데이터 분석 방법론
DESCRIPTION
제 2 회 고에너지물리 여름학교. 데이터 분석 방법론. 2002. 6. 25 경북대학교 고에너지물리연구소 조기현. 목 차. 고에너지물리 데이터 처리 방법론 Fitting 결론. 고에너지 물리. Goal. 물질의 궁극적구조의 그사이 상호작용의 연구로 우주의 기원에 대한 이해. 고에너지 물리. 방향. What is World Made of?. Atom Electron Nucleus Proton, neutron quarks. - PowerPoint PPT PresentationTRANSCRIPT
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데이터 분석 방법론
2002. 6. 25경북대학교
고에너지물리연구소 조기현
제 제 2 2 회 고에너지물리 여름학교회 고에너지물리 여름학교
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2고에너지물리연구센터 2002. 6. 25. CENTER FOR HIGH ENERGY PHYSICS
고에너지물리
데이터 처리 방법론
Fitting
결론
목 차
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고에너지 물리고에너지 물리
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4고에너지물리연구센터 2002. 6. 25. CENTER FOR HIGH ENERGY PHYSICS
고에너지 물리
물질의 궁극적구조의 그사이 상호작용의 연구로 우주의 기원에 대한 이해Goal
High Energy Physics
Unsolved Problems
Antimatter in SpaceDark Matter
Unsolved Problems
Higgs partic le(s)Q uark M ixing
etc .
Unsolved Problems
Leptoquarks
Unsolved Problems
SUSY P artic les
Symmetry and C onservation Laws
C harge C onjugation- P arity Vio lation
Existence of New Interac tions
Standard M odel(quark - lepton,
Interactions)
Unification ofInteractions
Supersym m etry
Extended M odelsCom posite M odels
방향
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5고에너지물리연구센터 2002. 6. 25. CENTER FOR HIGH ENERGY PHYSICS
What is World Made of?What is World Made of?
– Atom• Electron• Nucleus
– Proton, neutron• quarks
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6고에너지물리연구센터 2002. 6. 25. CENTER FOR HIGH ENERGY PHYSICS
How to know any of this?How to know any of this?(Testing Theory)(Testing Theory)
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7고에너지물리연구센터 2002. 6. 25. CENTER FOR HIGH ENERGY PHYSICS
How to detect?How to detect?
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8고에너지물리연구센터 2002. 6. 25. CENTER FOR HIGH ENERGY PHYSICS
How do we experiment How do we experiment with tiny particles? with tiny particles?
(Accelerators)(Accelerators) Accelerators solve two problems:
– High energy gives small wavelength to detect small particles.
– The high energy create the massive particles that the physicist want to study.
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9고에너지물리연구센터 2002. 6. 25. CENTER FOR HIGH ENERGY PHYSICS
Europe - In 2007, the LHC will b
e completed at CERN - Two big experiments (A
TLAS, CMS) in collab. of HEP institutes and physicists all over the world
- CERN, IN2P3(France), and INFN(Italy) are preparing HEP Grid for it.
USA- The BaBar Exp at SLAC- The Run II of the Tevatr
on at Fermilab (CDF and D0)
- The CLEO at Cornell - The LHC experiments at
CERN (ATLAS, CMS)- The RHIC exp at BNL- The Super-K in Japan- The HEP Grid in the ES
NET program
Japan- Belle at KEK- Super-K, Kamiok
a- LHC at CERN (A
TLAS)- The RHIC at BNL
(USA)- They are now wor
king for it.
World-wide High Energy Physics Experiment
Korea We have most of these world-wide experimental programs…
한국이 국제 공동연구로 참여 중
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10고에너지물리연구센터 2002. 6. 25. CENTER FOR HIGH ENERGY PHYSICS
Europe CERN
Germany DESY
US FNAL
US BNL
Space Station (ISS)
Japan KEK
China IHEP
Korea CHEP
연구내용
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11고에너지물리연구센터 2002. 6. 25. CENTER FOR HIGH ENERGY PHYSICS
Where is Fermilab?Where is Fermilab?
