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Statistics
David Forrest
University of Glasgow
May 5th 2009
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The Problem
We calculate 4D emittance from the fourth root of a
determinant of a matrix of covariances...We want to
measure fractional change in emittance with 0.1%
error. The problem is compounded because our data is
highly correlated between two trackers.
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How We Mean To ProceedWe assume that we will discover a formula that takes the form Sigma=K*(1/sqrt(N)) where K is some constant or parameter to be determined. How do we determine K?
1) First Principles: do full error propagation of cov matrices → difficult calculation
2) Run a large number of G4MICE simulations, using the Grid, to find the standard deviation for every element in the covariance matrix → Toy Monte Carlo to determine error on emittance
3) Empirical approach: large number of simulations to plot versus 1/sqrt(N), identifying K (this work)
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What I’m Doing
• 3 absorbers (Step VI), G4MICE, 4D Transverse Emittance
• I plot 4D Transverse Emittance vs Z for some number of events N, for beam with input emittance .
• I calculate the fractional change in emittance .• I repeat ~500 times and plot distribution of all
for each beam.• Carried out about 15,000 simulations on Grid (8
beams x 1700 simulations/beam plus repeats)
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8pi – N=1000 events
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Checks - X, X’before after
2.5pi
0.2pi
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Checks - X, X’before after
10.0pi
8.0pi
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Checks – beta functionExpected beta in absorbers ~420mm, solenoid 330 mm after matching
0.2
4.0
2.5
10.0
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ResultsEvents rms Sims
0.2
1000 0.0897 0.00417 1.726 0.1102 450
2000 0.0613 0.00330 1.735 0.0818 261
10000 0.0329 0.00183 1.731 0.0340 242
1.5
1000 0.0168 0.00065 0.084 0.0169 545
2000 0.0106 0.00037 0.0802 0.0110 545
10000 0.0054 0.00018 0.0803 0.0054 545
2.5
1000 0.0117 0.00051 -0.0022 0.0124 421
2000 0.0083 0.00033 -0.0034 0.0092 426
10000 0.0040 0.00025 -0.0050 0.0040 320
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Results-2Events rms Sims
3.0
1000 0.0114 0.00048 -0.022 0.0117 513
2000 0.0079 0.00031 -0.025 0.0081 437
10000 0.0036 0.00016 -0.026 0.0036 323
4.0
1000 0.0095 0.00037 -0.046 0.0097 440
2000 0.0066 0.00020 -0.050 0.0068 545
10000 0.0032 0.00015 -0.051 0.0031 340
6.0
1000 0.0073 0.00034 -0.071 0.0079 358
2000 0.0064 0.00023 -0.072 0.0067 500
10000 0.0026 0.00017 -0.072 0.0028 176
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Results-3Events rms Sims
8.0
1000 0.0091 0.00031 -0.081 0.0093 549
2000 0.0071 0.00035 -0.083 0.0070 426
10000 0.0031 0.00012 -0.081 0.0032 547
10.0
1000 0.0097 0.00037 -0.081 0.0102 540
2000 0.0069 0.00025 -0.085 0.0068 541
10000 0.0033 0.00016 -0.085 0.0034 359
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0.2
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1.5
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2.5
15
3.0
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4.0
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6.0
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8.0
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10.0
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K values
Beam K K C C
0.2 2.533 0.188 0.00727 0.00325
1.5 0.481 0.024 0.00042 0.00036
2.5 0.351 0.023 0.00052 0.00043
3.0 0.356 0.019 0.00051 0.00031
4.0 0.282 0.015 0.00038 0.00027
6.0 0.247 0.016 0.00024 0.00029
8.0 0.287 0.014 0.00025 0.00022
10.0 0.293 0.016 0.00041 0.00028
CN
K 1
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=K/sqrt(N)
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Sans pencil beam
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Physical Meaning (J Cobb)
• There is a physical meaning to this K value
• By usual error formula, assumingno correlations:
• So without correlations, we have this factor, normally >1, eg if f=-0.08, we get a factor of 1.29
Nf
Nf
N
N
f
f
in
outf
outin
inoutin
outf
in
out
in
inout
in
outin
inout
12)1(
2)1(
2
1
1
2
2
2
22
222
2
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Physical Meaning
• However, there are correlations between input emittance and output emittance, so we include a correlation factor, kcorr. The sim I measure includes this also.
2
1)1(2
out
incorr
corrnocorrcorrsim
Kk
Nkfk
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Correlation factor
Preliminary
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How many muons do we need?• We want to measure to an error of 0.1%
Beam (pi mrad) No correlations (106 events)
With correlations (105 events)
0.2 15 64
1.5 2.4 2.3
2.5 2.0 1.2
3.0 2.0 1.3
4.0 1.8 0.8
6.0 1.7 0.6
8.0 1.7 0.8
10.0 1.7 0.8
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Conclusions
• We need of order 105 muons to achieve 0.1% error on fractional change in emittance
• Simulations in place for doing a toy Monte Carlo study, to propagate errors from elements of covariance matrix
