On the start value problem of the general track fit

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# On the start value problem of the general track fit - PowerPoint PPT Presentation

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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### On the start value problemof 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
• 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)

• 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¶

K. Lyons

number of events [a.u.]

a [degrees]

¶ Atmospheric muon simulation (K. Lyons)

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
• 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¶

number of events

a [degrees]

¶ Atmospheric muon simulation (same as before)

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
• 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

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)
• 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)
• 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
• 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