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ЕРЕВАНСКИЙ ФИЗИЧЕСКИЙ ИНСТИТУТ Арарат А. Варданян

ПРИМЕНЕНИЕ ИСКУССТВЕННЫХ НЕЙРОННЫХ СЕТЕЙ ДЛЯ ОБРАБОТКИ ЭКСПЕРИМЕНТАЛЬНЫХ ДАННЫХ ФИЗИКИ КОСМИЧЕСКИХ ЛУЧЕЙ ВЫСОКИХ ЭНЕРГИИ "

АВТОРЕФЕРАТ

диссертации на соискание учёной степени кандидата фязико-математячесХИх наук по специальности 01.04.16 - физика атомного ядра, элементарных частиц я космических лучей

ЕРЕВАН 2000

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Тема диссертации утверждена в Ереванском физическом институте

Научный руководитель: д.ф-м.н., проф. A.A. Чшшнгарян

Официальные оппоненты: д.ф-м.н., проф. Э.А. Мамиджанян

к.ф-м.н. С. С. Остапченко

Ведущая организация: Ереванский государственный университет

Защита диссертации состоится 18 июля 2000г., в 1630 ч. на заседании специализированного совета 024 ЕрФИ (375036 Ереван, ул. Братьев Алиханян 2).

С диссертацией можно ознакомиться в библиотеке ЕрФИ.

Автореферат разослан 18 июня 2000г.

Научный секретарь спец. совета: .ЛЬ^^/к'рЬ ЧА. Маргарян

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General Characteristic of the Work

Motivation

The ambiguity of interpretation of the results of experiments with Cosmic Rays (CR) is connected with both significant gaps in our knowledge of the parameters of hadron-nuclear interactions at superaccelerator energies and indefiniteness of the primary cosmic ray composition, as well as with strong fluctuations of all shower characteristics. The extra difficulties are due to indirect experiments and hence, due to the use of Monte - Carlo simulations of development and detection of different components of nuclear electromagnetic cascades.

To make the conclusions about the investigated physical phenomenon more reliable and significant, it is necessary to further develop a unified theory of statistical inference, based on nonparametric models, in which various statistical approaches (density estimation, Bayesian decision making, error rate estimation, feature extraction, sample control during handling, Neural Net (NN) models, etc...) would be used.

Goals

Development of a unified methodology of the data analysis in framework of Monte Carlo Statistical Inference (MCSI) using advanced nonparametric techniques in order to be able to address reliably the most difficult and most important problem of high energy astroparticle physics data analysis - event by event analysis of CR interactions, determination of the type and energy of Primary CR (PCR) particles for each registered shower. Innovations consist in:

• As compared to the earlier used methods of Extensive Air Showers (EAS) data treatment, in the proposed work the object of analysis is each particular event (a point in the multivariate space of measured parameters) rather than alternative distributions of model and experimental data.

• Implementation of procedures like the Cross-Validation (CV), Ensembles of Networks and Final Prediction Error, in order to obtain a stable and reliable results on general nature of investigated phenomenon using finit simulation samples, rather than the particular (may be spurious) solution of the problem overfitted to the data on hand.

• An introduction of individual event weight in order to compensate the deficiency of more interesting high energy events in CR flux due to steep energetic spectrum. This allowed to make an unbiased estimation of the primary energy in the wide range 5 x 1014 — 1016eV and to decrease the relative error of estimation up to less than 25%.

Scientific and practical value.

Several methods are proposed, developed and implemented in Neural Information technologies to control the learning from examples, to gain the high efficiency

of Neural classification and estimation procedures, to estimate the method performance and to stabilize the obtained results. The handling of EAS simulated data proves, that the proposed methodology allows:

• to determine with ~ 70% efficiency the type of the primary and to estimate its energy with ~ 25% relative error of the estimation.

• to obtain energetic spectra of three species of CR and to make detailed analysis of the primary CR mass composition in the knee region.

