A general assistant tool for the checking results from Monte Carlo simulations Koi, Tatsumi...
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Transcript of A general assistant tool for the checking results from Monte Carlo simulations Koi, Tatsumi...
A general assistant tool A general assistant tool for the checking results for the checking results
from Monte Carlo simulationsfrom Monte Carlo simulations
A general assistant tool A general assistant tool for the checking results for the checking results
from Monte Carlo simulationsfrom Monte Carlo simulationsKoi, TatsumiKoi, TatsumiSLAC/SCCSSLAC/SCCS
Contents• Motivation• Precision and Accuracy• Central Limit Theorem• Testing Method• Current Status of Development• Summary
Motivation• After a Monte Carlo simulation, we get an answer. Ho
wever how to estimate quality of the answer.
What we must remember is• Large number of history does not valid result of simul
ation.• Small Relative Error does not valid result of simulatio
n
Motivation (Cont.)• To provide “statistical information to as
sist establishing valid confidence intervals for Monte Carlo results” for users, something like MCNPs did.
Subject of this study• Precision of the Monte Carlo
simulation• Accuracy of the result is NOT a
subject of this study
At first we have to define Precision and Accuracy of simulations
True Value Mote Carlo Results
AccuracyPrecision
Precision and Accuracy• Precision: Uncertainty caused by statistical
fluctuation• Accuracy: Difference between expected value
and true physical quantity.
Subject of this study (Cont.)
• Precision of the Monte Carlo simulation is subject of this study.
• To state accuracy of simulations, we should consider details of simulation, i.e., uncertainties of physical data, modeling of physical processes, variance reduction techniques and so on.
• To make a generalized tool, we have to limit subjects only for precision.
Accuracy is a subject for most of presentations in this workshop.
Principal of this study is
Central Limit Theorem
Central Limit Theorem• Every data which are influenced by
many small and unrelated random effects has normally distribution.
• The estimated mean will appear to be sampled from normal distribution with a KNOWN standard deviation when N approaches infinity.
N
Central Limit Theorem (Cont.)
• Therefore, We have to check that N have approached infinity in the sense of the CLT, or not.
• This corresponds to the checking the complete sampling of interested phase space has occurred, or not.
This is not a simple static test
butcheck of results from nature of Monte Carlo
simulations
Checking Values• Mean
• Variance and Standard Deviation
• Relative error
• Variance of Variance
11
2
2
N
xxS
N
ii
N
iixN
x1
1
N
SSwherex
SR xx
22
2
,
4
2
x
x
S
SSVOV
Checking Values (Cont.)
• Figure of Merit
• Scoring EfficiencyRintrinsic and Refficiency
• Shift
• SLOPEFit to the Largest history scores
TRFOM
2
1
NSxxSHIFT i23
2
N
historiesZERONONofnumberq
What we check?• Behavior of MEAN• Values of R• Time profile of R• Values of VOV• Time profile of VOV• Time profile of FOM• Behavior of FOM• Value of SLOPE• Value of SHIFT
• Effect of the largest history score occurs on the next history.– MEAN– R (Rintrinsic and Refficiency)
– VOV– FOM– SHIFT
Boolean Answer
Numeric Answer
Of cause, Boolean check is carried out
mathematically (statistically)
valuebehavior
time profile
For behaviors and time profiles check
• Derive Pearson’s r from data (results and theoretical values)– r=1(-1), perfect positive (negative)
correlation– r=0, uncorrelated
• null hypothesis is set to uncorrelated• Then, follows student t
distribution of degree of freedom • Checking significance of r with null hypothesis.• Rejection level of null hypothesis is 68.28% (1σ)
212 rNrt 2N
Example• Checking value: Observable Energy of Sampling Calorimeter.• Material
– Pb (Lead)-Scinitillator• Thickens
– Pb: 8.0 mm/layer, Sci: 2.0 mm/layer• Layers
– 120 layers– 1 m x 1 m – interaction surface
• Beam– Electon 4 GeV
• Range Cuts– 1 mm
Pb
8mm
2mm
Sci.
・・・・・・・・e-
SD
9.2
9.3
9.4
9.5
9.6
9.7
9.8
9.9
10
0 20 40 60 80 100 120
SD
Example 100 histories
mean
77
77.5
78
78.5
79
79.5
80
0 20 40 60 80 100 120
mean
R
00.0020.0040.0060.0080.01
0.0120.0140.0160.0180.02
0 20 40 60 80 100 120
R
VOV
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0 20 40 60 80 100 120
vov
SD
VOV
MEAN
R
Does not pass most of Boolean tests
SD
99.29.49.69.810
10.210.410.610.8
11
0 200 400 600 800 1000 1200
SD
vov
00.00050.001
0.00150.002
0.00250.003
0.00350.004
0.00450.005
0 200 400 600 800 1000 1200
vov
R
0
0.001
0.002
0.003
0.004
0.005
0.006
0.007
0 200 400 600 800 1000 1200
R
mean
77
77.5
78
78.5
79
79.5
80
0 200 400 600 800 1000 1200
mean
Example 1,000 histories
SD
VOV
MEAN
R
Does not pass some of Boolean tests
vov
00.000050.0001
0.000150.0002
0.000250.0003
0.000350.0004
0.000450.0005
0 2000 4000 6000 8000 10000 12000
vov
R
00.00020.00040.00060.00080.001
0.00120.00140.00160.00180.002
0 2000 4000 6000 8000 10000 12000
R
SD
99.29.49.69.810
10.210.410.610.8
11
0 2000 4000 6000 8000 10000 12000
SD
mean
77
77.5
78
78.5
79
79.5
80
0 2000 4000 6000 8000 10000 12000
mean
Example 10,000 histories
SD
VOV
MEAN
R
Does not pass one of Boolean tests (SLOPE check)
How to apply Energy Spectrum estimation
etc.• Checking each
confidence level of P1, P2, P3, P4,,,,
• Of course, scoring efficiency becomes low.
P1
P2
P3
P4
E
V/E
Unfortunately, this tool does not work well with
some deterministic variance reduction
techniques.This is come from
limitation of CLT (means some variance are
required for distribution), so that we can not over
come.
And some simulations becomes deterministic
without awaking of users.Please check your
simulation carefully.
Current Status of Development
• Most part of developments has been done.
• Following items are remained under development.– Output of testing result– Class or function for minimization of
multi dimensional functions
We want to include this tool in Geant4
butwhat category is suite for this
tool?
Run, SD, Hits and its collections,
Tally??
Summary• We have successfully developed a gene
ral assistant tool for the checking the results from Monte Carlo simulations like MCNPs.
• Through this tool, users easily know the confidence intervals for Monte Carlo results.