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UDC 004.93'1:519.237:51-72
Methods of e/w Identification with the Transition Radiation Detector in the CBM Experiment
E. P. Akishina, T. P. Akishina, O. Yu. Derenovskaya, V. V. Ivanov
Laboratory of Information Technologies Joint Institute for Nuclear Research Joliot-Curie 6, 141980 Dubna, Moscow region, Russia
A problem of e/w identification using n-layered transition radiation detector (TRD) in the CBM experiment is considered. With this aim, we elaborated algorithms and implemented various approaches. We discuss the characteristic properties of the energy losses by electrons and pions in the TRD layers and special features of applying artificial neural networks (ANN) and statistical methods to the problem under consideration. A comparative analysis is performed on the power of the statistical criteria and ANN.
Key words and phrases: general statistical methods, multivariate analysis, pattern recognition, CBM experiment, transition radiation detector TRD.
1. Introduction
The CBM Collaboration [1,2] builds a dedicated heavy-ion experiment to investigate the properties of highly compressed baryon matter as it is produced in nucleus-nucleus collisions at the Facility for Antiproton and Ion Research (FAIR) in Darmstadt, Germany. Fig. 1 depicts a general layout of the CBM experiment. There is a Silicon Tracking System (STS) inside the dipole magnet. Ring Imaging Cherenkov (RICH) is designed to detect electrons. TRD arrays identify electrons with momentum above 1 GeV/c. TOF provides time-of-flight measurements needed for hadrons identification. ECAL measures electrons, photons and muons.
Figure 1. Schematic view of the CBM experimental setup
The measurement of charmonium is one of the key goals of the CBM experiment. For detecting J/-0 meson in its dielectron decay channel the main task is the e/w separation. One of the most effective detectors for solving this problem is the Transition Radiation Detector.
The problem of particle identification (PID) using n-layered TRD consists in the following: having a set of energy losses measured in n layers of the TRD, one has to estimate to which particle, e or this set is relative. To estimate the efficiency of particle identification, we used different approaches.
Received 28th November, 2009.
2. Traditional Statistical Criteria: MV and LFR
Methods
__n
In the mean value (MV) method the PID is based on a variable AE = -1- £ AEi
n i=l
(where AEi is a particle energy loss in the z-th TRD layer and n is the number of layers in the TRD). Fig. 2 shows distributions of variable AE for n (a), e (b), and a summary distribution for e and n (c).
Figure 2. Distributions of variable AE for % (a) and for e (b) events; the summary
distribution for % and e events (c)
While applying the likelihood functions ratio (LFR) test [3] to the PID problem, the value
n n
L = L = Pe/(P„ + Pe) Pe = n Pe(AE,), P^ = n P* (AEi),
i=i i=i
is calculated for each event (see Fig. 3), where (AEi) is the value of the density function p-x in the case when the pion loses energy AEi in the i-th absorber, and pe(AEi) is a similar value for electron.
We have found that the distribution of ionization losses of pions in the TRD is well approximated by a log-normal density function (see Fig. 4)
fi(x) = exp-(ln, (1)
where a is the dispersion, jj, is the mean value, and A is a normalizing factor.
0 0.25 0.5 0.75 1 0 0.25 0.5 0.75 1
(a) (b)
10
10-
10-
ID
78
0.2
0.4
0.6
0.8
(c)
Figure 3. Distributions of variable L for % (a) and for e (b); the summary distribution
for % and e events (c)
The distribution of energy losses by electrons in the TRD is approximated with a good accuracy by the density function of a weighted sum of two log-normal distributions (see Fig. 5)
, ,, D/ a - ^ (in x-m)2 ¡3 - ^ (in N
f2(x) = B ^-exp + ^-exp ^ , (2)
1x 2x J
where a1 and a2 are dispersions, and are mean values, a and fi are contributions of the first and second log-normal distributions, correspondingly, and B is a normalizing factor.
3. Nonparametric Goodness-of-Fit -criterion [4, 5]
This test is based on the comparison of the distribution function F(x) corresponding to the preassigned hypothesis (H0) with empirical distribution function Sn(x):
{0, if x < x1; i/n, if Xi < X < Xi+1, i = 1,...,n — 1. 1, if Xn < X,
Here x1 < x2 < ... < xn is the ordered sample (variational series) of size n constructed on the basis of observations of variable x.