• the possibility of selecting mononuclear CR beams from experimental data and farther comparison of such beams with alternative simulations was investigated. These results give us hope that the study of the parameters of P — A and A —A interactions in the energy range 1-10 PeV is possible.

• developed methods are universal and can be used for high energy physics data analysis as well, as for other practical applications.

Approbation of Work. The main results of dissertation were reported at:

1. 9th International Symposium on Very High Energy Cosmic Ray Interactions, August 19-23, 1996, Karlsruhe, Germany

2. IS471 European Cosmic Ray Symposium, August 26-30, 1996, Perpignan, France

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3. 5th International Workshop on New Computing Techniques for Physics Research, September 2-6, 1996, Lausanne, Switzerland

4. 25th International Cosmic Ray Conference, July 30 - August 6, 1997, Durban, South Africa

5- 2nd International Workshop ANI 98, May 29 - June 5, 1998, Nor-Amberd, Armenia

6. 10"1 International Symposium on Very High Energy Cosmic Ray Interactions July 12-17, 1998, Assergi, Italy

7. 16ih European Cosmic Ray Symposium, July 20-24, 1998, Alcala de Hectares, Spain

8. 3rd International Workshop ANI 99, May 30 - June 4, 1999, Nor-Amberd, Armenia.

9. 26th International Cosmic Ray Conference, August 17-25, 1999, Sail Lake City, Utah, USA.

10. 17iA European Cosmic Ray Symposium, July 24-28, 2000, Lodz, Polan

The structure of the dissertation

The dissertation consists of an introduction, four chapters, conclusion, ap pendixes of acronyms and notations and bibliography. It contains 110 page of a text, 48 figures and 24 tables. The bibliography contains 90 references.

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the so-called Random Search (RS) algorithm, the second one is the deterministic algorithm based on Sobol's pseudo-random numbers. The RS algorithm implements the following steps:

The choice of the learning algorithm depends on many problem dependent factors, like the complexity of the problem, availability of the training set, real-time requirements, the cost of the losses (required quality of the solution). Hence it is not possible to give a unique decision on what strategy to follow. Nevertheless, the goal of each training algorithm is to achieve a good generalization performance relatively easy and fast. The ANI (Analysis and Non-parametric Inference) statistical analysis program package developed in Yerevan Physics Institute contains a special section of artificial neural network models. Several strategies are implemented for the NN training for estimation and classification tasks. The main algorithm for NN weights updating is

1. The initial values of NN weights are chosen randomly from Gaussian distribution with zero mean and small variance {¡i = 0 ; it « 10~2)

2. The random iteration step in multidimensional space of NN weights is performed from the initial point to modify the NN weights. Three submodes of random search are implemented:

• single mode - modifying the single random weight of NN;

• neuron mode - modifying all weights of randomly chosen neuron;

• multi mode - modifying all weights of neural network simultaneously (one of the two submodes of multi mode, the deterministic algorithm based on Sobol's pseudorandom numbers, was used)

The alternation of weights is performed according to following:

Wi = W-i + Q?_j * RNDM * A; i = l,JVjter (11)

were Wi is the vector of NN weights obtained at i-th iteration step, A is step size and RNDM is a random number from:

a) a pool uniformly distributed in [—0.5; 0.5] interval

b) standard Gaussian population

c) Cauchy population

The term introduces dependence of value of random step on the already achieved quality function and controls the degree of this dependence on achieved quality function; p=0,l,2.

The deterministic algorithm works as follows:

1. a multidimensional hypercube with dimension d=Nwcishts and side A is filled in such a way that the projection to any axis gives a uniformly distributed points (number of points in hypercube is equal to the number oi given iterations) - so-called pseudorandom numbers,

2. each "pseudorandom" point in this hypercube represents a vector of the NK weights, the network with such a weights is applied to calculate the qualitj function on a training sample, and the point which provides the smallest value of the quality function is taken as a vector of NN weights.

