unfolding of energy spectra: combunfold · daniel mazin magic software school, july 2016, kochel am...

Unfolding of energy spectra: CombUnfold.C

Daniel MazinICRR, U-Tokyo and Max-Planck-Institute for Physics, Munich

6th MAGIC Software School, July 2016, Kochel am See

Sabrina EineckeTU Dortmund

Unfolding of energy spectra: CombUnfold.C

Daniel MazinICRR, U-Tokyo and Max-Planck-Institute for Physics, Munich

6th MAGIC Software School, July 2016, Kochel am See

Sabrina EineckeTU Dortmund

7th MAGIC Software School, June 2018, La Palma

Daniel Mazin MAGIC software school, July 2016, Kochel am See !3

Acceptance

Resolution Background Indirect

Measurement

Etrue

Etrue

Eest Eest

Eest

Unfolding

Measurement process suffers from following properties • Indirect measurement • Limited resolution • Limited acceptance • Threshold effects • Background

Motivation

Daniel Mazin MAGIC software school, July 2016, Kochel am See !4

Finite energy resolution

• Variance within the air shower• Depth of Gammas follows certain probability• Interactions between particles follow certain probabilities• ...

• Detector not perfect• Reflection at mirrors• Detection efficiency• Limited measurement resolution of

electronics• ...

• Binning of observables• Systematic distortions like threshold effects

2 gammas with same energy:

2 gammas with different energy:

energy migration matrix

Motivation

Daniel Mazin MAGIC software school, July 2016, Kochel am See

Inverting of migration matrix

• Unfolding uses the migration matrix to convert excess events in Eest to excess events in Etrue

• We need to work with distributions, cannot access true energy of individual events

5

Estimated energy is not true energy!


Mathematical description

!6

Measured distribution of observable

Unknown distribution of the

sought

Migration matrix: response of detector, geometry, atmosphere, magnetic field, Zd,

Az, ...

Measured background distribution of observable

~A = M · ~B +~b



!7

Measured distribution of observable

Unknown distribution of the

sought

Migration matrix: response of detector, geometry, atmosphere, magnetic field, Zd,

Az, ...

~A = M · ~B +~b

Eest



!8

§  Simplification: (Solution methods for are the same as for )

§  System of equations:

§  nA = nB : one solution §  nA < nB : underconstrained case, infinite solutions §  nA > nB : overconstrained case, no exact solutions

~b = 0~A�~b = M · ~B ~A = M · ~B

0

B@A1...

AnA

1

CA =

0

B@M11 · · · M1nB

.... . .

...MnA1 · · · MnAnB

1

CA ·

0

B@B1...

BnB

1

CA

note: b=0 in CombUnfold because we work with excess distribution


Unfolding solutions

!9

§  In principle: §  Exact solutions §  Minimizing specific expressions

Least Squares expression

à  only valid for observables following a gaussian distribution à  not fulfilled for bins with low event numbers

Likelihood expression à  takes also Poisson statistics in account

L0(~a) =X

i

(gi(~a)� gi,m · ln gi(~a))!= min

with p(gi,m; gi): Poisson distr., gi,m: number of measured events in bin ~a: minimization parameter

�20 =

⇣~A�M · ~B

⌘T· V [ ~A]�1 ·

⇣~A�M · ~B

⌘!= min.


Forward (un)folding

!10

§  Assumption of parametric function for with parameters §  e.g. for power-law distribution:

§  Find best match between observed data and folded with §  Minimize (deviation between and with current )

§  Continous provides smooth solution §  No regularization needed

§  Not independent from model §  Not sensitive for distinct features

§  No real unfolding, use as cross-check!

f(x,~a)

f(x, a1, a2) = a1 · xa2

~a = (a1, a2, ..., an)

f(x) MM · f(x) ~a

~B


Example of where it will fail

!11

if you don’t know there is a line you will not find it in the forward folding method


Regularization

!12

§  Unfolding is ill-posed problem §  Strong fluctuations in solution §  Large deviations from true distribution

§  Reasons for fluctuations: §  Small and insignificant entries in matrix §  Amplification during unfolding §  Measurement inaccuracy §  Solution is dominated by noise

§  How to suppress these fluctuations? à Regularization!

