RooUnfold unfolding framework and algorithms

1
RooUnfoldunfolding frameworkand algorithms

• Tim Adye
• Rutherford Appleton Laboratory
• BaBar Statistics Working Group
• BaBar Collaboration Meeting
• 13th December 2005

2
Outline

• What is Unfolding?
• 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

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 ?. 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

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

5
What happens if you dont smooth
6
True Gaussian, with Gaussian smearing, systematic
translation, and variable inefficiency trained
using a different Gaussian
7
Double Breit-Wigner, with Gaussian smearing,
systematic translation, and variable inefficiency
trained using a single Gaussian
8
So why dont 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 doesnt include smearing
  (ie. its 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

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

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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

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1. Regularised Unfolding

• Use Maximum Likelihood to fit smeared bin
  contents to measured data, but include
  regularisation function
• where the regularisation parameter, a, controls
  the degree of smoothness (select a 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

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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

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2D Unfolding Example2D Smearing, bias, variable
efficiency, and variable rotation
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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!

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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

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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

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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

18
References - Overview

• G. Cowan, A Survey of Unfolding Methods for
  Particle Physics, Proc. Advanced Statistical
  Techniques in Particle Physics, Durham (2002)
• http//www.ippp.dur.ac.uk/Workshops/02/statistics/
  
• 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_Lec
  tures.html

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References - Techniques

• V. Blobel, Unfolding Methods in High Energy
  Physics,DESY 84-118 (1984) also CERN 85-02
• A. Höcker and V. Kartvelishvili, SVD Approach to
  Data Unfolding, NIM A 372 (1996) 469
• www.lancs.ac.uk/depts/physics/staff/kartvelishvili
  .html
• K. Tackmann, H. Lacker, Unfolding the Hadronic
  Mass Spectrumin B-gtXu l? Decays, BAD 894.
• G. DAgostini, A multidimensional unfolding
  method based on Bayes theorem, NIM A 362 (1995)
  487