The test statistics measures the "distance" between F(x) and Sn(x). Such statistics are known as non-parametric. We suggested a new class of non-parametric statistics
Figure 4. Distribution of pion energy losses in one layer of the TRD and its approximation by a log-normal function (1)
Figure 5. Distribution of electron energy losses in one layer of the TRD and its approximation by a weighted sum of two log-normal functions (2)
(with k > 3):
k n 2
= -
k + 1
i- 1
n
- F(Xi)
fc+1 i
- - - F (xi)
n
k+1 '
The goodness-of-fit criteria constructed on the basis of these statistics are usually applied for testing the correspondence of each sample to the distribution known apriori.
Energy losses for n have a form of Landau distribution. We use it as H0 to transform the initial measurements to a set of a new variable A:
A¿ =
AEj - AE',
Hi
^ - 0.225, i = 1, 2,...
(3)
AEi loss,
the energy loss in the i-th absorber,
i
AE:
mp
the value of most probable energy
& = 4"02 FWHM of distribution of energy losses for ■k.
In order to determine AE^ and FWHM of the distribution of pion energy losses in the i-th absorber, the indicated distribution was approximated by the density function of a log-normal distribution (see Fig. 4).
The obtained Xi, i = 1,... ,n are ordered due to their values (Xj, j = 1,... ,n) and used for calculation of :
k
n 2 k + 1
n ( E
3 = 1 I
j - 1
- )
fc+i
n - )
fc+1 '
(4)
Here the values of Landau distribution function <^(X) are calculated using the DSTLAN function (from the CERNLIB library). Fig. 6 shows the distributions n (a) and e (b); the summary distribution is shown in the (c).
4. Combined Method
This approach is based on a successive application of two statistical criteria: 1) the mean value method, and 2) the test.
The main idea of this scheme consists in the following:
— using the mean value method, we collect in the admissible region the main part of electrons together with a small admixture of pions,
- then we apply the test to the events selected in the admissible region; this way, we loose a small part of electrons together with additional suppression of pions.
5. Modified ukn Test
Fig. 7 shows the dE/dx distribution of e, and Fig. 8 presents the distribution of electron energy losses on the transition radiation (TR).
Fig. 9 shows the probability of events with a different number of TR counts. We see that the most probable value of TR counts in the TRD with 12 layers is 6, and we almost do not have the events with TR counts in all 12 layers.
Let us turn back to the distribution of electron energy losses on the transition radiation (see Fig. 8). Fig. 8 shows that when e passes the i-th layer with TR=0, then its energy loss follows the distribution of dE/dx losses (Fig. 7). In this case, it is practically impossible to distinguish electrons from pions on the basis of their energy losses. In the opposite case, when we have the TR count in the i-th layer, the electron energy loss will correspond to the sum of dE/dx + TR. Only such counts in TRD layers may permit us to distinguish electrons from pions.
When calculating , in formula (4), one uses a sample of Xi values (see Eq. 3), which are ordered due to their values. The Xi value is directly proportional to the energy loss by a particle registered in the i-th layer of the TRD. In this connection and taking into account that the most probable value of TR counts in the TRD with 12 layers is 6 (Fig. 9), we may use in the test only that part of {A^} sample which corresponds to indexes i > 6, i.e. to large values of particle energy losses.
ID 11
Entries 55900
Mean 4.498
RMS 4.215
xVndf1044. / 87
P1 0.6810± 0.2680E—02
P2 1.230 ± 0.2921 E-02
P3 0.2730E4-05 ± 116.9
dEdxfore with p = 1.5GeV/c
Figure 7. Distribution of electron energy losses on the ionization and its approximation by a log-normal function
ID 211
Entries 55900
Mean 5.335
RMS 6.567
0 5 10 15 20 25 30 35 40 45 50 TR 'or e ir A1
Figure 8. Distribution of electron energy losses on the transition radiation
300 55900 5.990
1.882_
/ 10 £E+05 ± 62.67
5.985 ± 0.8147E—02 1.849 ± 0.5603E—02
Figure 9. Distribution of events with different number of TR counts and its approximation by Gaussian distribution
Fig. 10 shows the distributions of values for GEANT samples: i = 7, n Nlayers — i + 1, k = 6.
6. Method Based on Artificial Neural Network
(ANN)
In [6] we applied a three-layered perceptron from the packages JETNET3.0 and ROOT to estimate the efficiency of PID. Initially the training patterns were formed using the set of energy losses AEi, i = 1,... ,n corresponding to the passage through the TRD pions or electrons.