§  Regularization = Requirement to

solution §  e.g. smoothness

§  Too strong: Loss of important

information §  Too weak: Too many unphysical

fluctuations

BUT: Regularization is unavoidable and has to be controlled!!!

Motivation: Explanation:


Regularization

!13

Too small

Too strong

Balanced

Fluctuations are suppressed


Regularization• Two additional terms:

• w is the weight of the regularization

• Reg is a measure of smoothness of B (true distribution)• Solution for B is obtained by minimizing:

• w is regularization parameter:

• Large w: no regularization; strong fluctuations

• Small w: strong regularization; smooth result

14


Regularization

!15

~p~B

~B

~BReg( ~B)

Reg( ~B) =X

j

pj · ln✓pj✏j

◆

Different choices of regularization applied to :

§  Tikhonov: Regularization term as square of an operator applied to à by using second derivative the curvature of is constrained

§  Schmelling: Regularizaiton term as minimum cross entropy à representing the deviation from normalized denoted as to normalized prior knowledge about ~✏

Reg( ~B) =X

j

✓d2B

dx2

◆2

j

=X

j

✓2 ·

✓Bj+1 �Bj

Bj+1 +Bj� Bj �Bj�1

Bj +Bj�1

◆◆2


Regularization

!16


Regularization in CombUnfold

!17


Regularization

18

Too small

Too strong

Balanced

(Fluctuations are suppressed à uncertainties become smaller) V [ ~B] =

✓I+

�2

w

◆�2

too strong regularization results in too small error bars


Regularization

19

real

exam

ple

from

Com

bUnf

old

Daniel Mazin MAGIC software school, July 2016, Kochel am See 20

HANDS-ON

CombUnfold.C


Combining spectra

• CombUnfold.C was designed to join several spectra and unfold them at once. Even if you have just a single flute output file, use this macro!

• Combining is done by proper averaging of collection areas and observation time

• Energy binnings have to be the same in different samples (rerun flute if needed)

21


Correlated fit• Due to unfolding, bins in true energy are

correlated

• Chi2 fit is more complicated: the correlations have to be taken into account using the covariance (error) matrix of the unfolding result

• Once the energy spectrum is unfolded, a correlated fit is performed inside CombUnfold.

• The results of the correlated fit should be quoted, not the simple fits to the unfolded spectra

22


program

• combunfold.rc (adjust to your data!, can rename it)

• Output of flute (Output_flute.root) is the input file for the unfolding

• CombUnfold.C (no changes needed)

• Enter root and execute:

• .x CombUnfold.C(“./combunfold.rc”)

23


Cookbook for CombUnfold

1. Adjust input file (output from flute, typically Output_flute.root) to MCombineDataForUnfolding.InputFiles

2. Run forward folding with

1.Automatic ranges in Eest and in Etrue

2.Power law function

3. Change Eest and Etrue to manual ranges and adjust the Eest and Etrue ranges (bin numbers)

4. Run forward folding with power law, check histograms

5. Eventually change the function to log parabola or broken power law

6. Run all 6 unfolding algorithms with the same setting as by step 5:check histograms for sanity, compare χ2 results and fit parameters

7. Eventually you need to adjust the regularization weight by hand

8. Optionally you can adjust the fitting range in energy 24

important slide!steps:


combunfold.rc

25

MCombineDataForUnfolding.NumFiles: 1

Number of files to be combined and unfolded:

Names of files to be combined and unfolded. These are flute output files:

MCombineDataForUnfolding.InputFiles[0]: /nfs/mydir/Output_flute.root

Number of iterations:# ------------------------------------------------------------------------- # Use this to set the number of spectrum iterations # ------------------------------------------------------------------------- MCombineDataForUnfolding.NSpectrumIterations: 1

step 1

keep here 1. it means initial unfolding to determine the tentative spectrum and 1 iteration using the tentative spectrum


combunfold.rc

# ----------------------- # FlagUnfold = 1 Schmelling (minimization by Gauss-Newton

method) # 2 Tikhonov (minimization by MINUIT) # 3 Bertero (appropriate for under-constrained case) # 4 Forward unfolding # 5 Schmelling (minimization by MINUIT) # 6 BerteroW (approproate for over-constrained case) MCallUnfold.FlagUnfold: 4

26

Choose the unfolding method!step 2


combunfold.rc

# If bin range for Eest should be selected automatically # set RangeAutoSelectA to 1 MCallUnfold.RangeAutoSelectA: 1

# If bin range for Etrue should be selected automatically # set RangeAutoSelectB to 1 MCallUnfold.RangeAutoSelectB: 1

27

Run first with forward folding and automatic range selectionstep 2


Adjust the function you fit the unfolded spectrum with

28

# Type of function to be fitted to the fluxes # or to be used in the Forward unfolding # F1Type = 1 power law (Npar = 3) # 2 for power law with cutoff (Npar = 4) # 3 for power law with VARIABLE POWER INDEX (Npar = 5) # 4 for power law with VARIABLE POWER INDEX and cutoff # (Npar = 5) # 5 for broken power law (Npar = 6) # 6 for broken power law # with variable alpha1 (Npar = 7) # 7 for broken power law # with a cutoff (Npar = 7) # 8 for broken power law with variable # alpha1 and cutoff (Npar = 8) MCallUnfold.F1Type: 1

step 2 (and 5)

Daniel Mazin MAGIC software school, July 2016, Kochel am See 29

remove comment markers at the function of your choice and check the comment markers for the other functions

MCallUnfold.F1Type: 3

# Optional : for the parameters of the function, # give starting values, initial step sizes, lower and upper limits # and indicate by fix=1 whether a parameter should be fixed # these are values for Type 1 # f0 alpha r MCallUnfold.Npar: 3 MCallUnfold.ParamVinit: 0.4e-10 -2.0 0.25 MCallUnfold.ParamStep: 1.e-12 0.2 0.0 MCallUnfold.ParamLimlo: 1.e-15 -10.0 0.0 MCallUnfold.ParamLimup: 1.e-7 10.0 0.0 MCallUnfold.ParamFix: 0 0 1

Adjust the function you fit the unfolded spectrum withstep 2 (and 5)


combunfold.rc

# If bin range for Eest should be selected automatically # set RangeAutoSelectA to 1 MCallUnfold.RangeAutoSelectA: 0

# If bin range for Etrue should be selected automatically # set RangeAutoSelectB to 1 MCallUnfold.RangeAutoSelectB: 0

30

change the energy range to manualstep 3


combunfold.rc

# ----------------------- # Bin range of Eest to be used in the unfolding # (unless selected automatically) MCallUnfold.nminAnmaxA: 7 25

# Bin range of Etrue to be used in the unfolding # (unless selected automatically) MCallUnfold.nminBnmaxB: 7 17

31

• Choose the ranges of Eest and Etrue • Needs to be adjusted!

Select bin ranges from the migration matrix plot in the OrigPlot tab:

step 3


Select bins in Eest and Etrue

32

step 3

• Eest: only bins containing more than ~10 events, no gaps

• Etrue: bins with acceptance of > 70%

• Etrue: bins with Aeff > 1000m2


combunfold.rc

33

1. place curser over the bin

2. read the bin numbers in the status display

step 3


Output Tabs• There 5 tabs for 2 iterations each (0th and 1st iterations). The sequence of

the plots is almost random

• Tab 1: Iter plots (goodness results as a function of the regularization weight)

• Tab 2: Input plots (input distribution, migration matrix etc in chosen Eest and Etrue ranges)

• Tab 3: Result Plots (distribution before and after unfolding, chi2 contributions of individual bins)

• Tab 4: Orig Plots (measured distribution, entire migration matrix, acceptance)

• Tab 5: Correlated Fit (fit with chosen function to the unfolded distribution, or the best spectral form for the forward folding)

34

step 4


Tab Iter plots

35

more regularization less regularization

step 4

• check smoothness • don’t select singularities • don’t select extremes


Tab Input plots

36

step 4

• check smoothness • check agreement


Tab result plots

37

step 4

• check agreement is reasonable • check chi2 contributions

open symbols: excess in Eest green markers: unfolded excess in Etrue you expect them to be different because of unfolding!


