Unfolding
neutrino cross sections using xsTool
Mark Rayner, University of Geneva
VANISH cross-section workshop, Valencia, 4 April 2014
[email protected]
Which unfolding methods could you use?
Which should you choose?
VaNISh, April 2014
Unfolding techniques for T2K xs analyses (MR)
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Which unfolding methods could you use?
Martin’s xsTool inheritance diagram
VaNISh, April 2014
Unfolding techniques for T2K xs analyses (MR)
?
?
?
3
Which unfolding method(s) should you choose?
Colin Anderson via Teppei
VaNISh, April 2014
Unfolding techniques for T2K xs analyses (MR)
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Why don’t we always unfold measurement errors?
a) It’s much easier to do a maximum likelihood fit
b) If we want to test the theory, we can add detector effects to Monte Carlo
VaNISh, April 2014
Unfolding techniques for T2K xs analyses (MR)
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Why don’t we always unfold measurement errors?
a) It’s much easier to do a maximum likelihood fit
But we don’t know how to parameterize the solution
b) If we want to test the theory, we can add detector effects to Monte Carlo
But we want our result to be useful in the long term,
and we want to compare results from different experiments
VaNISh, April 2014
Unfolding techniques for T2K xs analyses (MR)
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Nay, answer me: stand, and unfold yourself!
– Francisco
VaNISh, April 2014
Unfolding techniques for T2K xs analyses (MR)
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Nay, answer me: stand, and unfold yourself!
– Francisco
( R-1 n = Barnardo )
VaNISh, April 2014
Unfolding techniques for T2K xs analyses (MR)
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1. Bin by bin
Glen Cowan, Statistical Data Analysis (Oxford)
VaNISh, April 2014
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e.g. many early ATLAS results
Unfolding in ATLAS, Georgios Choudalakis, arXiv:1104.2962 [hep-ex]
VaNISh, April 2014
Unfolding techniques for T2K xs analyses (MR)
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Model dependant
VaNISh, April 2014
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Some generalities. The effect of resolution and efficiency on the true PDF.
measured
VaNISh, April 2014
true
Unfolding techniques for T2K xs analyses (MR)
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Some generalities. The effect of resolution and efficiency on the true PDF.
measured event distribution
measured
true
true event distribution
VaNISh, April 2014
Unfolding techniques for T2K xs analyses (MR)
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P( true bin 3  measured bin 1 )
true bins
=
measured
bins
R
Sums = efficiencies
in each true bin
VaNISh, April 2014
Unfolding techniques for T2K xs analyses (MR)
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Caveat emptor
Large bins?
Variable resolution and efficiency?
The response matrix is not model independent!
VaNISh, April 2014
Unfolding techniques for T2K xs analyses (MR)
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The bin contents themselves as ML parameters. What would the unbiased estimator be?
Minimum variance (‘RCF’) bound
Glen Cowan, Statistical Data Analysis (Oxford)
VaNISh, April 2014
Unfolding techniques for T2K xs analyses (MR)
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2. Matrix inversion
true
observed
unfolded
expected
y
x
x
Glen Cowan, Statistical Data Analysis (Oxford)
VaNISh, April 2014
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“If the matrix and/or the r.h.s. of a linear system is known with some level of
uncertainty,
and some singular values of the matrix are significantly smaller than others,
the system may be difficult to solve even if formally the matrix has full rank.
In many aspects such matrices behave like degenerate ones”
– Höcker and Kartvelishvili, SVD Approach to Data Unfolding,
arXiv:hep-ph/9509307
VaNISh, April 2014
Unfolding techniques for T2K xs analyses (MR)
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Starting with a highly structured true distribution hints at the cause of the instability
expected
true
Glen Cowan, Statistical Data Analysis (Oxford)
VaNISh, April 2014
Unfolding techniques for T2K xs analyses (MR)
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VaNISh, April 2014
Unfolding techniques for T2K xs analyses (MR)
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3. Iterative Bayesian Unfolding
Unsmearing matrix from Bayes theorem:
D’Agostini 1995
VaNISh, April 2014
Unfolding techniques for T2K xs analyses (MR)
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1)
2)
preferred: 3)
VaNISh, April 2014
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D'Agostini, 2010, Improved iterative Bayesian unfolding, arXiv:1010.0632 [physics.data-an]
“Empirically we learn that, in ‘normal’ cases, just two or three steps are
sufficient to recover quite accurately the true spectrum.”
VaNISh, April 2014
Unfolding techniques for T2K xs analyses (MR)
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D’Agostini 2010
“It is important to note that, contrary to other algorithms, the intermediate smoothing
acts as a regularization on the priors and not on the unfolded spectrum.”
