On the start value problem of the general track fit

1. On the start value problemof the general track fit M. de Jong

2. What is the problem? • General track fit is a non-linear problem • multiple solutions (local minima, saddle points, etc.) • requires iterative process • Probability density function non-Gaussian • only for small range of t (random background) • is not negative-definite (ARS token ring)¶ ¶ This could be solved by taking only first hit in each PMT (thesis R. Bruijn)

3. Traditional strategy • find start values phase space too large to scan • apply M-estimator fit enter regime where • apply Likelihood fit obtain ultimate angular resolution • Start values are obtained using a (linear) pre-fit • For an overview of the various (linear) pre-fits, see Karl Lyons’ talk at Colmar PAW

4. Angular resolution of pre-fits¶ K. Lyons number of events [a.u.] a [degrees] ¶ Atmospheric muon simulation (K. Lyons)

5. Alternative strategy • Scan part of the 5 dimensional phase space • grid of direction angles q and f • closed surface (W = 4p) • 3-parameter fit (x, y, t0) is linear¶ • Obtain complete set of solutions • detect (hidden) symmetries • e.g. local minima • select subset for subsequent fit(s) • subset should contain at least 1 good solution ¶ ANTARES-SOFT-2007-001

6. Procedure • choose grid angle (e.g. 5 degrees ≡ ~800 directions) • apply 1D clustering • make 3-parameter fit • remove outliers and repeat fit • sort solutions • limit subset to N (e.g. N = 10) • determine space angle between true track and each track in this subset Angular resolution ≡ smallest space angle between true track and each solution in subset

7. Angular resolution¶ number of events a [degrees] ¶ Atmospheric muon simulation (same as before)

8. comparison 5 degrees 5 degrees fewer events in tail a [degrees] a [degrees] • Median • Aart 6 degrees • Inertia Tensor 7 degrees • Direct Walk 9 degrees • Median • new method 4 degrees

9. cumulative distribution • Probabilities • 50% 3.6 degrees • 60% 4.2 degrees • 70% 5.3 degrees • 80% 7.8 degrees • 90% 19.0 degrees P(a ≤ amax degrees) amax [degrees]

10. Event classification if space angle between best quality solution and any other -but equally good- solution is larger than some number of degrees then event is classified as ambiguous

11. Event classification (II) • Discard event if there is 2nd solution, with: • P(c2,NDF) ≥ 0.01 • #hits ≥ #hits of best solution • Angular difference with best solution ≥ 20 degrees • Discard event if there is 2nd solution, with: • P(c2,NDF) ≥ 0.01 • #hits ≥ #hits of best solution – 1¶ • Angular difference with best solution ≥ 20 degrees ¶ This means that symmetry is broken by only 1 hit

12. cumulative distribution (II) • Probabilities • 50% 3.5 degrees • 60% 4.0 degrees • 70% 4.9 degrees • 80% 6.6 degrees • 90% 12.0 degrees all events class 1. (= 75%) P(a ≤ amax degrees) class 2. (= 45%) • Probabilities • 50% 3.2 degrees • 60% 3.5 degrees • 70% 3.9 degrees • 80% 4.7 degrees • 90% 6.5 degrees amax [degrees]

13. Summary • Alternative method to obtain start values • scan of directions within solid angle • List of solutions instead of ‘one-and-only’ • there is a solution in subset of 10 elements that is closer to true track than other available pre-fits • Detection of (hidden) symmetries • 90% of unambiguous events within 12 degrees from true track