The implementation of an algorithm based on a likelihood method to discriminate between real and ghost tracks is described. This algorithm is shown to perform slightly better than a cut on the $\chi^2$/ndof

Needham, M

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LHCb 2008-026LPHE 2008-004Identification of Ghost Tracks using aLikelihood MethodM. NeedhamLaboratoire de Physique des Hautes Energies,´Ecole Polytechnique Fe´de´rale de LausanneMay 28, 2008AbstractThe implementation of an algorithm based on a likelihood method todiscriminate between real and ghost tracks is described. This algorithm isshown to perform slightly better than a cut on the χ2/ndof.1 IntroductionThe high track densities expected in the LHCb detector present a challenging envi-roment for track reconstruction. During the pattern recognition step many pseudo-random combinations of hits, refered to as ghosts, are formed. The efficiency ofLHCb tracking algorithms are limited by the number of ghosts that can be tol-erated. This is particularly true for luminosities higher than the nominal LHCbrunning value of 2 × 1032 cm−2s−1. In this note a likelihood based method toidentify ghost tracks is presented. Such an approach is easy to implement andgeneric.The layout of this note is as follows. First, the algorithm is described. Next theperformance for long tracks is summarized for luminosities up to 20×1032 cm−2s−1and compared to a simple cut on the χ2/ndof. The performance for other tracktypes is then discussed. Finally, the implementation in the LHCb software is de-scribed in an appendix.2 AlgorithmGhost tracks are characterized by having a low probability of χ2 from the track fitand missing hits [1]. In addition, ghosts due to spillover from previous crossingscan be distinguished by having lower deposited energy in the silicon detectors. Tocreate a discriminant this information is combined into a log-likelihood using theapproach described in [2]. For each detector layer it is checked whether the trackis expected to give a hit. A binomial counting term is then calculated according tothe number of hits found (n) compared to the number expected (N). This gives acontribution to the log-likelihood:lnL = lnN !n!(N − n)! + nln²+ (N − n)ln(1− ²) (1)where ² is the product of the detector efficiency and the efficiency to add thehit to the track. For the silicon detectors similar counting terms are added tothe likelihood based on how many hits with the high threshold bit are observedcompared to the number expected based on the total number of clusters on thetrack. Table 1 - 2 summarize the efficiencies assumed for the various types oftracking hits.For the Outer Tracker the cell efficiency decreases towards the wall of the strawand a layer is insensitive to tracks that pass through the gap between two cells.2Term Probability (²) SourceVelo r 0.97 PatCheckerVelo φ 0.94 PatCheckerTT 0.9 LHCb note 2007-024IT 0.99 LHCb note 2007-024OT 0.94 PatCheckerTable 1: Probabilities assumed for the various counting terms.Term Probability (²) Spillover Probability (²) SourceVelo 0.99 0.10 Private codeTT 0.91 0.16 LHCb note 2007-024IT 0.94 0.06 LHCb note 2007-024Table 2: Probabilities assumed for the counting terms related the high thresholdbit. For comparison the corresponding numbers for spillover clusters are shown.Therefore, in layers with missing hits the track trajectory should be such that ithas large distance to the centre of the straw. This leads to a term in the likelihoodof the form [2]:L = 0.261 + exp(5.1r − 11.87) (2)where r is the distance of the track from the wire. This contribution is shown inFig 1.Finally, the probability of χ2 given by the track fit is added to the likelihood.For real tracks this contribution has the form shown in Fig. 2. The accumulationof tracks with low probability of χ2 generates a long tail in the log-likelihooddistribution. Potentially, this spoils the performance of the algorithm. In [1] it wassuggested to de-weight this contribution by a factor that is empirically determinedto be around three. Another approach, which will be investigated here, is to usethe shape of the distribution in Fig. 2. This can be modeled by a β function :β(p) =1Z(u1, u2)pu1−1(1− p)u2−1with u1 = 0.63 and u2 = 0.86. The contribution of the χ2 to the likelihood isthen:L =∫ p(χ2)0β(p(χ2))dp (3)3Distance from wire r  (cm)Figure 1: Histogram of the distance of the track trajectory from the wire, for OTcells that did not give a hit; the parametrization used in the likelihood calculationis superimposed.)2χp(0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1050001000015000200002500030000Mean   0.4108RMS    0.3153Skewness  0.2605Kurtosis  -1.256Figure 2: Probability of χ2 for long tracks. The dotted line is the result of a fit toa β function.3 ResultsThe performance studies were done using the following data samples:• A sample of 24000 Bd → J/ψ(µ+µ−)KS(pi+pi−) events generated at thedefault LHCb luminosity of 2× 1032 cm−2s−1.• Samples of 500 inclusive b events generated at luminosities of 5, 8, 10 and420 ×1032 cm−2s−1.All the samples were reconstructed with Brunel v31r11.3.1 Long Track StudiesFor these studies the definitions of long track acceptance, efficiency and ghost rategiven in [1] were used. Fig. 3 shows the likelihood distribution for real long trackswithin the LHCb acceptance and ghosts. A clear separation is observed. From thisplot it seems that there is a component of the ghost’s that look like the real tracks.This is consistent with the observation [3] that there is some fraction of the ghostrate due to real tracks, which are not correctly associated to Monte Carlo truth.likelihood-100 -80 -60 -40 -20 010210310410Figure 3: Likelihood distribution for reconstructed long tracks within the LHCbacceptance (solid line) and ghosts (points).Fig. 4 shows the efficiency versus ghosts rate for the various cuts on the likeli-hood method and also for a cut on the χ2/ndof. It can be seen that the differencein performance of the two methods is small. This shows that the χ2/ndof is al-ready a powerful discriminant and that the additional gain in performance fromthe likelihood technique is small. In Fig 5 the effect of re-weighting the χ2 contri-bution according to Equation 3 is shown. The effect of the re-weighting is small.Therefore, this approach will not be considered further in this note 1.1The approach used in [2] of simply dividing the χ2 contribution by three was also studied. Itwas also found to have only a small effect on the results.5Efficiency 0.72 0.74 0.76 0.78 0.8 0.82 0.84 0.86 0.88 0.9 0.92Ghost rate 00.020.040.060.080.10.120.14Figure 4: Efficiency versus ghost rate as a function of a cut on the χ2/ndof (blacktriangles) and the likelihood (red points). The points on the χ2/ndof graph fromleft to right correspond to cuts at 1.5, 2, 2.5, 3 , 4, 6, 8, 10, 15, ∞. Those on thelikelihood graph correspond to cuts at -10, -20, -30, -40, -50, -60, -70, -80, -90,-100 , −∞.Efficiency 0.72 0.74 0.76 0.78 0.8 0.82 0.84 0.86 0.88 0.9 0.92Ghost rate 00.020.040.060.080.10.120.14Figure 5: Efficiency versus ghost rate as a function of a cut on the likelihood (redpoints) and the case of reweighting the contribution from the track p(χ2) with a βfunction (black triangles).63.2 Performance as function of luminosityThe performance of the likelihood method and the χ2/ndof cut have also beencompared as a function of luminosity. Based on the studies in the previous sectionit was decided to apply a cut on the χ2/ndof at 6 or a cut on the likelihood at-60. In both cases at a luminosity of 2 × 1032 cm−2s−1 this reduces the ghostrate to 10.4 % for a 1 % loss in efficiency. The efficiency versus luminosity isshown in Fig. 6. For both methods, compared to the case of no cut, an increasingloss in efficiency is observed at higher luminosities. Due to the larger hit densitythere is an increased