Least - - squares squares methods methods
LeastLeast--squaressquares methodsmethods have a have a longlong traditiontradition in in track
track fittingfitting
AlreadyAlready from from bubblebubble chamberchamber experimentsexperiments theythey have have constituted
constituted thethe defaultdefault classclass ofof methodsmethods for for estimating
estimating tracktrack parametersparameters
In In earlyearly bubblebubble chamberchamber experimentsexperiments::
usualusual practicepractice to to neglectneglect effectseffects ofof multiple Coulomb multiple Coulomb scattering
scattering during theduring the estimationestimation (Jobes(Jobes and Shaylorand Shaylor, Rep. , Rep. ProgProg. . Phys. 1972)Phys. 1972)
errorserrors ofof multiple multiple scatteringscattering werewere afterwardsafterwards addedadded to to covariance
covariance matrixmatrix ofof estimatedestimated parametersparameters
Least
Least - - squares squares methods methods
ImportantImportant stepstep forwardforward due to due to LaurikainenLaurikainen et et al. al. (NIM 1972)(NIM 1972) ::
introducedintroduced thethe conceptconcept ofof breakpointsbreakpoints
fittedfitted deflectiondeflection angles angles explicitelyexplicitely
accuracyaccuracy ofof estimatesestimates improvedimproved
choicechoice ofof positionspositions ofof breakpointsbreakpoints somewhatsomewhat arbitrary
arbitrary
Least
Least - - squares squares methods methods
New New challengeschallenges hadhad to be to be facedfaced by by reconstructionreconstruction programs
programs whenwhen electronicelectronic experimentsexperiments camecame onon stage stage in in earlyearly 70’s70’s
Prime Prime exampleexample: : Split Split FieldField Magnet (SFM)Magnet (SFM) detectordetector at at thethe CERN CERN IntersectingIntersecting StorageStorage Rings (ISR)Rings (ISR)
preciseprecise treatmenttreatment ofof material material effectseffects in wire in wire chamberschambers waswas needed
needed
alsoalso importantimportant effectseffects from from crossingcrossing beam pipe at beam pipe at shallowshallow angles
angles
correctcorrect approachapproach pioneeredpioneered by by M. ReglerM. Regler (CERN 73(CERN 73--2 1973)2 1973)
Least
Least - - squares squares methods methods
SFM SFM detectordetector at CERN ISRat CERN ISR
Least
Least - - squares squares methods methods
excerpt
excerpt from from paperpaper by by NagyNagy et al.et al.
((NuclNucl. . PhysPhys. B 1979). B 1979)
correct
correct weightingweighting enabled
enabled veryvery preciseprecise cross
cross--sectionsection measurementsmeasurements
Least
Least - - squares squares methods methods
CorrectCorrect treatmenttreatment ofof multiple multiple scatteringscattering waswas essential
essential alsoalso for for experimentsexperiments at CERN SPSat CERN SPS
In In WA6 WA6 experimentexperiment at CERN proper at CERN proper treatmenttreatment ofof particleparticle deflectionsdeflections in airin air due to multiple due to multiple
scattering
scattering waswas neededneeded for reliable for reliable estimatesestimates ofof track
track parametersparameters
FormalismFormalism developeddeveloped by M. Regler by M. Regler waswas alsoalso used in
used in thisthis experimentexperiment ((FidecaroFidecaro et al., et al., NuclNucl. . PhysPhys. B . B
1980) 1980)
Least
Least - - squares squares methods methods
WA6 WA6
experiment experiment
at at CERNCERN
SPSSPS
Least
Least - - squares squares methods methods
perfectly
perfectly flatflat probability probability distribution distribution after
after cuttingcutting awayaway
nonnon--elasticelastic events events
Least
Least - - squares squares methods methods
In In thethe UA1UA1 Nobel Nobel prizeprize experimentexperiment, material , material effectseffects could
could be neglectedbe neglected for for particlesparticles aboveabove 200 GeV200 GeV/c/c due to
due to veryvery small small amountsamounts ofof material in material in CentralCentral Detector
Detector (M. (M. PimiPimiää, CERN, private communication), CERN, private communication)
UA1 UA1 CentralCentral DetectorDetector providedprovided
preciseprecise measurementsmeasurements perpendicularperpendicular to to fieldfield
bad bad resolutionresolution elsewhereelsewhere
TrackTrack findingfinding and fitting and fitting waswas donedone in in projectionprojection perpendicular
perpendicular to to fieldfield
Fast, Fast, approximateapproximate circlecircle fitfit due to V. due to V. KarimKarimäkiäki was was used as least
used as least--squares estimator squares estimator (NIM A 1991)(NIM A 1991)
Least
Least - - squares squares methods methods
In WA13 In WA13 experimentexperiment at CERN, a at CERN, a newnew formulationformulation ofof leastleast--squaressquares methodmethod waswas developeddeveloped
ThisThis methodmethod –– thethe so-so-calledcalled progressive progressive methodmethod due to P.
