G4ConvergenceTester and ExN03Con KOI, Tatsumi SLAC National Accelerator Laboratory 1Geant4...

Geant4 Collaboration Workshop 2010-10-6 1

G4ConvergenceTester and

ExN03Con

KOI, TatsumiSLAC National Accelerator Laboratory


"G4ConvergenceTester“and “ExN03Con”

• G4ConvergenceTester provides several information assisting user understanding of convergence level of his/her result, like MCNPs.

• “ExN03Con” is an example of the usage of G4ConvergenceTester.– examples/extended/analysis/N03Con/

• They have been included in Geant4 since v9.0 (2007-Jun)


How to Use#include "G4ConvergenceTester.hh“void ExN03RunAction::BeginOfRunAction(const G4Run* aRun){,,,,,, Eabs_tally = new G4ConvergenceTester(); Egap_tally = new G4ConvergenceTester(); Labs_tally = new G4ConvergenceTester(); Lgap_tally = new G4ConvergenceTester();}void ExN03RunAction::fillPerEvent(G4double EAbs, G4double EGap, G4double LAbs, G4double LGap){,,,,, Eabs_tally->AddScore( EAbs ); Egap_tally->AddScore( EGap ); Labs_tally->AddScore( LAbs ); Lgap_tally->AddScore( LGap );}void ExN03RunAction::EndOfRunAction(const

G4Run* aRun){,,,,, Eabs_tally->ShowResult(); Eabs_tally->ShowHistory(); Egap_tally->ShowResult(); Egap_tally->ShowHistory();

Labs_tally->ShowResult(); Labs_tally->ShowHistory(); Lgap_tally->ShowResult(); Lgap_tally->ShowHistory();}


Output ShowResult()values and logical tests

EFFICIENCY = 0.996338MEAN = 368.83VAR = 3627.76SD = 60.2309R = 0.0025516SHIFT = -86.9154VOV = 0.00310814FOM = 2201.43THE LARGEST SCORE = 431.408 and it happend at 1425th eventAffected Mean = 368.846 and its ratio to orignal is 1.00004Affected VAR = 3627.83 and its ratio to orignal is 1.00002Affected R = 0.00255121 and its ratio to orignal is 0.999846Affected SHIFT = -86.9072 and its ratio to orignal is 0.999905Affected FOM = 2201.43 and its ratio to orignal is 1

MEAN distribution is not RANDOMr follows 1/std::sqrt(N)r is monotonically decrease 1r is less than 0.1. r = 0.0025516VOV follows 1/std::sqrt(N)VOV is monotonically decrease 1FOM distribution is not RANDOMSLOPE is large enoughThis result passes 6 / 8 Convergence Test.


Output ShowHistory()trend

i/16 till_ith mean var sd r vov fom shift e r2eff r2int1 255 368.253 3351.08 57.8885 0.00982485 0.0426828 2381.54 -75.4414 1 0 9.61507e-052 511 366.976 3686.51 60.7166 0.00731197 0.0196636 2172.34 -76.4552 0.998047 3.82216e-06 4.95383e-053 767 366.078 3841.16 61.9771 0.00610909 0.0144748 2075.49 -82.2678 0.996094 5.10621e-06 3.21662e-054 1023 366.095 3715.83 60.9576 0.00520336 0.0103351 2133.71 -78.5195 0.99707 2.86943e-06 2.41791e-055 1279 367.283 3592.01 59.9334 0.00456103 0.00905412 2216.23 -80.3134 0.996875 2.44906e-06 1.83376e-056 1535 365.548 3973.11 63.0326 0.00439972 0.0073182 1990.73 -84.6719 0.996094 2.5531e-06 1.67918e-057 1791 365.964 4016.7 63.3774 0.00409098 0.00677035 1969.38 -89.0621 0.99442 3.13151e-06 1.35952e-058 2047 366.099 3942.82 62.7919 0.00379001 0.00596075 2005.12 -88.1626 0.994629 2.63677e-06 1.17204e-059 2303 366.693 3854.07 62.0812 0.00352709 0.00543865 2053.75 -88.5994 0.995226 2.08212e-06 1.03528e-0510 2559 367.151 3818.23 61.7918 0.00332634 0.00491706 2075.29 -88.5895 0.995313 1.83968e-06 9.22051e-0611 2815 367.211 3770.15 61.4015 0.00315099 0.00441452 2105.31 -87.405 0.995739 1.51974e-06 8.40546e-0612 3071 367.092 3835.82 61.934 0.003044 0.00398929 2069.86 -87.876 0.995768 1.38338e-06 7.87952e-0613 3327 367.861 3731.13 61.083 0.00287836 0.00377558 2133.28 -87.6999 0.996094 1.17836e-06 7.10409e-0614 3583 368.469 3692.83 60.7687 0.00275483 0.00360362 2160.84 -88.4698 0.995815 1.17267e-06 6.4143e-0615 3839 368.856 3610.19 60.0849 0.00262871 0.0033742 2213.45 -87.4147 0.996094 1.02124e-06 5.88707e-0616 4095 368.83 3627.76 60.2309 0.0025516 0.00310814 2202.06 -86.9154 0.996338 8.97356e-07 5.61173e-06


• Following slides are my presentation at 2005 Geant4 Collaboration workshop

A general assistant tool for the checking results

from Monte Carlo simulations

A general assistant tool for the checking results

from Monte Carlo simulationsKoi, TatsumiSLAC/SCCS

7Geant4 Collaboration Workshop 2010-10-6

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. However how to estimate quality of the answer.

What we must remember is• Large number of history does not valid result

of simulation.• Small Relative Error does not valid result of

simulation


Motivation (Cont.)• To provide “statistical information

to assist 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. 13Geant4 Collaboration

Workshop 2010-10-6

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 20Geant4 Collaboration

Workshop 2010-10-6

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

general 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.


G4ConvergenceTester and ExN03Con KOI, Tatsumi SLAC National Accelerator Laboratory 1Geant4...

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