Different search modes (single, neuron, mullti) give a possibility to perform different training strategies, obtaining quite different points of multidimensional space of neural net weights, and allow to investigate symmetries of nontrivial local minima in this space (due to very large dimensionality of NN weights space there exist many symmetries.

Both algorithms have many free parameters like the number of hidden layers in NN, number of hidden neurons in these layers, objective function to be minimized step size in Random Search algorithm and the size of the hypercube in deterministic algorithm, number of training epochs and number of "pseudorandom" points used in RS and deterministic algorithms respectively.

So, the comparative study of different learning strategies in terms of speec and quality is of great interest. As a performance measure on control data sel the geometrical mean per cent of the right classifications over 3 classes was used

Ri i R22Rz3 = RTrue, where Ra are the diagonal elements of the misclassificatior matrix.

An informative characteristic of the algorithm speed is the number of iterations after which the best point (minima in quality function) in the multidimensiona space of NN weights is found, and then the algorithm is converged (saturatioi of the quality function). Nevertheless, such benchmark could be used when th< Bayes risk is reached within the given number of iterations for all NN to b< compared. The figure 2 shows the classification performance dependence on stej size for different modes. The clear dependence of the different training strategic; on the step size is observed.

What we can conclude is that, in case of small step sizes the Bayes risk ii practically always reached by all three modes. Taking into account that the ver

¡large steps have no principal advantage over small ones, one can recognize that the neuron mode is the best in terms of quality versus speed ratio.

However, the different strategies can be used in one session of training, by the following scenario: the learning can be started using multi mode to scan a very large space of weights, to find a better point and then to switch to the neuron mode and continue the search in space near to the already obtained point with small steps, finally the single mode can be used to tune the NN weights for the given problem solution with high verbosity, of course, if the problem is complex snough and the saturation of the quality function is not available yet (the "global" minimum is not found). The use of the CV procedure for the overtraining effects controlling is assumed to be default here.

In third Chapter the general approach to the CR. experimental data analysis

s demonstrated on the simulated and KASCADE data. The EAS characteristics neasured by KASCADE experiment are explained, The detailed description of ill statistical methods and procedures are given and the application examples are ^resented.

To make the conclusions about the investigated physical phenomenon reliable tad accurate, we use a unified framework of statistical inference, based on non->arametric models, in which various statistical analysis methods (Bayesian deci-ion rules, Neural Networks models, Feature extraction, etc,...) are incorporated ANI program package.

Advocated approach consists in:

• use of the different statistical methods for simulated data validation and real data homogenity checks;

• one-dimensional and multidimensional studies of EAS characteristics based on simulated and experimetal data comparisons;

• correlation analysis for selection of the best subset of features for energy estimation and mass classification;

• comparison of the obtained results using alternative simulation models; the "best" model selection;

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Figure 2: The dependence^0} final performance from learning step size(the horizontal line represents the performance corresponding to Bayes risk)

• event-by-event analysis (primary energy estimation, primary type classification) of experimental showers;

• self-consistency and reliability checks;

• methodical and model errors estimation.

All statistical decisions and procedures are correct within the prechosen model. Thus a realistic simulation is the key problem of any physical inference in indirect experiments. Extensive air shower investigations are a classical example of such a situation. An adequate consideration of detector response and an identical reconstruction of experimental and simulated data are necessary steps of data analysis.

The EAS parameters measured by KASCADE are as shown in Table (1)

Table 1: EAS features detected by KASCADE experiment

Ne Number of electrons in EAS

NfT Truncated number of muons

(number of muons in the range of 40 to 200m)

Ss 9 Shower age associated with a Molier radius 89m

1V?» Number of muons in Central Detector (CD)

JVfc Number of hadrons

Tßmax The energy of most energetic hadrons

Egum Total energy of hadrons

Table 2: The best feature subsets according to the Bhattacharya distance, one-dimensional tests and correlation analysis_

Models 2 best next best worst

QGS Ne Ntr jvy £>89 Nh jgrriax

VENUS iVe Ntr Sm Nou . V Egum Nh jjjmax

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RO-+P -ßo-fo Ro-tfe

Rfe-*p

The above matrix presents accumulate a priori knowledge on the possibility of data classification into 5 categories. The overall index reflecting the "goodness" G of features used is following index of separability

In Conclusion the main results of the dissertation are presented concerning methodical achievements and data analysis results.

