1. Juande Zornoza UW Madison
Spectrum Reconstruction of Atmospheric Neutrinos with Unfolding Techniques
Juande Zornoza UW Madison

2. 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

3. 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.

4. 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:
Matrix notation
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

5. Dealing with instabilities
Regularization:
Solution with minimum curvature
Principle of maximum entropy
Iterative procedure, leading asymptotically to the unfolded distribution

6. Single Value Decomposition1
The response matrix is decomposed as:
U, V: orthogonal matrices
S: non-negative diagonal matrix (si, singular values)
This can be also seen as a minimization problem:
Or, introducing the covariance matrix to take into account errors:
1 A. Hoecker, Nucl. Inst. Meth. in Phys. Res. A 372:469 (1996)

7. SVD: normalization
Actually, it is convenient to normalize the unknowns
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

8. SVD: Rescaling Rotation of matrices:
allows to rewrite the system with a covariance matrix equal to I, more convenient to work with:

9. Regularization
Several methods have been proposed for the regularization. The most common is to add a curvature term add a curvature term
Other option: principle of maximum entropy

10. 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
Components of vector
L-curve

11. 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:
where
(Tikhonov factors)

12. Tikhonov factors The non-zero tau is equivalent to change di by
And this allows to find a criteria to find a good tau
Components of d
fun0
fun1 k
fun2
 = sk2

13. 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

14. Bayesian Iterative Method2
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:
prior guess: iterative approach
smearing matrix: MC
The expected number of events to be assigned to each of the causes is:
experimental data (simulated)
The dependence on the initial probability P0(Ei) can be overcome by an iterative process.
2 G. D'Agostini NIM A362(1995)

15. Reconsructed spectrum
Iterative algorithm
Choose the initial distribution P0(E). For instance, a good guess could be the atmospheric flux (without either prompt neutrinos or signal).
Calculate and
Compare to
Replace by and by
Go to step 2.
P(Xj|Ei)
Smearing matrix (MC)
Reconsructed spectrum
P(Ei|Xj)
n(Ei)
P(Ei)
no(E)
Po(E)
Initial
guess
n(Xj)
Experimental data

16. 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

17. 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

18. This is the end

19. 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)

20. Maximum entropy In general, we want to minimise
S can be the spikeness of the solution, or, the entropy of the system:
Probability of any event of being in bin i is fi/N.
Then, following the maximum entropy principle, we will minimize:
Useful for sharp gradients, i.e. when the solution it is not expected to be very smooth