probability to wrongly assign hits to a track. This leadsto a degradation of the fit quality which can be seen in Fig. 7 where the meanprobability of χ2 is observed to decrease as the luminosity increases.Fig. 8 shows the ghost rate versus luminosity. For luminosities above 1033 cm−2s−1the likelihood method gives a lower ghost rate than the χ2/ndof cut.] -1s-2cm32Luminosity [ 102 4 6 8 10 12 14 16 18 20 %707580859095100Efficiency (Default) cut)2χEfficiency (Efficiency (likelihood)Figure 6: Efficiency versus luminosity. Note the suppressed scale.3.3 Performance for other track typesSince the algorithm is generic it can be applied, without modification, to the out-put of other tracking algorithms. Fig 9 - 11 show the results for the Downstreamtracking [4], the Velo-TT tracking [5] and the standalone T-seeding based on theTsa reconstruction framework [2, 6, 7]. It can be seen that in all cases the al-gorithm allows the ghost rate to be reduced for a small loss in inefficiency. As7] -1s-2cm32Luminosity [ 102 4 6 8 10 12 14 16 18 20)2 χmean p(00.050.10.150.20.250.30.350.40.450.5Figure 7: Mean p(χ2) versus luminosity.] -1s-2cm32Luminosity [ 102 4 6 8 10 12 14 16 18 20 %0102030405060708090100Ghost rate (Default) cut)2χGhost rate (Ghost rate (likelihood)Figure 8: Ghost-rate versus luminosity.was seen for the case of the long tracking the power of the algorithm is dilutedby the tail in the probability of χ2 distribution. The biggest separation of poweris seen for the Tsa seeding and the least for the Velo-TT tracking. The fact thatpoor separation is seen in the latter case is not surprising. First, this track typeprovides limited information since it is formed only from Velo and TT hits. Inaddition, inside the algorithm the track fit is run and used to discriminate betweencompeting candidates. This means that internally the algorithm already cuts on8one of the quantities used in the likelihood. However, the fact that some separa-tion is seen indicates that the use of the available information by the algorithm isnot optimal. Therefore, the performance could be improved by implementing acompetition between candidates based on the likelihoodLikelihood-100 -80 -60 -40 -20 010210310410Efficiency 0.45 0.5 0.55 0.6 0.65 0.7 0.75Ghost rate 00.050.10.150.20.25Figure 9: Likelihood for ghosts (points) and real (solid) tracks for the DownstreamTracking algorithm (left) and efficiency versus ghost rate (right). The points fromleft to right correspond to cuts at -6, -8 , -10, -15, -20, -30, -40, -50, -60, −∞.4 SummaryIn this note an algorithm for discriminating between real and ghost tracks basedon a likelihood technique has been studied with simulated data. It has been shownthat this approach leads to an improvement in the discrimination between realand ghost tracks compared to a simple cut on the χ2/ndof of the track. However,the improvement is small because the χ2/ndof is already a powerful discriminant.With real data the power of the χ2/ndof may be reduced because of incorrectmodeling of hit resolution and material in the detector. In this case the likelihoodapproach, by adding more information could be more performant.9likelihood-100 -80 -60 -40 -20 0110210310Efficiency 0.075 0.08 0.085 0.09 0.095 0.1Ghost rate 00.020.040.060.080.10.120.140.160.18Figure 10: Likelihood for ghosts (points) and real (solid) tracks for the Velo-TTalgorithm (left) and efficiency versus ghost rate (right). The points from left toright correspond to cuts at -6, -8 , -10, -15, -20, -30, -40, -50, -60, −∞. NotaBene, the low efficiency of this algorithm is because VELO track segments usedin the long tracking are excluded before the algorithm is run whilst the efficiencyis normalized to all tracks in the Velo-TT acceptance.In both approaches the performance is limited by the tail in the probability ofχ2/ndof distribution. A better treatment of this tail would lead to improved per-formance. Two effects are known to contribute to this tail:• Wrongly resolved L/R signs in the Outer Tracker [1].