Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

TUnfold: an algorithm for correcting migration effects in high energy physics

Stefan Schmitt

TL;DR

TUnfold addresses the problem of correcting migration and background effects in multi-dimensional high-energy physics distributions by formulating unfolding as a constrained least-squares problem with Tikhonov regularisation. It offers two principled methods to choose the regularisation strength—the L-curve curvature and global-correlation minimisation—plus support for complex, multi-dimensional regularisation patterns, background subtraction, and systematic uncertainties. The framework integrates with ROOT via a set of C++ classes that organize core unfolding, background/systematics handling, correlation scans, and multidimensional binning structures. This combination yields robust, bias-controlled corrections suitable for differential cross sections and other observables where detector effects must be removed from measured data.

Abstract

TUnfold is a tool for correcting migration and background effects in high energy physics for multi-dimensional distributions. It is based on a least square fit with Tikhonov regularisation and an optional area constraint. For determining the strength of the regularisation parameter, the L-curve method and scans of global correlation coefficients are implemented. The algorithm supports background subtraction and error propagation of statistical and systematic uncertainties, in particular those originating from limited knowledge of the response matrix. The program is interfaced to the ROOT analysis framework.

TUnfold: an algorithm for correcting migration effects in high energy physics

TL;DR

TUnfold addresses the problem of correcting migration and background effects in multi-dimensional high-energy physics distributions by formulating unfolding as a constrained least-squares problem with Tikhonov regularisation. It offers two principled methods to choose the regularisation strength—the L-curve curvature and global-correlation minimisation—plus support for complex, multi-dimensional regularisation patterns, background subtraction, and systematic uncertainties. The framework integrates with ROOT via a set of C++ classes that organize core unfolding, background/systematics handling, correlation scans, and multidimensional binning structures. This combination yields robust, bias-controlled corrections suitable for differential cross sections and other observables where detector effects must be removed from measured data.

Abstract

TUnfold is a tool for correcting migration and background effects in high energy physics for multi-dimensional distributions. It is based on a least square fit with Tikhonov regularisation and an optional area constraint. For determining the strength of the regularisation parameter, the L-curve method and scans of global correlation coefficients are implemented. The algorithm supports background subtraction and error propagation of statistical and systematic uncertainties, in particular those originating from limited knowledge of the response matrix. The program is interfaced to the ROOT analysis framework.

Paper Structure

This paper contains 20 sections, 27 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The TUnfold algorithm
  3. Definitions
  4. Algorithm
  5. Normalisation of the matrix of migrations
  6. Choice of the regularisation strength
  7. L-curve scan
  8. Minimising global correlation coefficients
  9. Background subtraction
  10. Systematic uncertainties on the matrix of migrations
  11. Choice of regularisation conditions
  12. Regularisation of multi-dimensional distributions
  13. Regularisation on the density, multi-dimensional distributions
  14. Densities
  15. Derivatives
  16. ...and 5 more sections

Figures (2)

  • Figure 1: schematic view of migration effects and statistical fluctuations
  • Figure 2: example binning scheme with three nodes. The "generator" node is the root node. It has two child nodes, "signal" and "background". The "signal" node has a two-dimensional binning in two variables, pt and eta, whereas the background node has unconnected bins corresponding to various background sources.