20 mile west of Chicago
U.S.A
Fermilab
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12고에너지물리연구센터 2002. 6. 25. CENTER FOR HIGH ENERGY PHYSICS
Overview of FermilabOverview of Fermilab
Main Injectorand Recycler
p source
Booster
CDF
D0
Fixed Target
Experiment
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13고에너지물리연구센터 2002. 6. 25. CENTER FOR HIGH ENERGY PHYSICS
Fermi National Accelerator Laboratory
Highest Energy Accelerator in theWorld
Energy Frontier: CDF, D0 Search for New Physics (Higgs, SUSY, quark composites,…
Precision Frontier: charm, kaon, neutrino physics(FOCUS, KTeV, NUMI/MINOS,BOONE,…etc.
Connection to Cosmology: Sloan Digital sky survey, Pierre Auger,…
Largest HEP Laboratory in USA
2200 employees2300 users (researchers from univ.)Budget is >$300 million
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Data Data 처리 방법론처리 방법론
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15고에너지물리연구센터 2002. 6. 25. CENTER FOR HIGH ENERGY PHYSICS
Why do we do Why do we do experiments?experiments?
Parameter determination– To set the numerical values of some physical quantities– Ex) To measure velocity of light
Hypothesis testing– To test whether a particular theory is consistent with our
data– Ex) To check whether velocity of light has suddenly
increased by several percent since beginning of this year
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16고에너지물리연구센터 2002. 6. 25. CENTER FOR HIGH ENERGY PHYSICS
Type of DataType of Data
Real Data (on-site)– Raw Data : Detector Information– Reconstructed Data : Physics Information– Stream (Skim) Data : Selected interested physics
Simulated Data (on-site or off-site)– Physics generation : pythia, QQ, bgenerator, …– Detector Simulation : Fastsim, GEANT, …
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17고에너지물리연구센터 2002. 6. 25. CENTER FOR HIGH ENERGY PHYSICS
HEPKnowledge
ReactionSimulation
= EventGeneration
DetectorSimulation
Simulated Data
RealData
DataReduction
On-sites (Experimental sites)
Remote-sites (CHEP + participating institutions)
DataAnalysis
연구 방법
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18고에너지물리연구센터 2002. 6. 25. CENTER FOR HIGH ENERGY PHYSICS
오차 오차 (Error)(Error)
오차 (error)– 오차 : 계산치 또는 실제 값 사이의 차이– 실제값 (true value)
• 대체적으로 모름– 통계오차 (statistical error)
• 데이터의 통계적 요동에 의한 의한 오차– 계통오차 (systematic error)
• 장치를 옳 바로 보정하지 못하거나 관측자의 편견에 의한 오차
실험치의 표시 : 측정값 통계오차 계통오차Example ) m(top) = 175.9 4.8 5.3 GeV/c2 (CDF, 1998)
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19고에너지물리연구센터 2002. 6. 25. CENTER FOR HIGH ENERGY PHYSICS
Why estimate errors?Why estimate errors? To know how accuracy of the measurement Example
– 현재의 빛의 속도 측정값 c=2.998 X 108 m/sec – 새로운 빛의 속도 측정값 c=(3.09 0.15) X 108 m/sec– Case 1. If the error is 0.15, then it is consistent.
• Conventional physics is in good shape. • 3.09 0.15 is consistent with 2.998 X 108 m/sec
– Case 2 . If the error is 0.01, then it is not consistent.• 3.09 0.01 is world shattering discovery.
– Case 3. If the error is 2, then it is consistent.• However, the accuracy of 3.09 2 is too low. • Useless measurement
Whenever you determine a parameter, estimate the error or your experiment is useless.
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20고에너지물리연구센터 2002. 6. 25. CENTER FOR HIGH ENERGY PHYSICS
How to reduce errors?How to reduce errors?
