Spectrum Spectrum Reconstruction of Reconstruction of Atmospheric Atmospheric Neutrinos with Neutrinos with Unfolding Techniques Unfolding Techniques

Juande Zornoza UW MadisonJuande Zornoza UW Madison

IntroductionIntroduction We will review different approaches for We will review different approaches for

the reconstruction of the energy the reconstruction of the energy spectrum of atmospheric neutrinos:spectrum of atmospheric neutrinos: Blobel / Singular Value DecompostionBlobel / Singular Value Decompostion

actually, both methods are basically the same, actually, both methods are basically the same, with only differences in issues not directly related with only differences in issues not directly related with the unfolding but with the regularization and with the unfolding but with the regularization and so onso on

Singular Value DecompositionSingular Value Decomposition

Spectrum reconstructionSpectrum reconstruction In principle, energy spectra can be obtained using the In principle, energy spectra can be obtained using the

reconstructed energy of each event.reconstructed energy of each event.

However, this is However, this is not efficientnot efficient in our case because of the in our case because of the combination of two factors:combination of two factors: Fast decreaseFast decrease (power-law) of (power-law) of

the flux.the flux. Large fluctuationsLarge fluctuations in the in the

energy deposition.energy deposition.

For this reason, an alternative way has to be used: For this reason, an alternative way has to be used: unfolding techniquesunfolding techniques..

Spectrum unfoldingSpectrum 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:

( ) ( | ) ( )meas truef b R b y f y dyMatrix notation

Ay b

•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

Dealing with instabilitiesDealing with instabilities Regularization:Regularization:

Solution with minimum curvatureSolution with minimum curvature Solution with strictly positive curvatureSolution with strictly positive curvature Principle of maximum entropyPrinciple of maximum entropy

Iterative procedure, leading Iterative procedure, leading asymptotically to the unfolded distributionasymptotically to the unfolded distribution

Single Value Single Value DecompositionDecomposition11 The response matrix is decomposed as:The response matrix is decomposed as:

1 A. Hoecker, Nucl. Inst. Meth. in Phys. Res. A 372:469 (1996)

ˆ TA USV U, V: orthogonal matricesS: non-negative diagonal matrix (si, singular values)

This can be also seen as a minimization problem:This can be also seen as a minimization problem:

b xn

i

n

jiiij bxA

1

2

1

minˆ

minˆˆ 1 bxABbxAT

Or, introducing the covariance matrix to take into account errors:Or, introducing the covariance matrix to take into account errors:

SVD: normalizationSVD: normalization Actually, it is convenient to normalize the unknownsActually, it is convenient to normalize the unknowns

Ay b

xn

jijij bwA

1 AAijij contains the number of events, not the probability. contains the number of events, not the probability. Advantages:Advantages:

the vector w should be smooth, with small bin-to-bin variationsthe 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 binavoid weighting too much the cases of 100% probability when only one event is in the bin

/ inij j jw y y

SVD: RescalingSVD: Rescaling Rotation of matrices allows to rewrite the Rotation of matrices allows to rewrite the

system with a covariance matrix equal to system with a covariance matrix equal to I, more convenient to work with:I, more convenient to work with:

TB QRQ1

ij im mjmi

A Q Ar

1i im m

mi

b Q br

min)~~()~~( bwAbwA T

RegularizationRegularization Several methods have been proposed for the Several methods have been proposed for the

regularization. The most common is to add a regularization. The most common is to add a curvature term add a curvature termcurvature term add a curvature term

1

1

1 01 2 10 1 1 0

1 2 10 1

C

Other option: principle of maximum entropyOther option: principle of maximum entropy

min)()~~()~~( CwCwbwAbwA TT

RegularizationRegularization We have transformed the problem in the optimization of We have transformed the problem in the optimization of

the value of the value of , which tunes how much , which tunes how much regularizationregularization we we include:include: too large: too large: physical informationphysical information lost lost too small: too small: statistical fluctuationsstatistical fluctuations spoil the result spoil the result