Tab Orig plots

38

step 4


Tab Orig plots

39

step 4

• choose the range • choose the range • check the effective area • check the acceptance of

Etrue bins


Tab Correlated fit

40

step 4

• resulting fit parameters • check chi2 contributions

(not for forward folding)

here the result is not so good!


Proper choice of w• Variety of choices for regularization

terms (Reg)

• Following criteria are used to choose w:

• χ02 is not too high

• Noise component of the unfolded distribution (Trace (T)) should not be much higher that the noise component of the measurements (Trace (K))

• User can actually choose his own preferred weight w

41

step 7


combunfold.rc

42

# Criterion for choosing the best weight # FlagCriterion = 1 biggest increase of trace(Sigma), # 2 LSQ solution (very large weight, many iterations) # 3 chi2 = # of sign. measurements, # 4 chi2 = rank of G, # 5 Trace(Sigma)/Trace(C) = 1, # 6 D2bar minimal (only MC) # 7 # 8 choose the weight given by the bin IterBin MCallUnfold.FlagCriterion: 5

Default condition: noise in data equal to noise after unfolding

step 7


combunfold.rc

43

# Criterion for choosing the best weight # FlagCriterion = 1 biggest increase of trace(Sigma), # 2 LSQ solution (very large weight, many iterations) # 3 chi2 = # of sign. measurements, # 4 chi2 = rank of G, # 5 Trace(Sigma)/Trace(C) = 1, # 6 D2bar minimal (only MC) # 7 # 8 choose the weight given by the bin IterBin MCallUnfold.FlagCriterion: 8

# if FlagCriterion = 8 the weight is taken corresponding to bin IterBin MCallUnfold.IterBin: 9

In case there are reasons to change these conditions

step 7


Range of the fit

• # ----------------------- • # In MCorrelatedFit perform the correlated fit only in the range • # specified here by the user (GeV) • MCallUnfold.FitMinUser: 70 • MCallUnfold.FitMaxUser: 10000

44

step 8• Typically you don’t need to do that, the entire Etrue

range from the unfolding range is used for the fit. • However, you might want to exclude some bins from

the fit manually. • In the latter case, at the very end of combunfold.rc you

can specify the range where the fit should be performed


Output fileroot [0] TFile *f = new TFile("Unfolding_Output_combunfold.root")root [1] f->ls()

TFile** Unfolding_Output_combunfold.root TFile* Unfolding_Output_combunfold.root

KEY: TH1D CorrelFitResult;1 dN/dE for correlated fit KEY: TGraphAsymmErrors fGraph1;1 Gamma-ray flux after correlated fit (take these points for publication)

KEY: TGraphAsymmErrors fGraph2;1 Unused flux points after correlated fit KEY: TH1D CorrelFitResultE2;1 E2 * dN/dE

KEY: TGraphAsymmErrors fGraph1E2;1 SED (E2*dN/dE) after correlated fit KEY: TGraphAsymmErrors fGraph2E2;1 Unused Energy flux points (E2*dN/dE) after correlated fit

KEY: TF1 Func;1 KEY: TF1 FuncE2;1

KEY: TH1D ExcessEnergy;1 Original Measured Distributionroot [2]

45


Output file

• In the ROOT output file (e.g. Unfolding_Output_combunfold.root) you can find:

