ACAT05 May 22 - 27, 2005 DESY, Zeuthen, Germany

ACAT05 May 22 - 27, 2005

DESY, Zeuthen, Germany

ACAT05 May 22 - 27, 2005

DESY, Zeuthen, Germany

Search for the Higgs boson at LHC by using Genetic Algorithms


Mostafa MJAHEDEcole Royale de l’Air, Mathematics and Systems

Dept. Marrakech, Morocco

Mostafa MJAHEDEcole Royale de l’Air, Mathematics and Systems

Dept. Marrakech, Morocco

Introduction

Genetic Algorithms


Optimization of discriminant functions Optimization of Neural weights Hyperplans search Hypersurfaces search

Conclusion



M. Mjahed ACAT 05, DESY, Zeuthen, 26/05/2005 2

IntroductionIntroduction IntroductionIntroduction

IntroductionHiggs production at LHC

IntroductionHiggs production at LHC

Several mechanisms contribute to the production of SM Higgs boson in proton collisions

The dominant mechanism is the gluon fusion process, gg H

Introduction Decay ModesIntroduction Decay Modes

decay into quarks: H bb and H cc leptonic decay: H + -

gluonic decay: H g g decay into virtual W boson pair: H W +W -

Introduction Main discovery modes

Introduction Main discovery modes

MH < 2MZ:• H bb+X• H W+W- ll • H ZZ 4l• H

•The decay channel chosen:

H W+W- e+-, + e- , e+e-, +-.

•Signature:

• Two charged oppositely leptons with large transverse momentum PT

• Two energetic jets in the forward detectors.

• Large missing transverse momentum P’T

•Main background:

• tt production: with tt WbWb l j l j

• QCD W+W- +jets production

• Electroweak WWjj

HW+W-ll (1)


ll , ll: the pseudo-rapidity and the azimuthal angle differences between the two leptons

jj, jj: the pseudo-rapidity and the azimuthal angle differences between the two jets

•Mll , Mjj: the invariant mass of the two leptons and jets,

•Mnm (n,m = 1,2,3) some rapidity weighted transverse momentum

n, m = 1, 2,3, …

i rapidity of the leptons or jets, piT their transverse momentums.

HW+W-ll (2)

eventi

miT

ninm pM .

Main Variables


Genetic Algorithms Genetic Algorithms Genetic Algorithms Genetic Algorithms

Pattern RecognitionPattern Recognition

Measurement

Feature

Decision

Feature Extraction/

Feature Selection

Classification


Pattern Recognition Methods

Pattern Recognition Methods

Fuzzy LogicFuzzy Logic

Neural Networks

Neural Networks

Genetic Algorithms

Genetic Algorithms

Pattern Recognition MethodsPattern Recognition Methods

Statistical Methods

Statistical Methods

Connectionist Methods

Connectionist MethodsOthers

Methods

Others Methods

WaveletsWavelets

PCA

Decision Trees Decision Trees Discriminant Analysis

Discriminant Analysis

Clustering Clustering


• Based on Darwin’s theory of ”survival of the fittest”: Living organisms reproduce, individuals evolve/ mutate, individuals survive or die based on fitness

•The input is an initial set of possible solutions

• The output of a genetic algorithm is the set of ”fittest solutions” that will survive in a particular environment

•The process

• Produce the next generation (by a cross-over function)

• Evolve solutions (by a mutation function)

• Discard weak solutions (based on a fitness function)



•Preparation:

Define an encoding to represent solutions (i. e., use a

character sequence to represent a classe)

Create possible initial solutions (and encode them as

strings)

Perform the 3 genetic functions: Crossover, Mutate,

Fitness Test•Why Genetic Algorithms (GAs) ?

