1 Iterative dynamically stabilized (IDS) method of data unfolding (*) (*arXiv: ) Bogdan MALAESCU CERN PHYSTAT 2011 Workshop on unfolding

2 Outlook Introduction: main effects to deal with Additional problems in practice An iterative unfolding method A complex example Discussion and conclusions

3 Introduction: detector effects, folding and unfolding Example of transfer matrix (MC) A ij i j Folding: Unfolding of detector effects (acceptance corrected afterwards) Unfolding is not a simple numerical problem Must use a regularization method. Resolution + Distortion

4 Problems in practice: fluctuations due to background subtraction A “standard” unfolding could propagate large fluctuations into precise regions of the spectrum The uncertainties of the data points must be taken into account in the unfolding! (used to compute the significance of data-MC differences in each bin) Folding Unfolding Background subtraction

5 Problems in practice: transfer matrix simulation  perfect Key: use the significance of data-MC differences in each bin New structure (not simulated) MC  improved normalization MC  standard normalization Detector simulation (folding): systematic uncertainty New structures in data: must also be corrected for detector effects could bias MC normalization (needed in the unfolding, for data-MC comparison)

6 Ingredient for the unfolding procedure: a regularization function Used to “measure” significance in the (bin by bin) comparison of experimental data and MC simulation Allows one to perform a different treatment of fluctuations and significant new structures in data Important for the dynamical regularization of fluctuations Depends (monotonously) on the absolute data – MC difference, their uncertainties and a parameter (scale factor) Behavior at small/large parameter values is important, but the exact choice of the function is not critical Used at all the steps of the unfolding procedure, with different values for

7 Model for the test of the method Transfer matrix model: For the folding Fluctuated matrix used for the unfolding Reconstructed MC Generated MC Resolution effect Systematic transfer of events

8 Generated MC Data New Structures Data  Reconstructed MC Data  Generated MC Model for the test of the method Reconstructed MC Generated MC + New Structures  Truth Data

9 First estimation of the number of events in data, corresponding to structures simulated by MC: A better estimation: The same method at the level of (corrected spectrum/ generated MC) # data ev., in the bin k # background subtraction fluctuation ev., in the bin k ITERATIONS Ingredients for the unfolding procedure: the MC normalization procedure

10 Relative improvement of the normalization: (N D – N D MC )/N D The number of iterations is important only in the unstable region The size of the unstable region depends on the amplitude of fluctuations in background subtraction Study performed directly on data! 50 iterations ( at most ) λ N Choice λNλN StableUnstable Ingredients for the unfolding procedure: the MC normalization procedure

11 Ingredients for the unfolding procedure: one step of the unfolding method Folding: Unfolding matrix (like d’Agostini method): By construction: Unfolding: compare data and reconstructed MC spectra General equation Only approximate for spectra other than MC Fluctuation in background subtraction True MCSignificant difference (unfolded) Not significant difference (fixed) A ij i j

12 1 st step of the unfolding method Choice: (all differences between data and reconstructed MC spectra treated as not significant) Reconstructed MC Generated MC + New Structures  Truth Data Data New Structures Data  Reconstructed MC Data  Generated MC Corrected spectrum Corrected spectrum  generated MC If one would choose L =0 …

13 Ingredients for the unfolding procedure : Comparison of the corrected spectrum and generated MC: Estimation of large fluctuations in background subtraction: not significant deviations, with large uncertainties Transfer matrix improvement: use significant structures The folding matrix (P), describing detector effects, stays unchanged. Only the generated MC spectrum is improved. Normalization procedure

14 The Iterative Unfolding Method 1 st unfolding, where the large fluctuations due to background subtraction are kept unchanged 1)Estimation of large fluctuations due to background subtraction 2)Transfer matrix improvement (hence of the unfolding probability matrix) 3)Improved unfolding  Dynamical regularization: from the treatment of fluctuations in each bin, at each step of the procedure When should the iterations stop? Comparison of data and reconstructed MC Study the number of needed iterations, with toys  Choice of parameters used at different steps, with a model for data. One can (in general) give up some of the parameters (by performing a maximal unfolding & transfer matrix modification).

15 Results after iterations Data – improved reconstructed MC Estimation of background fluctuations Data  Reconstructed MC New structures

16 Unfolding Result New Structures Initial reconstructed MC Initial generated MC + New Structures  Truth Data Data Data  Initial reconstructed MC Data  Initial generated MC Corrected spectrum Corrected spectrum  Initial generated MC Statistical uncertainties propagated using pseudo-experiments (“toys”).

17 Discussion Studied but not discussed: N bins data  N bins result (rebinning in the unfolding or afterwards) Effect of rebinning on correlations Effect of regularization on uncertainties and correlations (see Kerstin’s talk) Treatment of bins with negative number of events (data) Empty bins in MC Preventing the existence of negative bins in the improved generated MC

18 Conclusion New general method for the unfolding of binned data Can treat problems that were not considered previously Dynamic regularization procedure, bin by bin at each step This method allows one to keep some control of bin to bin correlations in the unfolded spectrum Root code is available

19 Backup

20 Zoom on the narrow resonance region

21 Simplified example: Reduced effects of the transfer matrix Smoother « bias », without structures No « deeps » in the spectrum No important fluctuations from background subtraction Statistics reduced by a factor 20 A simple example for the use of the unfolding method Data uncertainties Data  Final reconstructed MC (after one iteration) Data  Initial reconstructed MC

22 Simplified unfolding method : Standard normalization for the MC No estimation of left fluctuations (from background subtraction) 1 st unfolding with λ = λ L ( = 1.5, justified by a study (see next)) One iteration with λ U = λ M =0 Effect of the 2 nd unfolding Effect of the 1 st unfolding Data uncertainties A simple example for the use of the unfolding method

23 Use (data – reconstructed MC) as bias with respect to the generated MC, in order to build « generated data » (toys) Folding with the matrix A ij (Do not) Fluctuate the folded data Unfolding with the matrix A’ ij (A ij fluctuated) Compare the result with the « generated data » A test with known « generated data » (before folding) No extra data fluctuations: test systematic effectsWith statistical data fluctuations: stability test Data uncertainties Data  Initial reconstructed MC Data  Final reconstructed MC (after one iteration)

24 Bias measurement after unfolding (without statistical fluctuations of folded data) Result – generated data (1 st step) Result – generated data (2 nd step) Bias measurement after unfolding (without statistical fluctuations of folded data) in large bins The 1 st unfolding provides a good result λ L = 1.5 : very small bias and reduced correlations with respect to the case λ L = 0 Data uncertainties A test with known « generated data » (before folding)

25 Diagonal uncertainties after the 1 st unfolding: larger in the non trivial case (less correlations between the bins) Uncertainties after 1 st unfolding λ L = 0 Uncertainties after 1 st unfolding λ L = 1.5 Data uncertainties A simple example for the use of the unfolding method