Upload
isaac-doyle
View
214
Download
0
Embed Size (px)
Citation preview
1
Iterative dynamically stabilized (IDS) method of data unfolding (*)
(*arXiv:0907.3791)
Bogdan MALAESCUCERN
PHYSTAT 2011Workshop 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)
Aij
ij
1
; ijij NBins
kjk
AP true spectrum data P
A
• 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
UnfoldingBackground 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 uncertaintyNew 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 l (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 l
2
( , , ) 1x
f x e
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 transferof 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:
2
2 2
MCd D
k k k kMC
MCD
k k kMC
Nd d B r
N
Nd d r
N
1
1 , ,n
MC MCD D k k k
kNN N f d d d
1
( )n
MC dD k k
k
N d B
• 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: (ND – ND
MC)/ND
•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)
λN Choice λN
StableUnstable
Ingredients for the unfolding procedure: the MC normalization procedure
11
Ingredients for the unfolding procedure: one step of the unfolding method
1
; ijij n
kjk
AP true spectrum data P
A
1
1 , ,, , k k k kj
n
j k k k kj
Mu
Cd
jM k
jC
u f d d f dN
d d dt PN
B
1
ijij n
ikk
AP
A
Folding:
Unfolding matrix (like d’Agostini method):
1
1
n
i ik kk
n
j kj kk
r P t
t P r
By construction:
Unfolding: compare data and reconstructed MC spectra
General equation
Only approximate for spectra other than MC
Fluctuation in background subtraction
True MC Significant difference (unfolded)
Not significant difference (fixed)
Aij
ij
12
1st step of the unfolding methodL 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 lL=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
1 , ,uj j j j
MCD
j j j
S
MC
B f u u u
Nu u t
N
• Transfer matrix improvement: use significant structures
The folding matrix (P), describing detector effects, stays unchanged. Only the generated MC spectrum is improved.
, , , 1;MCij ij j j j ijMC
D
MCu D
j j j jMC
M
NA A f u u u P pour i n
N
Nu u B t
N
Normalization procedure
14
The Iterative Unfolding Method• 1st 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 -Initialreconstructed MC
Data -Initialgenerated 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 - Finalreconstructed MC(after one iteration)
Data -Initialreconstructed MC
22
Simplified unfolding method:• Standard normalization for the MC• No estimation of left fluctuations (from background subtraction)• 1st unfolding with λ = λL ( = 1.5, justified by a study (see next))• One iteration with λU= λM=0
Effect of the 2nd unfolding
Effect of the 1st 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 Aij • (Do not) Fluctuate the folded data• Unfolding with the matrix A’ij (Aij 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 -Initialreconstructed MC
Data - Finalreconstructed MC(after one iteration)
24
Bias measurement after unfolding (without statistical
fluctuations of folded data)
Result – generated data (1st step)
Result – generated data (2nd step)
Bias measurement after unfolding (without statistical
fluctuations of folded data) in large bins
•The 1st unfolding provides a good result•λL = 1.5 : very small bias and reduced correlations with respect to the case λL = 0
Data uncertainties
A test with known « generated data » (before folding)
25
• Diagonal uncertainties after the 1st unfolding: larger in the non trivial case (less correlations between the bins)
Uncertainties after 1st unfolding λL = 0
Uncertainties after 1st unfolding λL = 1.5
Data uncertainties
A simple example for the use of the unfolding method