Adaptive methods methods
LeastLeast--squaressquares methodmethod is is optimaloptimal whenwhen
tracktrack modelmodel is linearis linear
probabilityprobability densitiesdensities areare GaussianGaussian
TrackTrack modelmodel cancan oftenoften be be linearizedlinearized
ProbabilityProbability densitydensity functionsfunctions ((pdfpdf) ) areare oftenoften notnot Gaussian
Gaussian::
measurementmeasurement errorserrors cancan have longhave long tailstails or or eveneven be closebe close to to flatflat
multiple multiple scatteringscattering errorserrors have have GaussianGaussian corecore butbut longlong, , single
single--scatteringscattering tailstails
Adaptive
Adaptive methods methods
Proper Proper treatmenttreatment ofof suchsuch effectseffects leads to leads to methodsmethods going
going beyondbeyond pure pure leastleast--squaressquares
VeryVery interestinginteresting specialspecial case:case:
pdfspdfs areare restrictedrestricted to Gaussianto Gaussian mixturesmixtures
ResultingResulting estimatorestimator:: GaussianGaussian--sumsum filter (GSF) filter (GSF)
(Fr(Frühwirthühwirth, CPC 1997), CPC 1997)
ResemblesResembles in in thisthis case a case a setset ofof KalmanKalman filters filters runningrunning in in parallelparallel
BothBoth measurementmeasurement errorserrors and material and material uncertainties
uncertainties areare in general in general GaussianGaussian mixturesmixtures
Adaptive
Adaptive methods methods
VeryVery thoroughthorough studystudy ofof multiple multiple scatteringscattering in in context
context ofof tracktrack reconstructionreconstruction has has beenbeen performed
performed by by FrFrühwirthühwirth and and ReglerRegler (NIM A (NIM A
2001) 2001) ::
full full pdfpdf obtained by obtained by successive (numerical) successive (numerical) convolutions
convolutions of single-of single-scattering densityscattering density
precise, precise, twotwo--component Gaussiancomponent Gaussian--mixture mixture approximation
approximation of density obtained and of density obtained and
parameterized as function of material thickness parameterized as function of material thickness
Adaptive
Adaptive methods methods
comparison
comparison withwith simulation
simulation comparison
comparison withwith MoliereMoliere
Adaptive
Adaptive methods methods
ReconstructionReconstruction ofof electronselectrons lends lends itselfitself to to treatment
treatment by GSFby GSF::
energyenergy loss loss distributiondistribution is is highlyhighly nonnon--GaussianGaussian
First First attemptattempt mademade by Frühwirthby Frühwirth and and Frühwirth-Frühwirth -Schnatter
Schnatter (CPC 1998)(CPC 1998)
“proof“proof--of of ––principle” type of studyprinciple” type of study
Adaptive
Adaptive methods methods
residual residual
distributions distributions
ofof inverse inverse momentum momentum
ofof GSF GSF
and and KalmanKalman filterfilter
Adaptive
Adaptive methods methods
More More recentlyrecently, , GaussianGaussian--mixturemixture approximations
approximations ofof thethe BetheBethe--HeitlerHeitler distributions
distributions have have beenbeen calculatedcalculated (Frühwirth(Frühwirth, CPC , CPC 2003)
2003)
These have been used in a more These have been used in a more realistic realistic implementation of a GSF
implementation of a GSF in the CMS tracker in the CMS tracker at CERN
at CERN (Adam et al., Proc. CHEP’03 2003)(Adam et al., Proc. CHEP’03 2003)
Results indicate that Results indicate that improvements can be improvements can be mademade with respect to standard with respect to standard KalmanKalman filterfilter
Adaptive
Adaptive methods methods
example
example ofof estimated
estimated pdfpdf from GSF from GSF
residual residual distributions distributions
ofof inverseinverse momentum momentum
Adaptive
Adaptive methods methods
probability probability distributions distributions ofof estimatedestimated
inverse inverse momentum momentum
KFKF GSFGSF
Adaptive
Adaptive methods methods
Neural Neural networksnetworks becamebecame veryvery popularpopular toolstools for data for data analysis
analysis in in thethe 80’s80’s
A A HopfieldHopfield neural netneural net waswas developeddeveloped for patternfor pattern recognition
recognition in in trackingtracking detectorsdetectors
→ Denby→ Denby--PetersonPeterson neural networkneural network (Denby(Denby, CPC 1988; , CPC 1988;
Peterson
Peterson, NIM A 1989), NIM A 1989)
MethodMethod waswas implementedimplemented in ALEPH in ALEPH experimentexperiment at at CERN and
CERN and claimedclaimed to to yieldyield resultsresults compatiblecompatible withwith standard
standard tracktrack findingfinding (Stimpfl(Stimpfl--AbeleAbele & & Garrido, CPC 1991)Garrido, CPC 1991)
Adaptive
Adaptive methods methods
event
event withwith generatedgenerated links links and and convergedconverged resultresult
Adaptive
Adaptive methods methods
AlternativesAlternatives to to meanmean--fieldfield annealingannealing for for minimization
minimization ofof energyenergy have have beenbeen foundfound inferior
