Iterative Unfolding Method

Mathematics – Statistics Theory

Scientific paper

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Proceedings of ACAT-2011 conference (Uxbridge, United Kingdom), 8 pages, 5 figures

Scientific paper

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 numerical ill-posedness of this task, various methods were invented which, given some assumptions on the initial probability distribution, try to regularize the problem. Most of these methods definitely introduce bias on the estimate of the initial probability distribution. We propose a linear iterative method (motivated by the Neumann series / Landweber iteration known in functional analysis), 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. Convergence is proved under certain quite general conditions, which hold for practical applications such as convolutions, calorimeter response functions, momentum reconstruction response functions based on tracking in magnetic field etc. The method can be seen to be asymptotically unbiased. The proof of convergence also provides explicit formulae for the propagation of the three error terms: residual error (distance from the limit), statistical error, and systematic error. These can be used to define an optimal stopping criterion, and error estimates there. We provide a numerical C library for the implementation of the method, and also discuss its relation to other known approaches.

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