Multivariate Statistical Methods in Particle Physics粒子物理中的多元统计方法

MachineLearningandMultivariateStatisticalMethodsinParticlePhysicsGlenCowanRHULPhysicspp.rhul.ac.uk/~cowanRHULComputerScienceSeminar17March,2021OutlineQuickoverviewofparticlephysicsattheLargeHadronCollider(LHC)MultivariateclassificationfromaparticlephysicsviewpointSomeexamplesofmultivariateclassificationinparticlephysics

NeuralNetworks BoostedDecisionTrees SupportVectorMachinesSummary,conclusions,etc.TheStandardModelofparticlephysicsMatter...+gaugebosons...photon(g),W±,Z,gluon(g)+relativity+quantummechanics+symmetries...=StandardModel25freeparameters(masses,couplingstrengths,...).IncludesHiggsboson(notyetseen).Almostcertainlyincomplete(e.g.nogravity).Agreeswithallexperimentalobservationssofar.ManycandidateextensionstoSM(supersymmetry,extradimensions,...)TheLargeHadronColliderCounter-rotatingprotonbeamsin27kmcircumferenceringppcentre-of-massenergy14TeVDetectorsat4ppcollisionpoints: ATLAS CMS LHCb(bphysics) ALICE(heavyionphysics)generalpurposeTheATLASdetector2100physicists37countries167universities/labs25mdiameter46mlength7000tonnes~108electronicchannelsAsimulatedSUSYeventinATLAShighpTmuonshighpTjetsofhadronsmissingtransverseenergyppBackgroundeventsThiseventfromStandardModelttbarproductionalsohashighpTjetsandmuons,andsomemissingtransverseenergy.→caneasilymimicaSUSYevent.LHCeventproductionratesmostevents(boring)interestingveryinteresting(~1outofevery1011)mildlyinterestingLHCdataAtLHC,~109ppcollisioneventspersecond,mostlyuninteresting doquicksifting,record~200events/sec singleevent~1Mbyte 1“year〞107s,1016ppcollisions/year 2109eventsrecorded/year(~2Pbyte/year)Fornew/rareprocesses,ratesatLHCcanbevanishinglysmall e.g.Higgsbosonsdetectableperyearcouldbe~103 →'needleinahaystack'ForStandardModeland(many)non-SMprocesseswecangeneratesimulateddatawithMonteCarloprograms(includingsimulationofthedetector).AsimulatedeventPYTHIAMonteCarlopp→gluino-gluino...MultivariateanalysisinparticlephysicsForeacheventwemeasureasetofnumbers:x1=jetpT

x2=missingenergyx3=particlei.d.measure,...followssomen-dimensionaljointprobabilitydensity,whichdependsonthetypeofeventproduced,i.e.,wasitE.g.hypothesesH0,H1,...Oftensimply“signal〞,“background〞FindinganoptimaldecisionboundaryInparticlephysicsusuallystartbymakingsimple“cuts〞: xi<ci xj<cjMaybelatertrysomeothertypeofdecisionboundary:H0H0H0H1H1H1TheoptimaldecisionboundaryTrytobestapproximateoptimaldecisionboundarybasedonlikelihoodratio:orequivalentlythinkofthelikelihoodratioastheoptimalstatisticforatestofH0vsH1.Ingeneralwedon'thavethepdfsp(x|H0),p(x|H1),...Rather,wehaveMonteCarlomodelsforeachprocess.

UsuallytrainingdatafromtheMCmodelsischeap. Butthemodelscontainmanyapproximations: predictionsforobservablesobtainedusingperturbation theory(truncatedatsomeorder);phenomenologicalmodeling ofnon-perturbativeeffects;imperfectdetectordescription,...TwodistincteventselectionproblemsInsomecases,theeventtypesinquestionarebothknowntoexist.

Example:separationofdifferentparticletypes(electronvsmuon) Usetheselectedsampleforfurtherstudy.Inothercases,thenullhypothesisH0means"StandardModel"events,andthealternativeH1means"eventsofatypewhoseexistenceisnotyetestablished"(todosoisthegoaloftheanalysis).