inferior (Diehl(Diehl et al., NIM A 1997)et al., NIM A 1997)
efficiency efficiency evolution
evolution ofof energy
energy
Adaptive
Adaptive methods methods
MethodMethod relatedrelated to to HopfieldHopfield netnet butbut withwith explicit
explicit referencereference to to tracktrack modelmodel::
ElasticElastic Arms Arms algorithmalgorithm ((OhlssonOhlsson et al., CPC 1992)et al., CPC 1992)
AttemptAttempt to to speed speed upup methodmethod by by formulatingformulating it it as as singlesingle--tracktrack algorithmalgorithm has has beenbeen mademade
(Fr(Frühwirthühwirth and and StrandlieStrandlie, CPC 1999), CPC 1999)
Adaptive
Adaptive methods methods
Shown in this paper:Shown in this paper:
standard standard gradientgradient--descentdescent based minimization based minimization not precise not precise enough
enough
advanced, timeadvanced, time--consuming consuming quasiquasi--Newton methods Newton methods required
required
non-non-linearlinear minimizationminimization couldcould equivalentlyequivalently be be formulatedformulated as as iteratively
iteratively reweightedreweighted leastleast--squaressquares procedureprocedure
fitting part offitting part of optimizationoptimization cancan in principlein principle be be donedone by anyby any least
least--squaressquares estimatorestimator, including, including KalmanKalman filterfilter
ResultingResulting algorithm:algorithm:
→ Deterministic→ Deterministic AnnealingAnnealing Filter (DAF)Filter (DAF)
Adaptive
Adaptive methods methods
DecisiveDecisive advantagesadvantages withwith respectrespect to standard to standard formulation
formulation ofof algorithmalgorithm::
material material effectseffects cancan straightforwardlystraightforwardly be be takentaken intointo accountaccount
inhomogeneousinhomogeneous magneticmagnetic fieldsfields cancan be be dealtdealt withwith
nono needneed for for tedioustedious, , numericalnumerical minimizationminimization
GeneralizationGeneralization ofof DAF DAF whichwhich fitsfits severalseveral trackstracks concurrently
concurrently has has alsoalso beenbeen developeddeveloped
→ → MultiMulti TrackTrack Filter (MTF)Filter (MTF) (Strandlie(Strandlie and Frühwirthand Frühwirth, CPC , CPC 2000)
2000)
Results from Results from application to simulated data from application to simulated data from ATLAS TRT
ATLAS TRT showed showed excellent robustness with excellent robustness with respect to ambiguities and noise
respect to ambiguities and noise
Adaptive
Adaptive methods methods
track
track pair inpair in RPhiRPhi--projectionprojection
including
including correctcorrect measurements
measurements andand fitted
fitted trackstracks
Adaptive
Adaptive methods methods
DAF and MTF have DAF and MTF have alsoalso beenbeen implementedimplemented in in standard
standard reconstructionreconstruction program in CMS program in CMS tracker
tracker (Winkler(Winkler, , PhDPhD ThesisThesis 2003)2003)
SystematicSystematic comparisonscomparisons to standard, to standard, combinatorial
combinatorial KalmanKalman filter have filter have beenbeen mademade
ClearClear improvementsimprovements in in resolutionresolution ofof tracktrack parameters
parameters seenseen in ”in ”difficultdifficult” ” situationssituations, , suchsuch as as reconstruction
reconstruction ofof highhigh--energyenergy narrownarrow jetsjets
Adaptive
Adaptive methods methods
impact
impact parameter parameter resolution
resolution
probability probability distributions distributions
Adaptive
Adaptive methods methods
DAF b
DAF b--taggingtagging efficiency
efficiency
MTF MTF probabilityprobability distributions distributions
Discussion Discussion
BoundariesBoundaries betweenbetween tracktrack findingfinding and and tracktrack fitting
fitting
BoundariesBoundaries betweenbetween tracktrack fitting and fitting and physics
physics analysesanalyses
LeastLeast--squaressquares methodsmethods
Discussion Discussion
BoundariesBoundaries betweenbetween tracktrack findingfinding and and tracktrack fitting
fitting::
during during eraera ofof bubblebubble chamberchamber experimentsexperiments tracktrack finding
finding waswas movingmoving from manual to from manual to automaticautomatic
for for electronicelectronic detectorsdetectors fakefake rate rate waswas sometimessometimes so so highhigh thatthat an intermediatean intermediate stepstep hadhad to be to be
introduced introduced
withwith inventioninvention ofof KalmanKalman filter filter thisthis stepstep waswas againagain integrated
integrated
Discussion Discussion
in in recentrecent HLT applicationsHLT applications tracktrack findingfinding is is stopped
stopped veryvery earlyearly
adaptive algorithmsadaptive algorithms postponepostpone final final assignmentassignment far far intointo tracktrack fitfit procedureprocedure
BoundariesBoundaries betweenbetween tracktrack findingfinding and and tracktrack fitting have
fitting have movedmoved from from beingbeing veryvery clearclear to to being
being veryvery fuzzyfuzzy!!!!
