A Kalman Filter for HADES This presentation will probably involve audience discussion, which will create action items. Use PowerPoint to keep track of these action items during your presentation In Slide Show, click on the right mouse button Select “Meeting Minder” Select the “Action Items” tab Type in action items as they come up Click OK to dismiss this box This will automatically create an Action Item slide at the end of your presentation with your points entered. A Kalman Filter for HADES Erik Krebs 17/11/2011 HADES Collaboration Meeting XXIII

Overview Theory of the Kalman filter. Using the Kalman filter with the segment fitter. Using the MDC wire information with the Deterministic Annealing Filter. Comparison with Current Tracking. Summary.

The Kalman Filter Set of mathematical equations to estimate the state of system perturbed by process noise. Least-squares estimator. Optimal estimator for linear systems. Works recursively on inaccurate measurements. Advantages of the Kalman Filter Recursive Approach: Measurements are processed as they arrive. Useful for real-time applications. Takes multiple scattering and energy loss (ionization and radiation) into account. Only small matrices need to be inverted.

Prediction (time update) Filter (measurement update) Initialization Initial estimate of state vector x0 and covariance matrix C0. k = { 1, .., n } Prediction (time update) Transport state vector xkk-1 Propagate covariance Ckk-1. Filter (measurement update) Take measurement mk into account. Compute state xk and covariance Ck. Smooth Backwards Update filtered state vectors xk and covariances Ck using all available measurements. The state vector x fully describes the internal state of the system. Xkk-1 : estimate of state vector before processing the measurement at time k. Xk: estimate of state vector after processing the measurement at time k. Xkn: smoothed estimate of state at time k using all available measurements.

Example for Prediction and Filter Steps

HADES Tracking Now Intermediate Planned Candidate Search Segment fitter Segment fitter MetaMatch MetaMatch MetaMatch Spline task Spline task Spline task Runge-Kutta task Kalman filter Kalman filter / Det. Annealing Filter

Segment Fitting and Kalman Filter Simulated Au+Au events at 1.25 AGeV with Geant. Four segment hits as measurements. Position resolution of segment hits Δx = 200 μm and Δy = 100 μm. Smeared Geant momentum by 10%. Cut tracks with 2 > 100.

Momentum Resolution Protons

Segment Fitter and Kalman Filter Positrons

Use Wire Information Challenge: Two measurements from neighbouring cells for the same hit. Fake hits. Kalman filter can’t handle competing hits. One Solution: Deterministic Annealing Filter. Extension of the Kalman filter. Measurements are assigned weights (probabilities). Iterate these steps: Run Kalman filter and smooth back using current weights. Recalculate weights according to distance of the measurements to the Kalman filter estimates. Lower annealing factor. Annealing avoids local minima.

Comparison with Current Tracking Protons Current tracking. Annealing filter with wire hits.

Comparison with Current Tracking Protons Current tracking. Annealing filter with wire hits.

Comparison with Current Tracking Positrons Electrons Current tracking. Det. Annealing filter.

Summary Kalman filter for segment and wire hits has been implemented. Includes multiple scattering and energy loss. Kalman filter needs good initail values. Det. Annealing Filter better than Runge-Kutta fit for high momenta and high theta tracks. Open Issues: Trace to META and vertex. Particle hypothesis. Initial momentum estimate (→ Spline?). Test with real data.

Thank You.

Fit Quality Current tracking. Annealing filter with wire hits.

Ionization Loss Protons Energy loss in Kalman filter Energy loss in Geant

Radiation and Ionization Loss for Positrons Energy loss in Kalman filter Energy loss in Geant

Protons Kalman filter Runge-Kutta with wire hits fit Det. Seg. fit Annealing Filter (DAF) Seg. fit and Kalman filter Encountered errors during reconstruction or 2 > 100

Protons KF + wire hits RK Fit Seg. Fit + KF DAF

Protons KF + wire hits RK Fit Seg. Fit + KF DAF

Positrons RK Fit Seg. Fit + KF DAF

Positrons RK Fit Seg. Fit + KF DAF

Electrons RK Fit Seg. Fit + KF DAF

Electrons RK Fit Seg. Fit + KF DAF