1. 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 Decompostion  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  Singular Value Decomposition

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 → f true (y) Measured quantity: b, which follows pdf→f meas (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 (in our case, b will be xlow) The response matrix inversion gives useless solutions, due to the effect of statistical fluctuations, so two alternative methods have been tried: -Singular Value Decomposition -Iterative Method based on Bayes’ Theorem

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

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

7. SVD: normalization  Actually, it is convenient to normalize the unknowns  A ij 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  Components of vector  Maximum curvature of the L-curve 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 equivalet to change d i by fun2 fun0 fun1 Components of d  = s k 2 k  And this allows to find a criteria to find a good tau

13. Bayesian Iterative Method 2 If there are several causes (E i ) which can produce an effect X j and we know the initial probability of the causes P(E i ), the conditional probability of the cause to be E i when X is observed is: The dependence on the initial probability P 0 (E i ) can be overcome by an iterative process. The expected number of events to be assigned to each of the causes is: 2 G. D'Agostini NIM A362(1995) 487-498 prior guess: iterative approach smearing matrix: MC experimental data (simulated)

14. Reconsructed spectrum Iterative algorithm 1.Choose the initial distribution P 0 (E). For instance, a good guess could be the atmospheric flux (without either prompt neutrinos or signal). 2.Calculate and. 3.Compare to. 4.Replace by and by. 5.Go to step 2. P(X j |E i ) P o (E) P(E i |X j ) n(X j ) n(E i )P(E i ) Initial guess n o (E) Experimental data Smearing matrix (MC)

15. Differences between SVD and Blobel  Different curvature term  Selection of optimum tau  B-splines used in the standard Blobel implementation  …

16. 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). (higher orders)(first order)

17. For IceCube  Several parameters can be investigated:  Number of channels  Number of hits  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

18. 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