TIME-05 Workshop, 3-7 October 2005, Zurich 1

Probabilistic Data Probabilistic Data Association Filter for the Association Filter for the fast tracking in ATLAS fast tracking in ATLAS Transition Radiation TrackerTransition Radiation Tracker

Dmitry Emeliyanov, Dmitry Emeliyanov,

Rutherford Appleton Rutherford Appleton

LaboratoryLaboratory

TIME-05 Workshop, 3-7 October 2005, Zurich 2

ATLAS Transition Radiation ATLAS Transition Radiation TrackerTracker

Barrel

End-caps

• TRT is a part of the ATLAS Inner Detector• Provides stand-alone electron identification

• Straw drift tubes (4mm) – large number (~36) hits per track• Robust pattern recognition• Input for the Level 2 Trigger

TRT

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Combined Trigger tracking Combined Trigger tracking strategystrategy

• One of the possible strategies is to combine: – stand-alone track finding in Pixel

and SCT trackers (see M.Sutton’s talk on Friday) with

– subsequent propagation of found tracks into the TRT using concurrent “hit-to-track” association and track fitting – a.k.a. track following

• The TRT tracking must be robust:– high occupancy in TRT (15-50%)– intrinsic “left/right” ambiguities of

drift tube hits

• and fast (it’s trigger !) – recursive, “single-pass” algorithm required

Pixel

Semi-Conductor Tracker

TRT

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Probabilistic Data Association Probabilistic Data Association FilterFilter

• Proposed by Y.Bar-Shalom in 1973 for radar tracking, adapted for many applications (robotics, computer vision)

• Main advantage: PDAF is a recursive algorithm computationally similar to the Kalman filter with almost linear execution time

• Basic requirements/assumptions:– Measurements (hits) are arranged into groups (“scans”=layers)– Assumption 1: There is a single track-of-interest and at most

one hit per layer originates from this track– Assumption 2: All the other hits are caused by the background– Assumption 3: A track can be undetected in some layers

• The PDAF consists of the two main blocks: data association and track update

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The Probabilistic Data The Probabilistic Data AssociationAssociation

• Extrapolate track to the next TRT layer

• Set up a validation region around the extrapolated track (typical size ~ 8 mm)

• Using PDAF assumptions create a set of association hypotheses explaining the hit pattern observed inside the validation region

• Evaluate probabilities of “hit-to-track” association hypotheses

Track

Validation region

TRT hits

Hypo Hit Hit

B B

T B

B T

0H

1H

2H

1 2

hit assignment

1m 2m

iP iH

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Evaluation of hypothesis Evaluation of hypothesis probabilityprobability

• According to Bayes’ theorem, the posterior probability is:

Bp,Tp

iiii AhphHPP || is the observed hit pattern, is the hit assignment postulated by hypothesis h iA

iH• Prior hypothesis (=assignment) probability depends on track position w.r.t. straw and background model• Assuming hits to be statistically independent, the joint p.d.f. conditioned on assignment is the product of the hit p.d.f.s: )()()(| 11 iBiTiBi mpmpmpAhp

i

• are p.d.f.s of track- and background-induced hits

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The prior probability The prior probability calculationcalculation

• The probabilistic model for the background:– Uniform p.d.f. - the same probability for all drift

radii:

– Straws “produce” BG hits with constant probability– A straw hit by BG track is not sensitive to the real track

Bp

B

• With this model, the prior probability is: )(1 iPDBi

Straw is NOT hit by BG track

Hit efficiency

Probability of track passing through straw

straw

Track p.d.f.

