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Spectrum
Reconstruction of
Atmospheric
Neutrinos with
Unfolding Techniques
Juande Zornoza
UW Madison
Introduction
 We will review different approaches for
the reconstruction of the energy
spectrum of atmospheric neutrinos:
 Blobel / Singular Value Decomposition
 actually, both methods are basically the same,
with only differences in issues not directly related
with the unfolding but with the regularization and
so on
 Iterative Method based on Bayes’ theorem
Spectrum reconstruction
 In principle, energy spectra can be obtained using the
reconstructed energy of each event.
 However, this is not
efficient in our case
because of the
combination of two factors:
 Fast decrease (power-law) of
the flux.
 Large fluctuations in the
energy deposition.
 For this reason, an alternative way has to be used:
unfolding techniques.
Spectrum unfolding
•Quantity to obtain: y, which follows pdf→ftrue(y)
•Measured quantity: b, which follows pdf→fmeas(b)
•Both are related by the Fredholm integral equation of first kind:
f meas (b)   R(b | y) ftrue ( y )dy
Matrix notation
Ây  b
•The response matrix takes into account three factors:
-Limited acceptance
-Finite resolution
-Transformation
•The response matrix inversion gives useless solutions, due to the
effect of statistical fluctuations
Dealing with instabilities
 Regularization:
 Solution with minimum curvature
 Principle of maximum entropy
 Iterative procedure, leading
asymptotically to the unfolded distribution
Single Value Decomposition1
 The response matrix is decomposed as:
Aˆ  USV T
U, V: orthogonal matrices
S: non-negative diagonal matrix (si, singular values)
 This can be also seen as a minimization
problem:
2


ˆ
  Aij xi  bi   min



i 1  j 1

nb
nx
 Or, introducing the covariance matrix to take
into account errors:

 

ˆAx  b T B 1 Aˆ x  b  min
1
A. Hoecker, Nucl. Inst. Meth. in Phys. Res. A 372:469 (1996)
SVD: normalization
 Actually, it is convenient to normalize the
unknowns
Ây  b
w j  y j / yini j
nx
A w
j 1
ij
j
 bi
 Aij contains the number of events, not the
probability.
 Advantages:
 the vector w should be smooth, with small bin-to-bin
variations
 avoid weighting too much the cases of 100%
probability when only one event is in the bin
SVD: Rescaling
 Rotation of matrices:
B  QRQ
T
1
Aij   Qim Amj
ri m
bi 
1
Qimbm

ri m
allows to rewrite the system with a
covariance matrix equal to I, more
convenient to work with:
~T ~
~
~
( Aw  b ) ( Aw  b )  min
Regularization
 Several methods have been proposed for the
regularization. The most common is to add a
curvature term add a curvature term
~T ~
~
~
( Aw  b ) ( Aw  b )   (Cw)T Cw  min
 1  

 1
C  0




1
2  
1
0
1
1
1
2  
0
1



0 

1 
1   
 Other option: principle of maximum entropy
Regularization
 We have transformed the problem in the
optimization of the value of , which tunes how
much regularization we include:
  too large: physical information lost
  too small: statistical fluctuations spoil the result
In order to optimize the
value of :
Evaluation using MC
information
Maximum curvature of
the L-curve
T
Components of vector d  U b
L-curve
Solution to the system
 Actually, the solution to the system with
the curvature term can be expressed as
a function of the solution without
curvature:
ny
w'  
i 1
di
f i vi
si
A '  AC 1
w '  Cw
where
d
( )
i
si2
 di 2
si  
2
1

if
s
s
i  
fi  2
 2
si   si /  if si2  
2
i
(Tikhonov factors)
Tikhonov factors
 The non-zero tau is equivalent to change di by
 1 if si2  
si2
fi  2
 2
si   si /  if si2  
 And this allows to find a criteria to find a good tau
fun0
fun1
fun2
Components of d
k
 = sk2
Differences between SVD and Blobel
 More simplified implementation
 Possibility of different number of bins in y
and b (non square A)
 Different curvature term
 Selection of optimum tau
 B-splines used in the standard Blobel
implementation
Bayesian Iterative
2
Method
•If there are several causes (Ei) which can produce an effect Xj and we
know the initial probability of the causes P(Ei), the conditional probability of
the cause to be Ei when X is observed is:
P( Ei | X j ) 
smearing matrix: MC
P( X j | Ei ) P0 ( Ei )
 P( X
l 1
j
| El ) P0 ( El )
prior guess:
iterative
approach
•The expected number of events to be assigned to each of the
nX
causes is:
nˆ ( Ei )   n( X j ) P( Ei | X j )
j 1
experimental data (simulated)
•The dependence on the initial probability P0(Ei) can be overcome by
an iterative process.
nˆ ( Ei )
ˆ
P( Ei )  nE
 nˆ( Ei )
i 1
2
G. D'Agostini NIM A362(1995) 487-498
Iterative algorithm
1. Choose the initial distribution P0(E). For instance, a good guess
could be the atmospheric flux (without either prompt neutrinos or
signal).
2. Calculate nˆ ( E ) and Pˆ ( E ).
3. Compare nˆ ( E ) to n0 ( E ).
4. Replace P0 ( E ) by Pˆ ( E ) and n0 ( E ) by nˆ ( E ).
5. Go to step 2.
P(Xj|Ei)
Smearing matrix (MC)
no(E)
Initial
guess
P(Ei|Xj)
Reconsructed
spectrum
n(Ei)
Po(E)
n(Xj)
Experimental data
P(Ei)
For IceCube
 Several parameters can be investigated:




Number of channels
Number of NPEs
Reconstructed energy
Neural network output…
 With IceCube, we will have much better
statistics than with AMANDA
 But first, reconstruction with 9 strings will be
the priority
Remarks
 First, a good agreement between data and MC
is necessary
 Different unfolding methods will be compared
(several internal parameters to tune in each
method)
 Several regularization techniques are also
available in the literature
 Also an investigation on the best variable for
unfolding has to be done
 Maybe several variables can be used in a
multi-D analysis
This is the end
B-splines
 Spline: piecewise continuous
and differentiable function that
connects two neighbor points
by a cubic polynomial:
from H. Greene PhD.
 B-spline: spline functions can be expressed by a
finite superposition of base functions (B-spilines).
(first order)
(higher orders)
Maximum entropy
 In general, we want to minimise  ln L  S
 S can be the spikeness of the solution, or, the
entropy of the system: S    P ln P
 Probability of any event of being in bin i is fi/N.
N
i 1
i
i
 Then, following the maximum entropy principle,
we will minimize:
 ln L    fi / N ln( fi / N )
i
 Useful for sharp gradients, i.e. when the
solution it is not expected to be very smooth