통계오차 (statistical error)– 같은 측정을 반복한다 .– N : the expected number of observation = Sqrt(N) : the spread
계통오차 (systematic error)– No exact formulae– Ideal case : All such effects should be absent.– Real world : An attempt to be made to reduce it
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21고에너지물리연구센터 2002. 6. 25. CENTER FOR HIGH ENERGY PHYSICS
How to solve systematic How to solve systematic errors?errors?
Use constraint condition– Ex) Triangle
Calibrations Energy and momentum conservation
– E(after) – E(before) = 0– |P(after)| - |P(before)| = 0
How small of the systematic error?– Systematic errors should be smaller than statistical
errors
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22고에너지물리연구센터 2002. 6. 25. CENTER FOR HIGH ENERGY PHYSICS
The meaning of The meaning of (error) (error)
분포 (distributions) x -> n(x)– Discrete
• ex) # of times n(x) you met a girl at age x– Continuous :
• ex) Hours sleep each night (x), # of people sleeping for time.
=> For an even larger number of observation and with small bin size, the histogram approach a continuous distribution.
평균 (Mean) 과 분산 (Variance)
가우스 분포 (Gaussian distribution) – Data 양이 많을 때– Error 계산에서 중요
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23고에너지물리연구센터 2002. 6. 25. CENTER FOR HIGH ENERGY PHYSICS
Tracking PerformanceTracking Performance
Ks
Hit Resolution~200 m
Goal : 180 m
p
COT tracksResidual distance (cm)
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24고에너지물리연구센터 2002. 6. 25. CENTER FOR HIGH ENERGY PHYSICS
평균평균 (Mean)(Mean) 과 분산과 분산(Variance)(Variance)
실제값(True Value)
측정값(Measuremen
t)평균
(Mean) 분산
(variance) 2 s2 표준편차
(standard deviation)
s
실제로는 참값을 알 수 없는 경우가 대부분임
x
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25고에너지물리연구센터 2002. 6. 25. CENTER FOR HIGH ENERGY PHYSICS
평균평균 (Mean)(Mean) 과 분산과 분산(Variance)(Variance)
평균– N 개의 데이터가 (x1, x2, x3,… xN) 값을 가질때
분산
– 실제값을 모르므로
N
xx i
N
xs i
22 )(
1
)( 22
N
xxs i
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26고에너지물리연구센터 2002. 6. 25. CENTER FOR HIGH ENERGY PHYSICS
Accuracy (Accuracy (
측정의 정확도를 나타냄
N
s
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27고에너지물리연구센터 2002. 6. 25. CENTER FOR HIGH ENERGY PHYSICS
Gaussian DistributionGaussian Distribution
• 대부분 실험의 경우 Data 양이 많을 때• Gaussian distribution is the fundamental in error treatment.
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28고에너지물리연구센터 2002. 6. 25. CENTER FOR HIGH ENERGY PHYSICS
The normalized function
Mean () Width () Width () is smaller, distribution is narrower. Properties
Gaussian Distribution (cont’d)Gaussian Distribution (cont’d)
}2/)(exp{2
1 22
xy
68.0)(
dxxf
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29고에너지물리연구센터 2002. 6. 25. CENTER FOR HIGH ENERGY PHYSICS
Gaussian Distribution (cont’d)Gaussian Distribution (cont’d)
• Mean () is same as zero.
• However width ( ) is different.
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30고에너지물리연구센터 2002. 6. 25. CENTER FOR HIGH ENERGY PHYSICS
CDF Secondary CDF Secondary Vertex TriggerVertex Trigger
NEW for Run 2 -- level 2 impact parameter trigger Provides access to hadronic B decays
Data from commissioning run
COT defines track SVX measures (no alignment or calibrations) at level 1 impact parameter
~ 87 m
d (cm)
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31고에너지물리연구센터 2002. 6. 25. CENTER FOR HIGH ENERGY PHYSICS
Mn_fit 을 이용한
Gaussian fitting
- +
68.0)(
dxxf
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32고에너지물리연구센터 2002. 6. 25. CENTER FOR HIGH ENERGY PHYSICS
유효숫자 유효숫자 (Significant Figure)(Significant Figure)
측정값은 실험적으로 불확실한 범위 이내서만 의미를 갖는 값
유효 숫자 – 첫번째 불확실한 자리까지 포함 – LSD (least significant digit) 와 MSD(Most significant
digit) 사이의 모든 숫자• LSD
– 소수점이 없을 때 : 가장 오른쪽이 0 이 아닌 숫자 ex)23000– 소수점이 있을 때 : 가장 오른쪽 숫자 ex) 0.2300
• MSD : 가장 왼쪽의 0 이 아닌 숫자
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33고에너지물리연구센터 2002. 6. 25. CENTER FOR HIGH ENERGY PHYSICS
유효숫자 유효숫자 (Example)(Example)
•유효숫자 네 자리 : 1234, 123400, 123.4, 1000.
