Transcript Statistical Errors
Statistics
David Forrest University of Glasgow May 5 th 2009 1
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
2
highly correlated between two trackers.
How We Mean To Proceed
W e 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
s e
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 5
before
Checks X, X’
after 0.2pi
2.5pi
6
before
Checks X, X’
after 8.0pi
10.0pi
7
Checks – beta function
Expected beta in absorbers ~420mm, solenoid 330 mm after matching
0.2
p
2.5
p
4.0
p
10.0
p 8
Ds
Results
Events
0.2
p s 1000 2000 10000
1.5
p 1000 2000 10000
2.5
p 1000 2000 10000 0.0897
0.0613
0.0329
0.0168
0.0106
0.0054
0.0117
0.0083
0.0040
0.00417
0.00330
0.00183
0.00065
0.00037
0.00018
0.00051
0.00033
0.00025
de/e 1.726
1.735
1.731
0.084
0.0802
0.0803
-0.0022
-0.0034
-0.0050
rms Sims
0.1102
0.0818
0.0340
0.0169
0.0110
0.0054
0.0124
0.0092
0.0040
450 261 242 545 545 545 421 426 320 9
Events
3.0
p 1000 2000 10000
4.0
p 1000 2000 10000
6.0
p 1000 2000 10000 s 0.0114
0.0079
0.0036
0.0095
0.0066
0.0032
0.0073
0.0064
0.0026
Ds
Results-2
de/e
rms
0.00048
0.00031
0.00016
0.00037
0.00020
0.00015
0.00034
0.00023
0.00017
-0.022
-0.025
-0.026
-0.046
-0.050
-0.051
-0.071
-0.072
-0.072
0.0117
0.0081
0.0036
0.0097
0.0068
0.0031
0.0079
0.0067
0.0028
Sims
513 437 323 440 545 340 358 500 176 10
Events
8.0
p 1000 2000 10000
10.0
p 1000 2000 10000 s 0.0091
0.0071
0.0031
0.0097
0.0069
0.0033
Results-3
Ds de/e 0.00031
0.00035
0.00012
-0.081
-0.083
-0.081
0.00037
0.00025
0.00016
-0.081
-0.085
-0.085
rms
0.0093
0.0070
0.0032
0.0102
0.0068
0.0034
Sims
549 426 547 540 541 359 11
0.2
p 12
1.5
p 13
2.5
p 14
3.0
p 15
4.0
p 16
6.0
p 17
8.0
p 18
10.0
p 19
Beam 0.2
1.5
2.5
3.0
4.0
6.0
8.0
10.0
K
2.533
0.481
0.351
0.356
0.282
0.247
0.287
0.293
d
K
0.188
0.024
0.023
0.019
0.015
0.016
0.014
0.016
K values
C
0.00727
0.00042
0.00052
0.00051
0.00038
0.00024
0.00025
0.00041
s
K
d
C
0.00325
1
N
C
0.00036
0.00043
0.00031
0.00027
0.00029
0.00022
0.00028
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s De/e =K/sqrt(N) 21
Sans pencil beam
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Physical Meaning (J Cobb)
• There is a physical meaning to this K value • By usual error formula, assuming no correlations:
f
s
f
2 D e e
in
e e
out in
e
out
e
in
e
in
2 s e e
o u t out
• So without correlations, we have this factor, normally >1, eg if f=-0.08, we get a factor of 1.29
s e
in
e
in
s
f
2 s
f
2 e e
out in
( 1 s e
o u t
e
out f
) 2 2 2
N
2 1
N
1 e e
out in N
2 s e e
in in
( 1 1
f
) 2 2 2
N
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Physical Meaning
• However, there are correlations between input emittance and output emittance, so we include a correlation factor, k corr . The s sim I measure includes this also.
s
sim
k corr
s
nocorr
k corr
K
e
in
e
out
2 2 ( 1
f
)
k corr
1
N
24
Correlation factor
Preliminary 25
How many muons do we need?
• We want to measure to an error of 0.1% Beam (pi mrad) No correlations (10 6 events) 15 With correlations (10 5 events)
64
0.2
1.5
2.5
3.0
4.0
6.0
8.0
10.0
2.4
2.0
2.0
1.8
1.7
1.7
1.7
2.3
1.2
1.3
0.8
0.6
0.8
0.8
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Conclusions
• We need of order 10 5 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 27