• TGraphAsymmErrors *fGraph1 differential spectrum

• TGraphAsymmErrors *fGraph1E2 SED

• TF1 *Func fit to diff spectrum

• TF1 *FuncE2 fit to SED

46


For publications

• Spectral parameters are typically taken from Forward Folding

• For spectral points we typically take Bertero (method #3) or Tikhonov (method #2)

47

AGN

e+e-

UV/O/IR Photons

Stars and Dustin Galaxies

HE/VHE ϒ-Rays

Nikishov (1962), Jelley (1966), Gould & Schreder (1966)

dN/d

E

Energy

intrinsic

dN/d

E

Energy

measured

E�EEBL ⇡ 4(mec2)2 ⇡ 1MeV2

EEBL ⇠ eV ! E� ⇠ TeV

slide from M Raue

TeV constraints

from M. Meyer

attenuation

EBL correction

0.1 1 10 100 1000� [µm]

1

10

100

�I �

[nW

m�

2sr

�1 ]

this work - Dominguez+ 10Kneiske+ 04 - best fitFranceschini+ 08Kneiske & Dole 10Gilmore+ 10Aharonian+ 06Mazin & Raue 07 - realisticMazin & Raue 07 - extremeAlbert+ 08Schlegel+ 98Hauser+ 98Finkbeiner+ 00Lagache+ 00Gardner+ 00Gorjian+ 00Cambresy+ 01Madau & Pozzetti 01Metcalfe+ 03Chary+ 04Fazio+ 04; Franceschini+ 08Xu+ 05Matsumoto+ 05Frayer+ 06Bernstein+ 07Levenson & Wright 08Matsuura+ 10Hopwood+ 10Bethermin+ 10Berta+ 10Keenan+ 10

Dominguez et al, MNRAS, 410, 2556 (2010)


EBL correction# -------------------------------------------------------------------------

# Use this to define the file containing the attenuation factors for the

# absorption due to the extragalactic photon background

# if no file is specified no absorption correction will be applied

# -------------------------------------------------------------------------

MCallUnfold.AttFactorFile: exptau_z0.212_modelFranceschini.dat

51

Memo: 1. no file (commented out) means no EBL correction! 2. make sure the file has 50 entries 3. make sure your attend file covers the needed energy range! 4. make sure you use a correct attend file (ask me or Abelardo)


EBL correction

52


Tips• Run with all unfolding methods

• Check if the result is robust (means the resulting fit parameters are stable) by including / excluding few bins in estimated / true energy (each side), but keep the bin validity criteria in mind

• WIKI: http://magic.pic.es/priv/wiki/index.php/MAGIC_software:Unfolding

• WIKI: http://magic.pic.es/priv/wiki/index.php/CombUnfold

• Don’t hesitate to ask questions: [email protected] [email protected]

53

http://magic.pic.es/priv/wiki/index.php/MAGIC_software:Unfolding




Cookbook for CombUnfold

1. Adjust input file (output from flute, typically Output_flute.root) to MCombineDataForUnfolding.InputFiles

2. Run forward folding with

1.Automatic ranges in Eest and in Etrue

2.Power law function

3. Change Eest and Etrue to manual ranges and adjust the Eest and Etrue ranges (bin numbers)

4. Run forward folding with power law, check histograms

5. Eventually change the function to log parabola or broken power law

6. Run all 6 unfolding algorithms with the same setting as by step 5:check histograms for sanity, compare χ2 results and fit parameters

7. Eventually you need to adjust the regularization weight by hand

8. Optionally you can adjust the fitting range in energy 54

important slide!steps:


References

• MAGIC-TDAS 06-01, Wittek and Aliu: “The Unfolding Program in the Standard Analysis Chain”

• “Unfolding of differential energy spectra in the MAGIC experiment” J.Albert et al., NIM A 583 (2007) 494-506

• “Solving inverse problems with the unfolding program TRUEE: example in astroparticle physics”, Milke et al., NIM A 697 (2013) 133-147

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unfolding of energy spectra: combunfold · daniel mazin magic software school, july 2016, kochel am...

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