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

1. Probabilistic Data Association Filter for the fast tracking in ATLAS Transition Radiation Tracker Dmitry Emeliyanov, Rutherford Appleton Laboratory TIME-05 Workshop, 3-7 October 2005, Zurich

2. Barrel End-caps ATLAS Transition Radiation Tracker • TRT is a part of the • ATLAS Inner Detector • Provides stand-alone • electron identification TRT • Straw drift tubes (4mm) – large • number (~36) hits per track • Robust pattern recognition • Input for the Level 2 Trigger TIME-05 Workshop, 3-7 October 2005, Zurich

3. Combined Trigger tracking strategy • 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 TRT Semi-Conductor Tracker Pixel TIME-05 Workshop, 3-7 October 2005, Zurich

4. Probabilistic Data Association Filter • 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 TIME-05 Workshop, 3-7 October 2005, Zurich

5. The Probabilistic Data Association Validation region 1 2 • 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 TRT hits hit assignment TIME-05 Workshop, 3-7 October 2005, Zurich

6. Evaluation of hypothesis probability • According to Bayes’ theorem, the posterior probability is: is the observed hit pattern, is the hit assignment postulated by hypothesis • 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: • are p.d.f.s of track- and background-induced hits TIME-05 Workshop, 3-7 October 2005, Zurich

7. The prior probability calculation • 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 – drift tube radius • With this model, the prior probability is: straw Track p.d.f. Straw is NOT hit by BG track Probability of track passing through straw Hit efficiency TIME-05 Workshop, 3-7 October 2005, Zurich

8. The hit assignment probabilities The posterior probabilities of hit assignments (with L/R sign) – L/R residuals, – Gaussian p.d.f. of the residual The normalization coefficient is: constant C ~ spatial density of BG hits: The probability of the track not being detected (all hits are BG) TIME-05 Workshop, 3-7 October 2005, Zurich

9. combined residual: The PDAF algorithm 1. Track parameter update: hit residuals asso- ciation proba- bilities 2. Track covariance update: merging Kalman filter covariance correction variance of the residuals: extrapolation TIME-05 Workshop, 3-7 October 2005, Zurich

10. Similar algorithms: GSF and DAF • 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) TIME-05 Workshop, 3-7 October 2005, Zurich

11. e/p-ID with PDAF-based tracking • In general, electrons produce more transition radiation hits (with high energy deposits) than pions. For e/p 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: • total number of TRT layers • crossed by the track the sum over all validated TR hits TIME-05 Workshop, 3-7 October 2005, Zurich

12. PDAF performance evaluation • Studies have been done on ATLAS Monte Carlo data: • Signal events: electrons, =20 GeV • Background events: inelastic pp interactions • Three data sets: • Signal events only • Signal + <4.5> inel. interactions (Poisson) ~ “low lumi” regime • 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 TIME-05 Workshop, 3-7 October 2005, Zurich

13. The results • Tests have been done for two TRT regions: • Barrel (BR), • Endcap (EC), TIME-05 Workshop, 3-7 October 2005, Zurich

14. PDAF timing measurements • The CPU time per track is measured on 2.4 GHz Pentium 4 using the Kalman filter timing as a reference 5 ms PDAF number of hits in val.reg. 0 100 5 ms Kalman filter number of hits in val.reg. 0 100 TIME-05 Workshop, 3-7 October 2005, Zurich

15. Discussion • 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 % TIME-05 Workshop, 3-7 October 2005, Zurich

16. Conclusion • 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 TIME-05 Workshop, 3-7 October 2005, Zurich