Transcript Statistics Update David Forrest University of Glasgow May 5th 2009 The Problem We calculate 4D emittance from the fourth root of a determinant of a.

Statistics Update
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 1% error.
The problem is compounded because our data is
highly correlated between two trackers.
2
How We Mean To Proceed
We 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
3) Empirical approach: large number of
simulations to plot se 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 e.
• I calculate the fractional change in emittance De.
• I repeat ~500 times and plot distribution of all De
for each beam.
• Carried out about 15,000 simulations on Grid
(8 beams x 1700 simulations/beam plus repeats)
4
8pi – N=1000 events
De/e
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Results
s
Ds
de/e
rms
N
1k
0.0897
0.00417
1.726
0.1102
450
2k
0.0613
0.00330
1.735
0.0818
261
10k
0.0329
0.00183
1.731
0.0340
242
1k
0.0168
0.00065
0.084
0.0169
545
2k
0.0106
0.00037
0.0802
0.0110
545
10k
0.0054
0.00018
0.0803
0.0054
545
1k
0.0117
0.00051
-0.0022
0.0124
421
2k
0.0083
0.00033
-0.0034
0.0092
426
10k
0.0040
0.00025
-0.0050
0.0040
320
0.2p
1.5p
2.5p
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Results-2
s
Ds
de/e
rms
N
1k
0.0114
0.00048
-0.022
0.0117
513
2k
0.0079
0.00031
-0.025
0.0081
437
10k
0.0036
0.00016
-0.026
0.0036
323
1k
0.0095
0.00037
-0.046
0.0097
440
2k
0.0066
0.00020
-0.050
0.0068
545
10k
0.0032
0.00015
-0.051
0.0031
340
1k
0.0073
0.00034
-0.071
0.0079
358
2k
0.0064
0.00023
-0.072
0.0067
500
10k
0.0026
0.00017
-0.072
0.0028
176
3.0p
4.0p
6.0p
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Results-3
s
Ds
de/e
rms
N
1k
0.0091
0.00031
-0.081
0.0093
549
2k
0.0071
0.00035
-0.083
0.0070
426
10k
0.0031
0.00012
-0.081
0.0032
547
1k
0.0097
0.00037
-0.081
0.0102
540
2k
0.0069
0.00025
-0.085
0.0068
541
10k
0.0033
0.00016
-0.085
0.0034
359
8.0p
10.0p
8
0.2p
9
1.5p
10
2.5p
11
3.0p
12
4.0p
13
6.0p
14
8.0p
15
10.0p
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K values
Beam
K
dK
C
dC
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
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sDe/e=K/sqrt(N)
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Sans pencil beam
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Biggest Remaining Questions
• > unity for pencil beam – thoughts?
• Turn stochastics off and what do you get?
– I have run 400 simulations like this but
they failed, will investigate and try again.
• Can also look at transport matrices thanks
to Chris – haven’t had chance yet.
• Question last week about longitudinal
emittance. Haven’t looked into this.
• x-px, y-py, beta-z checks
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