• Wrongly assigned hits from nearby tracks (Section 3.2).Further work is needed to improve the treatment of both effects in the track fit.References[1] M. Needham. Performance of the LHCb Track Reconstruction Software.LHCb-note 2007-144.10Likelihood-100 -80 -60 -40 -20 0110210310410Efficiency 0.78 0.8 0.82 0.84 0.86 0.88 0.9 0.92 0.94Ghost rate 00.010.020.030.040.050.060.070.08Figure 11: Likelihood for ghosts (points) and real (solid) tracks (left) and effi-ciency versus ghost rate (right) for the Tsa seeding algorithm. The points fromleft to right correspond to cuts at -10, -15, -20, -25, -30, -35, -40, −∞[2] R. Forty M. Needham. Standalone Track Reconstruction in the T-Stations.LHCb-note 2007-022.[3] M. Needham. Classification of Ghost Tracks. LHCb-note 2007-129.[4] O. Callot. Downstream Pattern Recognition. LHCb-Note 2007-026.[5] O. Callot et. al. VELO-TT track reconstruction. LHCb-note 2007-010.[6] R. Forty and M. Needham. Updated Performance of the T seeding. LHCb-note 2007-023.[7] M. Needham. The Tsa Reconstruction Framework. LHCb-Note 2007-037.11A Implementation of the Likelihood toolThe algorithm described above has been implemented as a tool in the TrackToolspackage. The code fragments below illustrate how to get and use the tool:# i n c l u d e ‘ ‘ T r a c k I n t e r f a c e s / I T r a c k M a n i p u l a t o r . h ’ ’/ / g e t t h e t o o lI T r a c k M a n i p u l a t o r ∗ l i k T o o l = t o o l<I T r a c k M a n i p u l a t o r >( ‘ ‘ T r a c k L i k e l i h o o d ’ ’ ) ;/ / u se i tTrack aTrack ;double l i k = l i k T o o l−>e x e c u t e ( aTrack ) ;For a long track the calculation takes 0.5 ms on a 64 bit 2.8 GHz Opteron proces-sor.Internally the tool delegates the task of calculating the number of expected hitsto two tools: the IHitExpectation in the case of IT, TT and OT and the IVelo-Expectation tool for the VELO. These tools are described in Appendix B. Thequality of the results given by these tools depends on what States are present onthe Track. Therefore, the accuracy of the result will be higher in Brunel whereStates at the z positions of all measurements are present. In addition, the infor-mation on the number of hits in the Silicon detectors above the high threshold isextracted from the Measurements on the track — which are not stored on theDST. These considerations mean the algorithm is run in Brunel and the outputadded to the ExtraInfo map of the track class by the TrackAddLikelihood al-gorithm and stored on the DST. For completeness the number of hits expected ineach of the tracking detectors are also stored. The following code fragment showhow gets and use this information from the track:Track aTrack ;double l i k = l i k T o o l−>i n f o (LHCb : : Track : : L i k e l i h o o d , 9 9 9 9 ) ;i f ( l i k < c u t V a l u e ){/ / l i k e l y t o be a g h o s t}12B Expected Hit ToolsTwo interfaces have been written that allow the number of expected hits in thetracking detectors to be estimated. The first is used by the OT, IT and TT and hasthe interface below:c l a s s I H i t E x p e c t a t i o n : v i r t u a l p u b l i c IAlgToo l{p u b l i c :/∗ ∗ s m a l l s t r u c t r e t u r n i n g h i t i n f o . . . .∗ /t y p e d e f s t r u c t {unsigned i n t nExpec ted ;unsigned i n t nFound ;double l i k e l i h o o d ;} I n f o ;/ / / R e t r i e v e i n t e r f a c e IDs t a t i c c o n s t I n t e r f a c e I D & i n t e r f a c e I D ( ) { re turn I I D I H i t E x p e c t a t i o n ; }/∗ ∗ R e t u r n s number o f h i t s e x p e c t e d , from z F i r s t t o i n f∗ @param aTrack R e f e r e n c e t o t h e Track t o t e s t∗ @return u n s i g n e d i n t number o f h i t s e x p e c t e d∗ /v i r t u a l unsigned i n t nExpec ted ( c o n s t LHCb : : Track & aTrack ) c o n s t = 0 ;/∗ ∗ R e t u r n s number o f h i t s e x p e c t e d∗ @param aTrack