Unfolding Methods. Alberto Rotondi Università di Pavia

Transcription

1 Unfolding Methods Alberto Rotondi Università di Pavia 1

2 Folding is a common process in physics signal Apparatus response Observed signal x f ( x) ( x, y) g( y) f ( x) ( x, y) dx 2

3 Convolution is a linear folding signal Apparatus response Observed signal x z y x y z x f ( x) ( y x) g( y) f ( x) ( y x)dx 3

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5 Folding theorem The jacobian is auxiliary variable From the general theorem one obtains by integrating on the auxiliary variable hence which is the probability density 5

6 For independent variables: when Z is given by the sum Convolution theorem we have and we obtain When X 1 and X 2 are independent, we obtain the convolution integral In physics ( instrument function, f signal) 6

7 When Uniform*Gaussian where one has immediately 7

8 1D Unfolding values = R* + 8

9 2D Unfolding 9

10 Fourier Techniques 10

11 Image Deconvolution 11

12 original Gaussian smearing Poisson statistics Fourier (un)restored 12

13 original Poisson statistics Gaussian smearing Fourier restored 13

14 original Gaussian smearing Poisson statistics Fourier (un)restored 14

15 original Poisson statistics Gaussian smearing Fourier restored 15

16 original Gaussian smearing Poisson statistics Fourier (un)restored 16

17 PET: La PET: positron tomografia emission thomography a positroni 17

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19 Image restoration 19

20 Explanation: The smeared distributions of two input distributions cannot be distinguished if they agree on a large scale of x but differ by oscillations on a microscopic scale much smaller than the experimental resolution or to increase the DoF by using a parametric model P( ) P( ) P( 20 ')

21 The frequentist assumes P( ) 1 21

22 Regularization terms 22

23 What is MaxEnt??? 23

24 The iterative principle (26) ) 24

25 The iterative Principle without best fit Good! 25

26 The iterative Principle without best fit + smoothing 26

27 The iterative Principle without best fit Bad! 27

28 The iterative algorithm + best fit 28

29 The iterative algorithm + Best fit 29

30 The iterative algorithm + best fit + regularization 30

31 The iterative algorithm + best fit + MaxEnt regularization 31

32 The iterative algorithm + best fit + Tichonov regularization 32

33 The iterative algorithm + best fit + Tichonov regularization 33

34 The iterative algorithm + best fit + Tichonov regularization 34

35 ATHENA experimental set-up ATHENA apparatus Silicon micro strips CsI crystals 511 kev Charged tracks to reconstruct antiproton annihilation vertex. Identify 511 kev photons from e + -e - annihilations. Identify space and time coincidence of the two with ± 5 mm and 5 s resolution (Probability of a random coincidence: 0.6% per pbar annihilation without considering detection efficiency) 511 kev 35

36 From the ATHENA detector Pbar-only (with electrons) y cm 36 x

37 antihydrogen!!!!!!!!!! FIRST COLD ANTIHYDROGEN PRODUCTION & DETECTION (2002) M. Amoretti et al., Nature 419 (2002) 456 M. Amoretti et al., Phys. Lett. B 578 (2004) 23 SIGNAL ANALYSIS: opening angle xy vertex distribution radial vertex distribution 65 % +/- 10% of annihilations are due to antihydrogen between 2002 & 2004 more than 2 millions antihydrogen atoms have been produced that s about 2 x mg.. or Giga years for a gram ; ;

38 Hbar (MC) Annihilation vertex in the trap x-y plane BCKG (HotMixData) Cold Mix data Pbar vertex XY projection (cm) ML Fit Result x Hbar + (1-x) BCKG = Cold Mix Hbar percentage x = 0.65 ±

39 Cold Mix data Iteratve best fit method exp background Cold Mix The vertex algorithm resolution function is gaussian with 3 mm The 2D deconvolution reveals two different annihilation modes 39

40 The iterative algorithms + best fit + regularization iterative algorithms are used in unfolding (ill posed) problems they need a Bayesian regularization term when there are degrees of freedom, one can use a best fit of a signal+background function to the data in this case there are no Bayesian terms (pure frequentist approach) 40

41 end 41

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43 Regularization parameter 43

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46 Conclusions best fit minimization methods are crucial in physics. They are mainly frequentist they are based on the ML and LS algorithms (they are implemented in the ROOT-MINUIT framework) to judge the quality of the result, frequentists use the test bayesians use the hypothesis probability Bayesian a priori hypotheses should be used with 46 informative priors!!!

47 Image restoration 47

48 2D Unfolding 48

49 The iterative principle 49

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51 Unfolding techniques Statistica III 51