The main results of the presented dissertation in developing the MCSI consist

in:

1. development and implemention of the stopping rules to the NN training process in order to avoid the overtraining effects due to the use of the finite simulation samples;

2. implementation of the Cross-Validation procedure for the NN generalization capabilities estimation (Final Prediction Error), for the NN best architecture selection and for control of the training process;

3. introduction of the weights of individual events to compensate the deficiency of high energy events due to the steep falling energy spectra in PCR, and to increase the interpolation capabilities' of NN. This allows to obtain the unbiased estimation of the primary energy in wide energy interval;

4. introduction of the median committee of neural networks in analogy of the adaptive estimate of the probability density to obtain more reliable and stable results on neural clasification and estimation;

5. performing of the comparison of different nonparametric analysis methods, (particularly Bayesian and NN methods were compared), demonstration of the method independence of statistical inference;

6. comparisons of the performances of different training algorithms, the recommendations on usage of different stochastic training procedures were given.

The developed methods were applied for the EAS data analysis. The data from vorld biggest KASCADE experiment were treated on event-by-event basis.

The main physical results obtained from KASCADE data analysis are the fol-owing:

1. The classifcation of the PCR into three nuclear groups is performed, the dependences of the relative abundances of different nuclear groups on the primary energy is obtained;

• the accuracy of the classification into light, intermediate and heavy groups of nuclei is - for light group about 80%, for intermediate group about 60% and for heavy group about 75%,

• the reconstructed relative abundances in the "knee" region (2—4x 1015 eV) are as follows: light group {H,He) « 0.6 ± 0.02 , intermediate group (C, N,0) a 0.32 dfc 0.06 and heavy group (Si-Fe) « 0.08 ± 0.04, (only methodical errors are presented here, the QGSJet model was used),

• the abundance of the light group of nuclei is increasing up to the "knee" energies and above the "knee" is decreasing, while the abundances of intermediate and heavy groups of nuclei are decreasing in the energy range bellow the "knee" and increasing above the "knee" energies.

2. The estimation of energy of the PCR in the range of 1014 - 1016eV is performed and for the first time the energy spectra of three species of PCR are obtained;

• the relative error of the energy estimation is less than 25%,

• the "knee" feature is observed in all-particle and light group of nuclei spectra,

• the obtained slopes and "knee" positions for the diferential spectra of

three species of PCR and for all-particle spectra are the following:

7i 72 Eknee(xl0laeV)

all — part. -2.67 ± 0.008 -3.21 ±0.010 2.95 ±0.13

light -2.39 ± 0.009 -3.35 ±0.038 2.71 ± 0.26

intermed. -2.86 ± 0.016 -3.02 ± 0.083 3.69 ± 1.25

heavy -3.28 ± 0.037 -2.52 ± 0.126 4.48 ± 1.14

3. Using developed NN methodology the possibility to obtain pure CR beams and to make experiments with such beams was demonstrated;

• achieved purity of the light and heavy groups of nuclei is more than 90% while the efficiency of the selection is kept above 50%,

• characteristics of hadrons and muons measured by the KASCADE central detector were compared with alternative ones in QGS Jet model for wide primary energy range.

The main results of the thesis aie published in the references:

1. Chilingarian A. A., Roth M. and Vardanyan A. A. for the KASCADE collaboration, Towards Target-type Experiments with mononuclear Cosmic ray Beams, in Proc. of the Workshop ANI 99, vol. FZK Internal report 6472, p. 11, Nor-Amberd, 1999.

2. Chilingarian A. A., Roth M. for the KASCADE collaboration and Vardanyan A. A., Nonparametric methods for determining energy and elemental composition of cosmic rays from EAS observables, in 16th European CR Symposium, p. 571, Alcala de Henares, Spain, 1998.