Manysubtleissueshere,mainlyrelatedtotheheavyburden ofproofrequiredtoestablishpresenceofanewphenomenon. Typicallyrequirep-valueofbackground-onlyhypothesis below~10-7(a5sigmaeffect)toclaimdiscoveryof "NewPhysics".Discovering"NewPhysics"TheLHCexperimentsareexpensive

~$1010(acceleratorandexperiments)thecompetitionisintense

(ATLASvs.CMS)vs.Tevatronandthestakesarehigh:4sigmaeffect5sigmaeffectSothereisastrongmotivationtoextractallpossibleinformationfromthedata.Usingclassifieroutputfordiscoveryyf(y)yN(y)NormalizedtounityNormalizedtoexpectednumberofeventsexcess?signalbackgroundbackgroundsearchregionDiscovery=numberofeventsfoundinsearchregionincompatiblewithbackground-onlyhypothesis.p-valueofbackground-onlyhypothesiscandependcruciallydistributionf(y|b)inthe"searchregion".ycutExampleofa"cut-based"studyInthe1990s,theCDFexperimentatFermilab(Chicago)measuredthenumberofhadronjetsproducedinproton-antiprotoncollisionsasafunctionoftheirmomentumperpendiculartothebeamdirection:Predictionlowrelativetodataforveryhightransversemomentum."jet"ofparticlesHighpTjets=quarksubstructure?AlthoughthedataagreeremarkablywellwiththeStandardModel(QCD)predictionoverall,theexcessathighpTappearssignificant:Thefactthatthevariableis"understandable"leadsdirectlytoaplausibleexplanationforthediscrepancy,namely,thatquarkscouldpossessaninternalsubstructure.Wouldnothavebeenthecaseifthevariableplottedwasacomplicatedcombinationofmanyinputs.HighpTjetsfrompartonmodeluncertaintyFurthermorethephysicalunderstandingofthevariableledonetoamoreplausibleexplanation,namely,anuncertainmodellingofthequark(andgluon)momentumdistributionsinsidetheproton.Whenmodeladjusted,discrepancylargelydisappears:Canberegardedasa"success"ofthecut-basedapproach.Physicalunderstandingofoutputvariableledtosolutionofapparentdiscrepancy.NeuralnetworksinparticlephysicsFormanyyears,theonly"advanced"classifierusedinparticlephysics.Usuallyusesinglehiddenlayer,logisticsigmoidactivationfunction:NeuralnetworkexamplefromLEPIISignal:e+e-

→W+W-(often4wellseparatedhadronjets)Background:e+e-

→qqgg(4lesswellseparatedhadronjets)←inputvariablesbasedonjetstructure,eventshape,...nonebyitselfgivesmuchseparation.Neuralnetworkoutput:(Garrido,JusteandMartinez,ALEPH96-144)SomeissueswithneuralnetworksIntheexamplewithWWevents,goalwastoselecttheseeventssoastostudypropertiesoftheWboson.

Neededtoavoidusinginputvariablescorrelatedtothe propertiesweeventuallywantedtostudy(nottrivial).Inprincipleasinglehiddenlayerwithansufficientlylargenumberofnodescanapproximatearbitrarilywelltheoptimaltestvariable(likelihoodratio). Usuallystartwithrelativelysmallnumberofnodesandincrease untilmisclassificationrateonvalidationdatasampleceases todecrease.UsuallyMCtrainingdataischeap--problemswithgettingstuckinlocalminima,overtraining,etc.,lessimportantthanconcernsofsystematicdifferencesbetweenthetrainingdataandNature,andconcernsabouttheeaseofinterpretationoftheoutput.DecisiontreesOutofalltheinputvariables,findtheoneforwhichwithasinglecutgivesbestimprovementinsignalpurity:ExamplebyMiniBooNEexperiment,B.Roeetal.,NIM543(2005)577wherewi.istheweightoftheithevent.Resultingnodesclassifiedaseithersignal/background.Iterateuntilstopcriterionreachedbasedone.g.purityorminimumnumberofeventsinanode.Thesetofcutsdefinesthedecisionboundary.BoostingTheresultingclassifierisusuallyverysensitivetofluctuationsinthetrainingdata.Stabilizebyboosting: Createanensembleoftrainingdatasetsfromtheoriginaloneby updatingtheeventweights(misclassifiedeventsgetincreased weight). Assignascoreaktotheclassifierfromthekthtrainingsetbased onitserrorrateek:Finalclassifierisaweightedcombinationofthosefromtheensembleoftrainingsets:Particlei.d.inMiniBooNEDetectorisa12-mdiametertankofmineraloilexposedtoabeamofneutrinosandviewedby1520photomultipliertubes:H.J.Yang,MiniBooNEPID,DNP06Searchfornmtoneoscillationsrequiredparticlei.d.usinginformationfromthePMTs.BDTexamplefromMiniBooNE~200inputvariablesforeachevent(ninteractionproducinge,morp).