Discussion Discussion
In In thethe futurefuture therethere is is reasonreason to to believebelieve thatthat thisthis fuzziness
fuzziness willwill evolveevolve eveneven furtherfurther::
possiblepossible upgradeupgrade ofof LHCLHC to to tenten times design times design luminosity
luminosity willwill alsoalso be a be a challengechallenge for for reconstruction
reconstruction algorithmsalgorithms!!
alternatives to alternatives to combinatorialcombinatorial KalmanKalman filterfilter for for track
track findingfinding willwill have to be have to be consideredconsidered
interestinginteresting optionoption wouldwould be to be to applyapply thethe Deterministic
Deterministic AnnealingAnnealing Filter for Filter for thethe full full tracktrack reconstruction
reconstruction chainchain
Discussion Discussion
BoundariesBoundaries betweenbetween tracktrack fitting and fitting and physics
physics analysesanalyses::
withwith GaussianGaussian--sumsum filter filter thethe estimateestimate providedprovided by by track
track fitfit is a is a GaussianGaussian mixturemixture ratherrather thanthan single single Gaussian
Gaussian
vertexvertex fitfit usingusing full full informationinformation from from mixturemixture yieldsyields as as output output anotheranother GaussianGaussian mixturemixture (Fr(Frühwirthühwirth and and Speer, Proc. ACAT’03)
Speer, Proc. ACAT’03)
full informationfull information from such mixtures should be from such mixtures should be carried as
carried as far as far as possbilepossbile into further analysesinto further analyses
Discussion Discussion
Example: proper Example: proper calculationcalculation ofof sum sum ofof momenta momenta would
would be be convolutionconvolution ofof severalseveral GaussianGaussian mixtures
mixtures
SuchSuch output from output from reconstructionreconstruction requiresrequires analysis
analysis algorithmsalgorithms to have to have deeperdeeper insightinsight intointo reconstructedreconstructed objectsobjects thanthan beforebefore
ApplicationApplication ofof GaussianGaussian--sumsum filters filters willwill challenge
challenge traditionaltraditional boundariesboundaries betweenbetween reconstruction
reconstruction and and physicsphysics analysesanalyses
Discussion Discussion
LeastLeast--squaressquares methodsmethods::
baseline baseline tooltool for for bubblebubble chamberchamber tracktrack fittingfitting
furtherfurther developeddeveloped (proper (proper treatmenttreatment ofof material material effects
effects) ) afterafter inventioninvention ofof electronicelectronic experimentsexperiments
KalmanKalman filter filter carriedcarried leastleast--squaressquares approachapproach intointo LEP eraLEP era
GaussianGaussian--sumsum filters and adaptive filters and adaptive methodsmethods cancan alsoalso be be regardedregarded as as leastleast--squaressquares methodsmethods
→ → leastleast--squaressquares willwill be be appliedapplied alsoalso at LHCat LHC
Discussion Discussion
LeastLeast--squaressquares is is thethe commoncommon denominatordenominator for for tracktrack fitting from fitting from bubblebubble chamberschambers to to LHCLHC
ReasonReason to to believebelieve thatthat successsuccess ofof leastleast--squaressquares estimators
estimators willwill continuecontinue alsoalso in in futurefuture experiments
experiments