R e f e r e n c e t o t h e Track t o t e s t∗ @return I n f o i n f o i n c l u d i n g l i k e l i h o o d∗ /v i r t u a l I n f o e x p e c t a t i o n ( c o n s t LHCb : : Track & aTrack ) c o n s t = 0 ;} ;As can be seen two methods are defined in the interface. The first simply returnsthe number of hits expected in a given detector. The second returns a structure,which contains the number of expected hits, the number found and the likelihoodcontribution.For the VELO a second interface (IVeloExpectation) is used. This allows thenumber of r and φ hits that are expected to be calculated separately. For the VELOaccount must be taken of the fact that stations can be missed because the particle isthe product of a hyperon decay occurring after the start of the VELO acceptance.The interface below takes account of these requirements and in addition providesmethods that allow to calculate how many stations the particle traverses before thefirst hit seen on the track. The interface is as follows:c l a s s I V e l o E x p e c t a t i o n : v i r t u a l p u b l i c IAlgToo l{p u b l i c :/∗ ∗ Helper s t r u c t∗ /t y p e d e f s t r u c t {13unsigned i n t nR ;unsigned i n t nPhi ;} I n f o ;/ / / R e t r i e v e i n t e r f a c e IDs t a t i c c o n s t I n t e r f a c e I D & i n t e r f a c e I D ( ) { re turn I I D I V e l o E x p e c t a t i o n ; }/∗ ∗ R e t u r n s number o f h i t s e x p e c t e d , from z F i r s t t o endVe lo∗ @param aTrack R e f e r e n c e t o t h e Track t o t e s t∗ @return number o f h i t s e x p e c t e d ∗ /v i r t u a l i n t nExpec ted ( c o n s t LHCb : : Track & aTrack ) c o n s t = 0 ;/∗ ∗ R e t u r n s I n f o on h i t s e x p e c t e d , from z F i r s t t o endVe lo∗ @param aTrack R e f e r e n c e t o t h e Track t o t e s t∗ @return I n f o ∗ /v i r t u a l I V e l o E x p e c t a t i o n : : I n f oe x p e c t e d I n f o ( c o n s t LHCb : : Track & aTrack ) c o n s t = 0 ;/∗ ∗ R e t u r n s number o f h i t s e x p e c t e d , from z S t a r t t o z S t o p ∗∗ @param aTrack R e f e r e n c e t o t h e Track t o t e s t∗ @param z S t a r t −−> s t a r t o f scan range∗ @param z S t o p −−> end o f scan range∗ @return number o f h i t s e x p e c t e d ∗ /v i r t u a l i n t nExpec ted ( c o n s t LHCb : : Track & aTrack ,c o n s t double z S t a r t ,c o n s t double zS top ) c o n s t = 0 ;/∗ ∗ R e t u r n s I n f o on h i t s e x p e c t e d , from z S t a r t t o z S t o p∗∗ @param aTrack R e f e r e n c e t o t h e Track t o t e s t∗ @param z S t a r t −−> s t a r t o f scan range∗ @param z S t o p −−> end o f scan range∗ @return I n f o ∗ /v i r t u a l I V e l o E x p e c t a t i o n : : I n f oe x p e c t e d I n f o ( c o n s t LHCb : : Track & aTrack ,c o n s t double z S t a r t ,c o n s t double zS top ) c o n s t = 0 ;/∗ ∗ R e t u r n s number o f h i t s missed , from zBeamLine t o f i r s t H i t∗ @param aTrack R e f e r e n c e t o t h e Track t o t e s t∗ @return number o f h i t s m i s sed b e f o r e f i r s t h i t ∗ /v i r t u a l i n t nMissed ( c o n s t LHCb : : Track & aTrack ) c o n s t = 0 ;/∗ ∗ R e t u r n s number o f h i t s missed , from z t o f i r s t H i t∗ @param aTrack R e f e r e n c e t o t h e Track t o t e s t∗ @param z −−> s t a r t o f scan range∗ @return number o f h i t s m i s sed b e f o r e f i r s t h i t ∗ /v i r t u a l i n t nMissed ( c o n s t LHCb : : Track & aTrack ,c o n s t double z ) c o n s t = 0 ;/∗ ∗ R e t u r n s t r u e i f t r a c k p a s s s e s t h r o a g i v e n s e n s o r∗∗ @param aTrack R e f e r e n c e t o t h e Track t o t e s t@param sensorNum −−−> s e n s o r number∗ @return t r u e i f i n s i d e s e n s o r ∗ /v i r t u a l bool i s I n s i d e ( c o n s t LHCb : : Track & aTrack ,c o n s t unsigned i n t sensorNum ) c o n s t = 0 ;14

Identification of Ghost Tracks using a Likelihood Method

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