•유효숫자 네 자리 : 10.10, 0.0001010, 100.0, 1.010X103
•유효숫자 세 자리 : 1010 cf) 1010. ( 유효숫자 네 자리 )
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34고에너지물리연구센터 2002. 6. 25. CENTER FOR HIGH ENERGY PHYSICS
유효숫자 연산유효숫자 연산
덧셈 또는 뺄셈 – 마지막 결과의 소수부분의 자릿수는 셈에 포함된 측정값
중 가장 작은 소수점 아래– Example)
123+ 5.35-------- 128.35
1.0001 ( 유효숫자 5 자리 )
+ 0.0003 ( 유효숫자 1 자리 )
--------
1.0004 ( 유효숫자 5 자리 )
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35고에너지물리연구센터 2002. 6. 25. CENTER FOR HIGH ENERGY PHYSICS
유효숫자 연산 유효숫자 연산 (cont’d)(cont’d)
곱셈 및 나눗셈– 가장 적은 유효 숫자와 같게 – Example)
16.3 X 4.5 = 73.35 => 73
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36고에너지물리연구센터 2002. 6. 25. CENTER FOR HIGH ENERGY PHYSICS
오차의 전파 오차의 전파 II(Propagation of Errors)(Propagation of Errors)
두개 이상의 확률변수 (x1,x2, …) 로 된 함수 F(x1, x2, …) 표준편차는 다음과 같이 나타낼수 있다 . – 단 , 변수사이에 correlation 이 없을 때
...)()( )( 23
2
3
22
2
2
21
2
1
2 x
F
x
F
x
FF
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37고에너지물리연구센터 2002. 6. 25. CENTER FOR HIGH ENERGY PHYSICS
오차의 전파 오차의 전파 IIII(Propagation of Errors)(Propagation of Errors)
두개 이상의 확률변수 (x1,x2, …) 로 된 함수 F(x1, x2, …) 표준편차는 다음과 같이 나타낼수 있다 . – 단 , 변수사이에 correlation 이 있을 때
=> 앞으로 correlation 이 없는 경우만 고려
jijiji
F x
F
x
F )()(,
2
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38고에너지물리연구센터 2002. 6. 25. CENTER FOR HIGH ENERGY PHYSICS
Combining ErrorsCombining Errors
덧셈 또는 뺄셈 (F=x1+x2 or F= x1-x2)
Example) x1 = 100. 10.
+ x2 = 400. 20. ----------- F = 500. 22.
Example) 측정값의 오차
22
21 F
22sysstat
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39고에너지물리연구센터 2002. 6. 25. CENTER FOR HIGH ENERGY PHYSICS
Combining Errors (cont’d)Combining Errors (cont’d)
F=ax ( 단 , a 는 상수 )
Example) x =100. 10. a = 5 ------------ F = 500. 50.
aF
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40고에너지물리연구센터 2002. 6. 25. CENTER FOR HIGH ENERGY PHYSICS
Combining Errors (cont’d)Combining Errors (cont’d)
곱셈 (F=x1 • x2)
Example) x1 = 100. 10.
x2 = 400. 20. ----------- F = (400. 45. ) X 102
222
22121 )/()/( xxxxF
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41고에너지물리연구센터 2002. 6. 25. CENTER FOR HIGH ENERGY PHYSICS
Combining Errors (cont’d)Combining Errors (cont’d)
나눗셈 (F= x1 / x2)
Example) x1 = 100. 10.
