一种线性迭代展开方法
A Linear Iterative Unfolding Method
摘要：
在实验物理学中经常面临的一个任务是测量某个量的概率分布。通常，要测量的这个量被非理想的检测器响应或某些物理过程所模糊。从测量的分布中去除这种拖尾效应的过程称为展开，并且由于该任务的众所周知的数值病态行为，这是信号处理中的一个微妙问题。人们发明了各种方法，在初始概率分布上给出一些假设，试图使展开问题正规化。这些方法中的大多数肯定会在初始概率分布的估计中引入偏倚。我们提出了一种线性迭代法，它的优点是不需要对初始概率分布的假设，唯一的正则化参数是迭代的停止顺序，它可以用来选择引入的偏差和传播的统计和系统误差之间的最佳折衷。方法一致：在响应函数的一般条件下，证明了在没有测量误差的情况下，“逐箱”收敛到初始概率分布。这个条件适用于实际应用，如卷积，量热计响应函数，动量重建响应函数的基础上跟踪磁场等测量误差的存在下，明确的公式传播的三个重要的误差项：偏置误差，统计误差，和系统误差。这三个误差项之间的折衷可以用于定义最优迭代停止准则，并且可以在那里估计误差。我们提供了一个数值的C库的方法，其中包括自动统计误差传播以及实施。
英文摘要：
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.
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