due to P. BilloirBilloir –– includesincludes measurementsmeasurements recursively
recursively intointo thethe fitfit (NIM 1984)(NIM 1984)
EstimatesEstimates ofof tracktrack parameters parameters areare updatedupdated eacheach time time information
information from from newnew measurementmeasurement is is includedincluded
Least
Least - - squares squares methods methods
layout layout ofof
WA13 WA13
experiment experiment
Least
Least - - squares squares methods methods
During During developmentdevelopment ofof reconstructionreconstruction software for software for DELPHI
DELPHI experimentexperiment at LEP at LEP collidercollider at CERN, it at CERN, it waswas realized
realized by R. Frühwirthby R. Frühwirth that progressive method was that progressive method was equivalent to
equivalent to KalmanKalman filterfilter (NIM A 1987)(NIM A 1987)
Immediate implications:Immediate implications:
existence of Kalmanexistence of Kalman smoothersmoother
easy to obtain easy to obtain optimal estimates anywhere along the optimal estimates anywhere along the track
track
enables enables efficient outlier rejectionefficient outlier rejection, as track parameter , as track parameter
predictions are calculated from all other measurements in a predictions are calculated from all other measurements in a track
track
Least
Least - - squares squares methods methods
from Fr
from Frühwirth’sühwirth’s seminal paper seminal paper
Least
Least - - squares squares methods methods
AnotherAnother importantimportant effecteffect needingneeding treatmenttreatment during
during estimationestimation is is energyenergy lossloss
IonizationIonization loss is loss is oftenoften regardedregarded as as deterministic
deterministic correctioncorrection to to tracktrack modelmodel
ElectronsElectrons needneed specialspecial treamenttreament::
suffersuffer from from bremsstrahlungbremsstrahlung energyenergy lossloss
dominant dominant componentcomponent ofof energyenergy loss loss aboveabove ~100 ~100 MeV/cMeV/c
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Least - - squares squares methods methods
distribution
distribution ofof relative
relative energyenergy lossloss
strongly
strongly peakedpeaked withwith longlong tailtail
Least
Least - - squares squares methods methods
A A full full spectrumspectrum ofof electronelectron trackstracks waswas treatedtreated withinwithin unbiased
unbiased leastleast--squaressquares estimationestimation for for thethe first time by first time by Stampfer
Stampfer et al.et al. (CPC 1994)(CPC 1994)
BremsstrahlungBremsstrahlung waswas treatedtreated as as stochasticstochastic processprocess
First First twotwo momentsmoments ofof distributiondistribution due to modeldue to model by by Bethe
Bethe and and HeitlerHeitler (Proc(Proc. . RoyalRoyal Soc. London 1934)Soc. London 1934) includedincluded in in Kalman
Kalman filterfilter
MethodMethod originallyoriginally testedtested outout onon data from DELPHIdata from DELPHI and and shownshown to be to be superior to conventionalsuperior to conventional
treatments treatments
Least
Least - - squares squares methods methods
residual
residual andand pull
pull distributionsdistributions standard
standard approach approach
Kalman Kalman
filter filter