3. Chilingarian A. A., Roth M., Vardanyan A.A. for KASCADE collaboration,, A nonparametric approach for determination of energy spectrum and mass composition of cosmic rays from BAS observables, Nuclear Phys. B, T5A, 302, 1999.

4. Chiiingarian A. A., Ter-Aiitonyan S., Vardaiiyan A. A., et al., On the accuracy of the elemental composition determination on the mountain altitudes and sea level, Nuclear Phys. B, 52B, 240, 1997.

5. Chiiingarian A. A., Ter-Antonyan S., Vardanyan A. A., et al., On the nonparametric classification and regression methods for the multivariate EAS data analysis, Nuclear Phys. B, 52B, 237, 1997.

6. Chiiingarian A. A., Ter-Antonyan S., Vardanyan A. A., et al., Energy Spectra and Elemental Composition Determination on Mountain Altitude and Sea Level, Nuclear Phys. B, 60B, 117, 1998.

7. Roth M. for the KASCADE collaboration, Ter-Antonyan S., Vardanyan A. A., How to infer the primary energy spectrum from EAS observations demonstrated with KASCADE data, in Proc. 25th ICRG, vol. 4, p. 157, Durban, 1997.

8. Vardanyan A. A., Chiiingarian A. A. and Roth M. for the KASCADE collaboration, On the Possibility of Selecting Pure Nuclear Beams from Measurements of the KASCADE Experiment, in Proc. of the Workshop ANI 99, vol. FZK Internal report 6472, p. 23, Nor-Amberd, 1999.

9. Vardanyan A. A., Chiiingarian A. A., Ter-Antonyan S., The comparison of Bayesian and Neural techniques in problem of classification to multiple categories, NIM, A 389, 230, 1997.

10. Vardanyan A. A., Rx>th M., Chiiingarian A. A., KASCADE collaboration, Nonparametric determination of energy and elemental composition of cosmic rays from EAS observables, in Proc. ANI 98 Workshop, vol. FZK Internal report 6215, p. 19, Nor-Amberd, 1998.

11. Vardanyan A. A., Zazyan M., Ter-Antonyan S., Chiiingarian A. A., The reconstruction of the energy spectra and primary composition by EAS multivariate analysis, Izv. AN, 61(3), 1997.

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АННОТАЦИЯ

Целью настоящей работы является дальнейшее развитие единой методологии анализа данных для надежных выводов в наиболее трудной и одной из самых важных проблем астрофизики высоких энергий: определения типа и энергии частиц первичного космического излучения для каждого индивидуально зарегистрированного события.

В отличии от ранее используемых методов обработки Широких Атмосферных Ливней (ШАЛ), основанных на сравнении одномерных и двумерных модельных и экспериментальных распределений, в настоящей работе объектом анализа является каждое индивидуальное событие (точка в многомерном пространстве параметров ШАЛ).

Применение процедур, таких как перекрестная проверка, комитет (ансамбль) нейронных сетей и оценивание ошибки заключительного предсказания, позволило получить стабильные и надежные выводы. Разработанные методики позволяют исключить получение частных (может быть случайных), решений проблемы в случае использования ограниченных выборок.

Введете индивидуальных весов каждого события позволило получать несмещенные оценки первичной энергии в широком интервале 5 х 1014 — 1016 эВ и уменьшить относительные ошибки оценки до ~ 25%.

Развитые в диссертационной работе методы позволили:

а) определить тип первичной частицы с 70% эффективностью и оценить ее

энергию с точностью до ~ 25%;

б) впервые получить энергетические спектры трех групп космических лучей

и производить детальный анализ массового состава космических лучей

в области "излома".

в) впервые получить "пучки" ядер космического излучения, обогащенных

протонами и ядрами железа.

Развитые методы универсальны и могут быть использованы при анализе данных физики высоких энергий, а также для решения других актуальных проблем.