Eachindividualtreeisrelativelyweak,withamisclassificationerrorrate~0.4–0.45B.Roeetal.,NIM543(2005)577MonitoringovertrainingFromMiniBooNEexample:Performancestableafterafewhundredtrees.ComparisonofboostingalgorithmsAnumberofboostingalgorithmsonthemarket;differintheupdaterulefortheweights.BoosteddecisiontreecommentsBoosteddecisiontreeshavebecomepopularinparticlephysicsbecausetheycanhandlemanyinputswithoutdegrading;thosethatprovidelittle/noseparationarerarelyusedastreesplittersareeffectivelyignored.Anumberofboostingalgorithmshavebeenlookedat,whichdifferprimarilyintheruleforupdatingtheweights(e-Boost,LogitBoost,...).Somestudieshavelookedatotherwaysofcombiningweakerclassifiers,e.g.,Bagging(Boostrap-Aggregating),generatestheensembleofclassifiersbyrandomsamplingwithreplacementfromthefulltrainingsample.Notmuchexperienceyetwiththese.ThetopquarkTopquarkistheheaviestknownparticleintheStandardModel.Sincemid-1990shasbeenobservedproducedinpairs:SingletopquarkproductionOnealsoexpectedtofindsinglyproducedtopquarks;pair-producedtopsarenowabackgroundprocess.Usemanyinputsbasedonjetproperties,particlei.d.,...signal(blue+green)DifferentclassifiersforsingletopAlsoNaiveBayesandvariousapproximationstolikelihoodratio,....Finalcombinedresultisstatisticallysignificant(>5slevel)butnoteasytounderstandclassifieroutputs.SupportVectorMachinesMapinputvariablesintohighdimensionalfeaturespace:x

→

fMaximizedistancebetweenseparatinghyperplanes(margin)subjecttoconstraintsallowingforsomemisclassification.Finalclassifieronlydependsonscalarproductsoff(x):SoonlyneedkernelBishopch7UsinganSVMTouseanSVMtheusermustasaminimumchoose

akernelfunction(e.g.Gaussian) anyfreeparametersinthekernel(e.g.thesoftheGaussian) thecostparameterC(playsroleofregularizationparameter)Thetrainingisrelativelystraightforwardbecause,incontrasttoneuralnetworks,thefunctiontobeminimizedhasasingleglobalminimum.Furthermoreevaluatingtheclassifieronlyrequiresthatoneretainandsumoverthesupportvectors,arelativelysmallnumberofpoints.Theadvantages/disadvantagesandrationalebehindthechoicesaboveisnotalwayscleartotheparticlephysicist--helpneededhere.SVMinparticlephysicsSVMsareverypopularintheMachineLearningcommunitybuthaveyettofindwideapplicationinHEP.HereisanearlyexamplefromaCDFtopquarkanlaysis(A.Vaiciulis,contributiontoPHYSTAT02).signaleff.Summary,conclusions,etc.Particlephysicshasusedseveralmultivariatemethodsformanyyears: linear(Fisher)discriminant neuralnetworks naiveBayesandhasinthelastseveralyearsstartedtouseafewmore

k-nearestneighbour boosteddecisiontrees supportvectormachinesTheemphasisisoftenoncontrollingsystematicuncertaintiesbetweenthemodeledtrainingdataandNaturetoavoidfalsediscovery.Althoughmanyclassifieroutputsare"blackboxes",adiscoveryat5ssignificancewithasophisticated(opaque)methodwillwinthecompetitionifbackedupby,say,4sevidencefromacut-basedmethod.QuotesIlike“Allessolltesoeinfachwiemöglichsein,abernichteinfacher.〞 –A.Einstein“Ifyoubelieveinsom