1. RooUnfold unfolding framework and algorithms
Tim Adye
Rutherford Appleton Laboratory
BaBar Statistics Working Group
BaBar Collaboration Meeting
13th December 2005

2. Outline What is Unfolding? Overview of a few techniques
and why might you want to do it?
Overview of a few techniques
Regularised unfolding
Iterative method
RooUnfold package
Currently implements three methods with a common interface
Status and Plans
References
13th December 2005
Tim Adye

3. Unfolding In other fields known as “deconvolution”, “unsmearing”
Given a “true” PDF in μ, that is corrupted by detector effects, described by a response function, R, we measure a distribution in ν. In terms of histograms
This may involve
inefficiencies: lost events
bias and smearing: events moving between bins (off-diagonal Rij)
With infinite statistics, it would be possible to recover the original PDF by inverting the response matrix
13th December 2005
Tim Adye

4. Not so simple…
Unfortunately, if there are statistical fluctuations between bins this information is destroyed
Since R washes out statistical fluctuations, R-1 cannot distinguish between wildly fluctuating and smooth PDFs
Obtain large negative correlations between adjacent bins
Large fluctuations in reconstructed bin contents
Need some procedure to remove wildly fluctuating solutions
Give added weight to “smoother” solutions
Solve for µ iteratively, starting with a reasonable guess and truncate iteration before it gets out of hand
Ignore bin-to-bin fluctuations altogether
13th December 2005
Tim Adye

5. What happens if you don’t smooth
13th December 2005
Tim Adye

6. True Gaussian, with Gaussian smearing, systematic translation, and variable inefficiency – trained using a different Gaussian
13th December 2005
Tim Adye

7. Double Breit-Wigner, with Gaussian smearing, systematic translation, and variable inefficiency – trained using a single Gaussian
13th December 2005
Tim Adye

8. So why don’t we always do this?
If the true PDF and resolution function can be parameterised, then a Maximum Likelihood fit is usually more convenient
Directly returns parameters of interest
Does not require binning
If the response function doesn’t include smearing (ie. it’s diagonal), then apply bin-by-bin efficiency correction directly
If result is just needed for comparison (eg. with MC), could apply response function to MC
simpler than un-applying response to data
13th December 2005
Tim Adye

9. When to use unfolding
Use unfolding to recover theoretical distribution where
there is no a-priori parameterisation
this is needed for the result and not just comparison with MC
there is significant bin-to-bin migration of events
13th December 2005
Tim Adye

10. Where could we use unfolding?
Traditionally used to extract structure functions
Widely used outside PP for image reconstruction
Dalitz plots
Cross-feed between bins due to misreconstruction
“True” decay momentum distributions
Theory at parton level, we measure hadrons
Correct for hadronisation as well as detector effects
13th December 2005
Tim Adye

11. 1. Regularised Unfolding
Use Maximum Likelihood to fit smeared bin contents to measured data, but include regularisation function
where the regularisation parameter, α, controls the degree of smoothness (select α to, eg., minimise mean squared error)
Various choices of regularisation function, S, are used
Tikhonov regularisation: minimise curvature
for some definition of curvature, eg.
RooUnfHistoSvd by Kerstin Tackmann and Heiko Lacker
based on GURU by Andreas Höcker and Vakhtang Kartvelishvili
uses Singular Value Decomposition
RUN by Volker Blobel
Maximum entropy:
13th December 2005
Tim Adye

12. 2. Iterative method Uses Bayes’ theorem to invert
and using an initial set of probabilities, pi (eg. flat) obtain an improved estimate
Repeating with new pi from these new bin contents converges quite rapidly
Truncating the iteration prevents us seeing the bad effects of statistical fluctuations
Fergus Wilson and I have implemented this method in ROOT/C++
Supports 1D, 2D, and 3D cases
13th December 2005
Tim Adye

13. 2D Unfolding Example 2D Smearing, bias, variable efficiency, and variable rotation
13th December 2005
Tim Adye

14. RooUnfold Package
Make these different methods available as ROOT/C++ classes with a common interface to specify
unfolding method and parameters
response matrix
pass directly or fill from MC sample
measured histogram
return reconstructed truth histogram and errors
full covariance matrix
Easy to do with multiple dimensions (when supported)
This should make it easy to try and compare different methods in your analysis
Could also be useful outside BaBar!
13th December 2005
Tim Adye

15. RooUnfold Classes RooUnfoldResponse
response matrix with various filling and access methods
create from MC, use on data (can be stored in a file)
RooUnfold – unfolding algorithm base class
RooUnfoldBayes – Iterative method
RooUnfoldSvd – Inteface to RooUnfHistoSvd package
RooUnfoldBinByBin – Simple bin-by-bin method
Trivial implementation, but useful to compare with full unfolding
RooUnfoldExample – Simple 1D example
RooUnfoldTest and RooUnfoldTest2D
Test with different training and unfolding distributions
13th December 2005
Tim Adye

16. RooUnfold Status Available in CVS
Announced in Statistics HN
See README file for details of building and running
Interface can still be adjusted based on comments
I already have an idea for simplifying use in multi-dimensional case
13th December 2005
Tim Adye

17. Plans and possible improvements
So far this is mostly a programming exercise
Would be interesting to compare the different methods for some real analysis distributions
But YMMV
Add common tools, useful for all algorithms
Inputs and results in different formats
already supports histograms and ROOT vectors/matrices
Automatic calculation of figures of merit (eg. Â2)
can also use standard ROOT functions on histograms
Simplify selection of regularisation parameter
More algorithms?
Maximum entropy regularisation
Simple matrix inversion without regularisation
perhaps useful with large statistics
13th December 2005
Tim Adye

18. References - Overview
G. Cowan, A Survey of Unfolding Methods for Particle Physics, Proc. Advanced Statistical Techniques in Particle Physics, Durham (2002)
G. Cowan, Statistical Data Analysis, Oxford University Press (1998), Chapter 11: Unfolding
R. Barlow, SLUO Lectures on Numerical Methods in HEP (2000), Lecture 9: Unfolding
www-group.slac.stanford.edu/sluo/Lectures/Stat_Lectures.html
13th December 2005
Tim Adye

19. References - Techniques
V. Blobel, Unfolding Methods in High Energy Physics, DESY (1984); also CERN 85-02
A. Höcker and V. Kartvelishvili, SVD Approach to Data Unfolding, NIM A 372 (1996) 469
K. Tackmann, H. Lacker, Unfolding the Hadronic Mass Spectrum in B->Xu lν Decays, BAD 894.
G. D’Agostini, A multidimensional unfolding method based on Bayes’ theorem, NIM A 362 (1995) 487
13th December 2005
Tim Adye