• Many real life problems cannot be solved in polynomial amount of time using deterministic algorithm

• Sometimes near optimal solutions that can be generated quickly are more desirable than optimal solutions which require huge amount of time

• Problems can be modeled as an optimization one


• Crossover1 0 0 1 1 0 1 0

0 1 1 1 1 1 0 1

1 0 0 1 1 1 0 1

0 1 1 1 1 0 1 0

• Mutation 1 0 0 1 1 0 1 0

1 1 0 1 0 0 1 0

1122

33 44

22 11 22 44

Spin

•Roulette Wheel Selection

Genetic Functions Genetic Functions Genetic Functions Genetic Functions

GA ProcessGA ProcessGA ProcessGA Process

Initialize PopulationInitialize Population

Terminate?Terminate?Yes

Outputsolution

Outputsolution

No

Evaluate FitnessEvaluate Fitness

Perform selection, crossover and mutation




Optimization of discriminant functions Optimization of Neural weights Hyperplans search Hypersurfaces search

GAs for Pattern ClassificationGAs for Pattern Classification

Efficiency and Purity of ClassificationEfficiency and Purity of Classification

• Validation

Test Events

ClassificationC1 C2

C1: N1 N11 N12

C2: N2 N21 N22

Total M1 M2

Efficiency for Ci classification Efficiency for Ci classificationi

iii N

NE

Purity for Ci events Purity for Ci events

i

iii M

NP

Misclassification rate for Ci or Error Misclassification rate for Ci or Error

i

iji N

N

Search for the Higgs boson at LHC by using Genetic

Algorithms

Search for the Higgs boson at LHC by using Genetic

Algorithms

pp HX W+W-X l+ l-X

Signal

pp ttX WbWbX l j l jX pp qqX W+W- X



Events generated by the LUND MC PYTHIA 6.1 at s = 14 TeV

MH = 115 - 150 GeV/c2

10000 Higgs events and 10000 Background events are used

Background

Research of discriminating variables


VariablesVariables



jj , jj: the pseudo-rapidity and the azimuthal angle differences between the two jets

jj , jj: the pseudo-rapidity and the azimuthal angle differences between the two jets

Mll : the invariant mass of the two leptons

Mll : the invariant mass of the two leptons

Rapidity-impulsion weighted Moments Mnm : (n=1, …,6)

i rapidity:

Rapidity-impulsion weighted Moments Mnm : (n=1, …,6)

i rapidity:

)(.2

1

//

//

ii

iii pE

pELog

Jeti

miT

ninm pM .

Mjj : the invariant mass of the two jets

Mjj : the invariant mass of the two jets

ll, ll, jj, jj, Mll , Mjj, M11, M21, M31, M41

C

CHiggs CBack The most separating discriminant Function FHiggs / Back

between the classes CHiggs and CBack is :

FHiggs / Back = - 0.02+0.12ll +0.4jj +0.35 Mll +0.61 Mjj + 0.74M11 +1.04M21

The classification of a test event x0 is then obtained according to the condition:

if FHiggs / Back (xo) 0 then xo CHiggs else xo CBack

Optimization of Discriminant Functions (1)Optimization of Discriminant Functions (1)

Classification of test events

0.610

0.601

Efficiency

CBack

CHiggs

Test events

0.604

0.606

Purity

Classification

F(x ) = (gsignal - gback )T V-1 x = i xi

Discriminant Analysis

Optimization of Discriminant Functions (2)Optimization of Discriminant Functions (2)

5i4

i2i1

i0i 6

i3i I

• Discriminant Function

• GA Parameters

• Number of generations

• Selection, Crossover, Mutation

0.74 1.010.40.12-0.02 0.610.35 0.395

• Fitness

• GA Code Matlab GA Toolbox

Fitness Fn: Misclassification rate


Test Events

ClassificationEff. Purity Eff

CHiggs 0.601 0.606

0.6055 0.3945CBack 0.610 0.604

Optimization of Discriminant Functions (3) Optimization of Discriminant Functions (3)

Chromosome 1 ………………

Chromosome N

Generation of N solutions

Fitness Fn:


Terminate?Yes

Outputsolution

OutputsolutionNo





• Genetic Process

514

1211

101 6

131 I

5N4

N2N1

N0N 6

N3N N

Coefficients of F

Optimization of Discriminant Functions (4) Optimization of Discriminant Functions (4)

• Optimization Results

Number of generations =10000

CPU Time: 120 s

0.74 1.010.40.12-0.02 0.610.35 0.35• Optimal Disc. Fn

0.650

0.649

Purity

0.648

0.652

Eff.

