Undersmoothed uncertainty quantification for unfolding

Authors: Junhyung Lyle Kim, Rice University, and Mikael Kuusela, Carnegie Mellon University

This repository provides an extension for the unfolding software TUnfold V17.8 written by Stefan Schmitt. TUnfold implements Tikhonov regularization for unfolding smeared data from particle detectors. However, as demonstrated by Kuusela (2016), the resulting confidence intervals may seriously underestimate the uncertainty in the unfolded space, unless care is taken in the choice of the regularization strength tau. This extension provides a new function for choosing the regularization strength, UndersmoothTau, which aims at choosing tau so that the confidence intervals have nearly nominal coverage. UndersmoothTau is an implementation of the data-driven undersmoothing technique introduced in Kuusela (2016). Please refer to the references section for more details on the statistical methodology.

Documentation

Detailed documentation for UndersmoothedUnfolding can be found here.

New functions

Function	Input	Output	Description
UndersmoothTau	Initial tau, tolerance epsilon, max number of iterations	Undersmoothed tau	Undersmooths the initial tau (from L-curve for example) gradually until the minimum estimated coverage meets the target coverage, which is the nominal 68% minus the tolerance epsilon. This is the main function from the user's perspective.
ComputeCoverage	Estimate of the true histogram, tau	Estimated coverage	Computes the estimated coverage given an estimate of the true histogram and a regularization strength tau. Used by UndersmoothTau.

Example usage

UndersmoothTau is implemented so that it can be used with any initial estimate of tau. Below is example usage of UndersmoothTau with the ScanLcurve method provided in TUnfold.

TUnfold unfold = new TUnfold();          // construct a TUnfold object
unfold.ScanLcurve();                     // unfold using ScanLcurve method
TauFromLcurve = unfold.GetTau();         // retrieve tau chosen by ScanLcurve

// starting from tau chosen by ScanLcurve, undersmooth tau until the minimum estimated coverage  
// meets the target coverage, which is the nominal 68% minus the tolerance epsilon (0.01 in this example).
TauFromUndersmoothing = unfold.UndersmoothTau(TauFromLcurve, 0.01, 1000);
unfold.DoUnfold(TauFromUndersmoothing);   // unfold again with undersmoothed tau

See UndersmoothDemo.C for more details on how to use UndersmoothTau.

Demonstration

The simulation below compares the performance of the unfolded confidence intervals when the regularization strength is chosen using ScanLcurve provided by TUnfold and the algorithm UndersmoothTau provided by this extension. The tolerance epsilon was set to 0.01, so the intervals from UndersmoothTau should have 67% coverage. The top plots compare the binwise coverage of the methods. The coverage is estimated by repeating the unfolding 1,000 times with independent realizations of data. The bottom plots show one realization of the unfolded confidence intervals for each method. The confidence intervals provided by ScanLcurve are too short and suffer from drastic undercoverage, while the intervals provided by UndersmoothTau have nearly nominal coverage without being excessively long. See UndersmoothDemo.C for details of the simulation setup.

Installation

To install, simply set up the ROOT environment (tested with ROOT V6.18/00) and run make as below:

$ source /path/to/install-or-build/dir/bin/thisroot.sh
$ make

You should see an executable named UndersmoothDemo. Running the executable will generate UndersmoothDemo.pdf. (Running the demo takes a while. To obtain results faster reduce repeatNum in UndersmoothDemo.C.)

Known issues

• The current version is intended to be used with kEConstraintNone for the area constraint option when constructing the TUnfold object.

References

M. Kuusela. Uncertainty quantification in unfolding elementary particle spectra at the Large Hadron Collider. PhD thesis, École Polytechnique Fédérale de Lausanne (EPFL), 2016.

S. Schmitt. TUnfold, an algorithm for correcting migration effects in high energy physics. Journal of Instrumentation, 7:T10003, 2012.