DP

CB Rp /1 CR – drift tube radius

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The hit assignment The hit assignment probabilitiesprobabilities

The posterior probabilities of hit assignments (with L/R sign)

)()()( iiD iPC

– L/R residuals, – Gaussian p.d.f. of the residual

/i )(

The probability of the track not being detected (all hits are BG) CP 1

0

constant C ~ spatial density of BG hits: CB RC 2/

)()( /1/ iDi iPP

The normalization coefficient is:

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The PDAF algorithmThe PDAF algorithmhit residuals

asso-

ciation

proba-

bilities

Kalman filter

merging

covariance

correction

extrapolation

__

D0P

1P1P2P2P

1

1

2

2

S

X

~

X~

1. Track parameter update:__~

KXX

combined residual: //__

iiP 2. Track covariance update:

SPP 0

~

0 1TKDK

2___2//

iiPDvariance of the residuals:

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Similar algorithms: GSF and Similar algorithms: GSF and DAFDAF

• Gaussian Sum Filter (GSF):– GSF is equivalent to PDAF if all Gaussian components (track para-

meter estimates) are merged into a single Gaussian after each layer

– PDAF is faster because merging 1-dim residuals is faster than merging 5-dim track estimates

• Deterministic Annealing Filter (DAF):– DAF is an iterative algorithm which refines assignment

probabilities through passes of re-weighted forward/backward filtering with gradually lowered measurement variance (annealing)

– PDAF is a “single-pass” algorithm (no iterations/annealing) more suitable for trigger applications

– The PDAF employs more elaborated hit association method which explicitly allows for the “no-detection” hypothesis – a typical case in the TRT due to the spacing between straws (2.8 mm)

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eeID with PDAF-based ID with PDAF-based tracking tracking

• In general, electrons produce more transition radiation hits (with high energy deposits) than pions. For e/ separation, the tracking must provide:– - number of transition radiation (TR) hits on a track

– - total number of TRT hits on a track

• With PDAF, these numbers are estimated from association probabilities:

TRN

TN

K

kT kPKN

10 )(

K

k TRiiiTR kPkPN

1 }{

)()(the sum over all

validated TR hits

K - total number of TRT layers

crossed by the track

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PDAF performance evaluationPDAF performance evaluation

• Studies have been done on ATLAS Monte Carlo data:– Signal events: electrons, =20 GeV – Background events: inelastic pp interactions

• Three data sets:1. Signal events only2. Signal + <4.5> inel. interactions (Poisson) ~ “low lumi” regime3. Signal + <23> inel. interactions (Poisson) ~ “high lumi” regime

• Performance criteria:• Quality of “hit-to-track” association – average

assignment probabilities for the true and background hits– hits are identified according to Monte Carlo truth information

• Robustness of the PDAF – the performance versus occupancy

Tp

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The resultsThe results

Data Set Signal“Low-lumi”

“High-lumi”

TRT region BR EC BR EC BR EC

Average occupancy 0.010.03

0.17

0.16

0.6 0.5

Average , true hits 0.98 0.93 0.97 0.93 0.95

0.93

Average , BG hits 0.52 0.52 0.50 0.49 0.490.49

Ratio 0.23 0.23 0.23 0.24 0.250.25

iP

iP

• Tests have been done for two TRT regions:– Barrel (BR), – Endcap (EC),

7.05.21.1

TTR NN /

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PDAF timing measurementsPDAF timing measurements• The CPU time per track is measured on 2.4 GHz Pentium

4 using the Kalman filter timing as a reference

PDAF

Kalman filter

Algo-rithm

Average time per

track

PDAF 2.2 ms

KF 1.5 ms

0

5 ms

5 ms

100

0 100][01.01.1~ msNhits

][018.02.1~ msNhits

number of hits in val.reg.

number of hits in val.reg.

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DiscussionDiscussion

• Performance: The PDAF identifies true hits with 93-97% efficiency and provides 50% background rejection

• Lower efficiency in the endcap TRT regions is due to the large amount of material traversed by a track before TRT – higher probability of energy loss via Bremsstrahlung

• The ratio of TR hits/all hits calculated using PDAF association probabilities is almost independent from occupancy – good for reliable electron identification

• Timing: Computational requirements of the PDAF are comparable with those of the standard Kalman filter

• The probabilistic data association takes 30% of CPU time, track update – 20 %, track extrapolation – 50 %

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ConclusionConclusion

• The new algorithm based on the Probabilistic Data Association Filter (PDAF) has been developed for track reconstruction in ATLAS Transition Radiation Tracker

• The tests on simulated data have shown that the PDAF is an efficient and not too slow alternative to the Kalman filter and can be used in concurrent track finding and fitting algorithms