Classification

0.65

Eff

CBack0.35

CHiggs

Test Events


Optimization of Neural Weights (1)Optimization of Neural Weights (1)


Neural Analysis

NN Architecture: (10, 10, 10, 1)

i

ioh

i hwo 22

11 i

ioh

i hwo 22

11

)( 121

2 j

ijhh

jii hwfh )( 121

2 j

ijhh

jii hwfh

)( 11

jij

hxjii xwfh )( 1

1 j

ijhx

jii xwfh

if O1(x) 0.5 then x CHiggs else x CBack

Test Evts

ClassificationEff. Pur. Eff

CHiggs 0.654 0.663

0.661 0.338

CBack 0.669 0.659

ll ll jj jj Mll M0 M11 M21 M31 M41

O1

h1

h2

Optimization of Neural Weights (2)Optimization of Neural Weights (2)

•Connection Weights + Thresholds

• GA Parameters

• Total number of parameters to be optimized : 230

• Fitness: Misclassification rate,

Wijxh1, i

h1 I Wijh1h2, i

h2 Wijh2o

100 +10 I 100 + 10 10

• Optimization Results

Number of generations =1000

CPU Time: 6 mn

Test Evts


CHiggs 0.691 0.696

0.695 0.305

CBack 0.699 0.693

Hyperplan search (1)Hyperplan search (1)

Hyperplan Hj

• Genetic Process

0j 1

j 2

j 3

j 4

j 5

j 6

j 7

j 8

j 9

j 10

j

• Generation of N hyperplans Hj, j=1: N=20

• 11 parameters to optimize

• Number of generations =10000

• CPU Time: 4 mn

H(x) = 0 + i xi = 0 , i=1, 10

Classification rule

if H(x) 0 then x CHiggs else x CBack


Terminate?Yes

Outputsolution

OutputsolutionNo





Hyperplan search (2)Hyperplan search (2)

• Hyperplan search Results

Test Evts


CHiggs 0.661 0.655

0.656 0.344CBack 0.651 0.657


Same results than Discriminant functions optimization

0.650

0.649

Pur.

0.648

0.652

Eff.

Classification

0.65

Eff

CBack0.35

CHiggs

Test Events

Hypersurface search (1)Hypersurface search (1)

Hypersurface j

Initialize Population

Terminate?Yes

OutputsolutionNo

Evaluate Fitness


Evaluate Fitness

• Genetic Process

• Generation of N hyperplans Sj, j=1: N=20

• 31 parameters to optimize

• Number of generations =10000

• CPU Time: 6 mn

Classification rule

if S(x) 0 then x CHiggs else x CBack

)()( 3210

10 iiii

iii xxxxS

I j i=0:10 j i

j i=1:10 i j i=1:10

• Hypersurface search Results

Test Evts


CHiggs 0.689 0.696

0.691 0.309CBack 0.693 0.693


Same results as NN weights optimization

Hypersurface search (2)Hypersurface search (2)


0.693

0.696

Pur.

0.699

0.691

Eff.

Classification

0.695

Eff

CBack0.305

CHiggs

Test Evts

ConclusionConclusion

MethodsMethods

Importance of Pattern Recognition Methods

The improvement of an any identification is subjected to the multiplication of multidimensional effect offered by PR methods and the discriminating power of the proposed variables.

Genetic Algorithms Method allows to minimize the classification error and to improve efficiencies and purities of classifications.

The performances are in average 3 to 5 % higher than those obtained with the other methods.

Discriminant Functions Optimization : comparative to hyperplan search approach

Neural Weights Optimization : comparative to hypersurface search approach

CC

CHiggsCHiggs CBack

CBack


Variables Variables

Conclusion (continued)Conclusion (continued)

Characterisation of Higgs Boson events:

Other variables should be examined

Physics Processes Physics Processes

Other processes should be consideredOther processes should be considered

Detector effects should be added to the simulated events Detector effects should be added to the simulated events


Documents

ACAT05 May 22 - 27, 2005 DESY, Zeuthen, Germany