VaNISh, April 2014
Unfolding techniques for T2K xs analyses (MR)
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This is now a popular method in collider physics
Unfolding in ATLAS, Georgios Choudalakis, arXiv:1104.2962 [hep-ex]
Good to have an objective criteria for choosing the number of iterations
VaNISh, April 2014
Unfolding techniques for T2K xs analyses (MR)
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This is now a popular method in collider physics
Unfolding in ATLAS, Georgios Choudalakis, arXiv:1104.2962 [hep-ex]
Good to test model independence!
VaNISh, April 2014
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4. Iterative, dynamically stabilized unfolding
Abstract
We propose a new iterative unfolding method for experimental data, making use
of a regularization function. The use of this function allows one to build an
improved normalization procedure for Monte Carlo spectra, unbiased by the
presence of possible new structures in data. We are able to unfold, in a
dynamically stable way, data spectra which can be strongly affected by
fluctuations in the background subtraction and simultaneously reconstruct
structures which were not initially simulated. This method also allows one to
control the amount of correlations introduced between the bins of the unfolded
spectrum, when the transfers of events correcting the systematic detector effects
are performed.
An iterative, dynamically stabilized method of data unfolding, Bogdan Malaescu,
arXiv:0907.3791 [physics.data-an]
VaNISh, April 2014
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5. Tikhonov Regularization
Solution: matrix inversion
Regularization strength
Z = the 2nd-order finite difference operator
from a discrete approximation of the second derivative
mt
y
c.f. Colin Anderson’s MiniBooNE thesis
VaNISh, April 2014
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6. MaxEnt Regularization
An alternative regularization function:
Popular in image reconstruction and astrophysics
See Jaynes’ superb book, and also Cowan
VaNISh, April 2014
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7. Regularization via Singular Value Decomposition
Höcker and Kartvelishvili, SVD Approach to Data Unfolding, arXiv:hep-ph/9509307
Orthogonal
UT = U-1
true bins
measured
bins
R
=
U
Orthogonal
VT = V-1
T
s1
s2
S
V
LEFT singular singular
vectors
values
RIGHT singular
vectors
VaNISh, April 2014
Unfolding techniques for T2K xs analyses (MR)
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rotate by VT
rotate by U
scale by S
s1
s2
unit disk
VaNISh, April 2014
Unfolding techniques for T2K xs analyses (MR)
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rotate by VT
rotate by U
scale by S
s1
s2
unit disk
VaNISh, April 2014
Unfolding techniques for T2K xs analyses (MR)
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rotate by VT
rotate by U
scale by S
s1
s2
unit disk
VaNISh, April 2014
Unfolding techniques for T2K xs analyses (MR)
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rotate by VT
rotate by U
scale by S
s1
s2
unit disk
VaNISh, April 2014
Unfolding techniques for T2K xs analyses (MR)
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rotate by VT
rotate by U
scale by S
s1
s2
unit disk
VaNISh, April 2014
If (b1-b2) is not statistically significant, and ε is small
 solution dominated by random fluctuations
Unfolding techniques for T2K xs analyses (MR)
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wj should be smooth if MC is good
- and Reduce effect of MC stat error on A
VaNISh, April 2014
Unfolding techniques for T2K xs analyses (MR)
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wj should be smooth if MC is good
- and Reduce effect of MC stat error on A
Covariance is the unit matrix
VaNISh, April 2014
Unfolding techniques for T2K xs analyses (MR)
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And finally we’re ready to regularize!
The solution turns out to be of the form
linearly independent combinations of the data bi
with unit variance
So we see which the smallest si with a statistically significant di and choose τ = si2
 We smoothly filter out the statistically instability
VaNISh, April 2014
Unfolding techniques for T2K xs analyses (MR)
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Response matrix
truth
unfolded x
data b
Höcker and Kartvelishvili, SVD Approach to Data Unfolding, arXiv:hep-ph/9509307
VaNISh, April 2014
Unfolding techniques for T2K xs analyses (MR)
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di (ordered by decreasing si)
Deviation from the truth
Höcker and Kartvelishvili, SVD Approach to Data Unfolding, arXiv:hep-ph/9509307
VaNISh, April 2014
Unfolding techniques for T2K xs analyses (MR)
40
Which unfolding methods could you use?
Martin’s xsTool inheritance diagram
VaNISh, April 2014
Unfolding techniques for T2K xs analyses (MR)
?
?
?
41
VaNISh, April 2014
Unfolding techniques for T2K xs analyses (MR)
42
Which unfolding method(s) should you choose?
Colin Anderson via Teppei
VaNISh, April 2014
Unfolding techniques for T2K xs analyses (MR)
43