Td U b

In order to optimize the value In order to optimize the value of of ::Evaluation using Evaluation using MCMC informationinformationComponents of vectorComponents of vectorMaximum curvature of the Maximum curvature of the L-L-curvecurve

L-curve

Solution to the systemSolution to the system Actually, the solution to the system with Actually, the solution to the system with

the curvature term can be expressed as the curvature term can be expressed as a function of the solution without a function of the solution without curvature:curvature:

1

'yn

ii i

i i

dw f v

s

wherewhere

(Tikhonov factors)(Tikhonov factors)

2

2

22

2

if if

/1

i

i

ii

ii s

sss

sf

2

2)(

i

iii ssdd

1'A AC

'w Cw

Tikhonov factorsTikhonov factors The non-zero tau is equivalet to change dThe non-zero tau is equivalet to change dii by by

fun2

fun0fun1

Components of d

= sk2k

2

2

22

2

if if

/1

i

i

ii

ii s

sss

sf

And this allows to find a criteria to find a good tauAnd this allows to find a criteria to find a good tau

Bayesian Iterative MethodBayesian Iterative Method22

10

0

)()|()()|(

)|(

lllj

iijji EPEXP

EPEXPXEP

1

ˆ( ) ( ) ( | )Xn

i j i jj

n E n X P E X

•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:

•The dependence on the initial probability P0(Ei) can be overcome by an iterative process.

•The expected number of events to be assigned to each of the causes is:

En

ii

ii

En

EnEP

1

)(ˆ

)(ˆ)(ˆ

2 G. D'Agostini NIM A362(1995) 487-498

prior guess: iterative approachsmearing matrix: MC

experimental data (simulated)

Reconsructed spectrum

Iterative algorithmIterative algorithm

)(ˆ En )(ˆ EP)(ˆ En

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 and .3. Compare to .4. Replace by and by .5. Go to step 2.

)(0 En)(0 EP )(ˆ EP )(0 En )(ˆ En

P(Xj|Ei)

Po(E)

P(Ei|Xj)

n(Xj)

n(Ei) P(Ei)

Initialguess

no(E)

Experimental data

Smearing matrix (MC)

Differences between SVD and Differences between SVD and BlobelBlobel

Different curvature termDifferent curvature term Selection of optimum tauSelection of optimum tau B-splines used in the standard Blobel B-splines used in the standard Blobel

implementationimplementation ……

B-splinesB-splines Spline: piecewise continuous Spline: piecewise continuous

and differentiable function that and differentiable function that connects two neighbor points connects two neighbor points by a cubic polynomial: by a cubic polynomial:

from H. Greene PhD.

B-spline: spline functions can be expressed by a B-spline: spline functions can be expressed by a finite superposition of base functions (B-spilines).finite superposition of base functions (B-spilines).

(higher orders)(first order)

For IceCubeFor IceCube Several parameters can be investigated:Several parameters can be investigated:

Number of channelsNumber of channels Number of hitsNumber of hits Reconstructed energyReconstructed energy Neural network output…Neural network output…

With IceCube, we will have much better With IceCube, we will have much better statistics than with AMANDAstatistics than with AMANDA

But first, reconstruction with 9 strings will be But first, reconstruction with 9 strings will be the prioritythe priority

RemarksRemarks First, a good agreement between data and MC First, a good agreement between data and MC

is necessaryis necessary Different unfolding methods will be compared Different unfolding methods will be compared

(several internal parameters to tune in each (several internal parameters to tune in each method)method)

Several regularization techniques are also Several regularization techniques are also available in the literatureavailable in the literature

Also an investigation on the best variable for Also an investigation on the best variable for unfolding has to be doneunfolding has to be done

Maybe several variables can be used in a Maybe several variables can be used in a multi-D analysismulti-D analysis