In spite of the fact that the distribution of the energy losses by electrons significantly differs from the character of the distribution of the energy losses by pions, for such a choice of input data the training process was going on very slow (see bottom
Figure 10. Distributions of modified ^ values calculated for n (a) and for e (b) events;
the summary distribution (c)
curve in 11), there were large fluctuations (against the trend) of the efficiency of events identification by the network. Moreover, one cannot reach the needed level of pions suppression.
o it= 0.8 LLI
0.7 0.6 0.5 0.4 0.3 0.2 0.1
50 100 150 200 250 300 350
N epoch
Figure 11. The efficiency of pion/electron identification by the MLP for original (bottom curve) and transformed (top curve) samples
Aiming to improve the situation, we decided to apply as input sets the sets of variable A (3): see top curve in Fig. 11. Fig. 12 shows distributions of the MLP output signals obtained at the training (b) and testing (d) stages; the left plots show the distributions of errors between the target value and the MLP output signal at the training (a) and testing (c) stages.
35000 30000 25000 20000 15000 10000 5000 0
ID 201
Entries 105600
Mean -o.: 32E-03
RMS o.; 93E-02
-0.01 -0.005 0 0.005 0.01
(c)
-1 -0.5 0
(d)
0.5 1
Figure 12. Distributions of errors (a and c); the distributions of the MLP output signals
(b and d)
7. Conclusion
Table 1 shows the results of comparison of the given methods for different particle (pion and electron) momenta.
Table 1
Comparison of the given methods: pion suppression factor for different momenta
p, GeV/c 1 2 3 4 5 7 9 11
LFR 1273 1315 1581 1480 936 861 800 749
MV 15 17 17 16 16 16 16 16
104 96 73 55 44 35 29 25
MV + ukn 284 271 249 242 203 198 157 159
mod 296 621 628 776 650 745 588 537
MV + mod 311 515 518 493 438 606 398 413
root 1219 1400 1112 1446 730 1054 610 882
jetnet 1857 1837 1378 1713 1446 1317 1045 1089
This table demonstrates that the MV method and the w^ criterion do not provide a required level of the pion suppression 100-150). The main cause is that the electron energy losses are not described by a single distribution, but bear a character of a composite hypothesis.
The criteria simple from a practical viewpoint (the modified w^ criterion and the composite criteria based on MV + ш1^ criterion) provide high levels of the pion suppression.
One succeeds in reaching the best pion suppression level using: a) ANN when transmitting from the initial energy losses in the TRD layers to a new variable typical for the ш1^ criterion, and b) LFR method with the energy losses approximated by a lognormal distribution for pions and by a weighted sum of two lognormal distributions for electrons.
References
1. Letter of Intent for the Compressed Baryonic Matter experiment. — http://www. gsi.de/documents/DOC-2004-Jan-116-2.pdf.
2. The CBM Collaboration. CBM Compressed Baryonic Matter Experiment. Technical Status Report: Techrep / GSI. — Darmstadt, 2005. — http://www.gsi.de/ onTEAM/dokumente/public/DOC-2005-Feb-447e.html.
3. Statistical Methods in Experimental Physics / W. T. Eadie, D. Dryard, F. E. James et al. — Amsterdam-London: North-Holland Pub. Comp, 1971.
4. ZrelovP. V.,IvanovV. V. The Relativistic Charged Particles Identification Method based on the Goodness-of-Fit w3-Criterion // Nucl. Instr. and Methods in Phys. Res. — 1991. — Vol. A310. — Pp. 623-630.
5. Electron/Pion Identification in the CBM TRD Applying a Goodness-of-Fit Criterion / E. P. Akishina, T. P. Akishina, V. V. Ivanov et al. // Particles and Nuclei, Letters. — 2008. — Vol. 5, No 2. — Pp. 202-218.
6. Акишина Т. П., Дереновская О. Ю., Иванов В. В. Об идентификации электронов и пионов с помощью многослойного перцептрона в детекторе переходного излучения эксперимента СВМ // Вестник РУДН, Серия «Математика. Информатика. Физика». — 2010. — № 1. — С. 100-109.
УДК 004.93'1:519.237:51-72
Методы идентификации е/ъ с помощью детектора переходного излучения в эксперименте СВМ
Е. П. Акишина, Т. П. Акишина, О. Ю. Дереновская, В. В. Иванов
Лаборатория информационных технологий Объединённый институт ядерных исследований ул. Жолио-Кюри, д.6, Дубна, Московская область, 141980, Россия
Рассмотрена задача идентификации электронов и пионов в детекторе переходного излучения (ТИВ) эксперимента СВМ. С этой целью разработаны разные алгоритмы и подходы. Обсуждаются характерные свойства энергетических потерь электронов и пионов в слоях ТИВ и особенности применения искусственных нейронных сетей (ИНС) и статистических методов для решения рассматриваемой проблемы. Проведен сравнительный анализ мощностей статистических критериев и ИНС.
Ключевые слова: статистические методы анализа данных, многомерные методы анализа данных, методы распознавания образов, эксперимент СВМ, детектор переходного излучения ТИВ.
