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On the start value problem of the general track fit. M. de Jong. 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

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what is the problem
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)

traditional strategy
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
angular resolution of pre fits
Angular resolution of pre-fits¶

K. Lyons

number of events [a.u.]

a [degrees]

¶ Atmospheric muon simulation (K. Lyons)

alternative strategy
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

procedure
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

angular resolution
Angular resolution¶

number of events

a [degrees]

¶ Atmospheric muon simulation (same as before)

comparison
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
cumulative distribution
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]

event classification
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

event classification ii
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

cumulative distribution ii
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]

summary
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