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)¶

( )f t

2

2

( )f t

t

( )0

f tt

¶ This could be solved by taking only first hit in each PMT (thesis R. Bruijn)

Traditional strategy

1. find start values phase space too large to scan2. apply M-estimator fit enter regime where3. apply Likelihood fit obtain ultimate angular resolution

( )0

f tt

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¶

[degrees]

num

ber o

f eve

nts

[a.u

.]

K. Lyons

¶ Atmospheric muon simulation (K. Lyons)

Alternative strategy

Scan part of the 5 dimensional phase space– grid of direction angles and • closed surface ( = 4)• 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

1. choose grid angle (e.g. 5 degrees ≡ ~800 directions)2. apply 1D clustering3. make 3-parameter fit4. remove outliers and repeat fit5. sort solutions6. limit subset to N (e.g. N = 10)7. 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¶

[degrees]

num

ber o

f eve

nts

¶ Atmospheric muon simulation (same as before)

comparison

[degrees] [degrees]

5 degrees5 degrees

Median• Aart 6 degrees• Inertia Tensor 7 degrees• Direct Walk 9 degrees

Median• new method 4 degrees

fewer eventsin tail

cumulative distribution

max [degrees]

P(≤

max

deg

rees

)

Probabilities 50% 3.6 degrees 60% 4.2 degrees 70% 5.3 degrees 80% 7.8 degrees 90% 19.0 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)

1. Discard event if there is 2nd solution, with:– P(2,NDF) ≥ 0.01– #hits ≥ #hits of best solution– Angular difference with best solution ≥ 20 degrees

2. Discard event if there is 2nd solution, with:– P(2,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)

all eventsclass 1. (= 75%)class 2. (= 45%)

max [degrees]

P(≤

max

deg

rees

)Probabilities 50% 3.5 degrees 60% 4.0 degrees 70% 4.9 degrees 80% 6.6 degrees 90% 12.0 degrees

Probabilities 50% 3.2 degrees 60% 3.5 degrees 70% 3.9 degrees 80% 4.7 degrees 90% 6.5 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