x2 = 400. 20. ----------- F = 0.250 0.028
222
22121 )/()/()/( xxxxF
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42고에너지물리연구센터 2002. 6. 25. CENTER FOR HIGH ENERGY PHYSICS
Combining results Combining results Using weighting factorUsing weighting factor
Cases– With different detection efficiencies– With different parts of apparatus– With different experiment
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43고에너지물리연구센터 2002. 6. 25. CENTER FOR HIGH ENERGY PHYSICS
Combining results Combining results Using weighting factor Using weighting factor
(cont’d)(cont’d)
평균– N 개의 데이터가 (x1, x2,. ..xk,… xN) 값을 가지고– Xk 에 대한 error 가 k 라고 하면
where weighting factor
Error :
kk
kk
w
xwx
2/1 kkw
kw/12
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44고에너지물리연구센터 2002. 6. 25. CENTER FOR HIGH ENERGY PHYSICS
Ex) World Average of sin(2Ex) World Average of sin(2) )
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45고에너지물리연구센터 2002. 6. 25. CENTER FOR HIGH ENERGY PHYSICS
Ex) BEx) B00 lifetime summary lifetime summary
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46고에너지물리연구센터 2002. 6. 25. CENTER FOR HIGH ENERGY PHYSICS
Ex) CDF BEx) CDF Bdd Mixing Mixing
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47고에너지물리연구센터 2002. 6. 25. CENTER FOR HIGH ENERGY PHYSICS
Upper LimitUpper Limit
Measurement (B = Bm )
Observation (Bm> 5)
– Signal is greater than 5 sigma of error. Evidence ( 3 < Bm < 5 )
– Signal is greater than 3 sigma of error, however less than 5 sigma.
Upper Limit (3 < Bm )– Signal is less than 3 sigma.
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48고에너지물리연구센터 2002. 6. 25. CENTER FOR HIGH ENERGY PHYSICS
Upper Limit BUpper Limit Bll (cont’d) (cont’d)
Method I. General Case
Measurement B = Bm Bl < Bm + 1.28 (90% CL) 1.64 (95% CL) 2.33 (99% CL)
Measurement B = Bm
Ex) Bl =(3 5) X 10-9 Bl < (3+1.28X5) X 10-9 at 90% CL
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49고에너지물리연구센터 2002. 6. 25. CENTER FOR HIGH ENERGY PHYSICS
Upper Limit BUpper Limit Bll (cont’d) (cont’d)
Method 2. Negative Bm – Background Subtracted– Example)
• Bm = (-1 1) X 10-9
• Bm = ( 0 1) X 10-9
– Upper Limit at 90 % CL Level • g is Gaussian (Mean is Bm , width is )
9.0
0
0
gdB
gdBlB
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50고에너지물리연구센터 2002. 6. 25. CENTER FOR HIGH ENERGY PHYSICS
Compare Upper Limit (90% Compare Upper Limit (90% CL) CL)
Bm Method 1 Method 24 5.3 5.33 4.3 4.32 3.3 3.31 2.3 2.4
0.5 1.8 2.00 1.3 1.6
-0.5 0.8 1.4-1 0.3 1.2-2 -0.7 0.8-3 -1.7 0.6-4 -2.7 0.5
Assume =1
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51고에너지물리연구센터 2002. 6. 25. CENTER FOR HIGH ENERGY PHYSICS
Ex) CP Asymmetry in CharmEx) CP Asymmetry in Charm(D(D++ K K--KK++))
Cabibbo Suppressed mode
Cabbibo Favored mode
D+ KK++
D KK+
D+ K ++
D K+
( ) ( )
( ) ( )CP
D DA
D D
)(
)()(
KDN
KKDND
)(
)()(
0
0
KDN
KKDND C.F.
A=0.0060.011 0.005
A < 0.025 at 95 %CL
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Fitting Fitting
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53고에너지물리연구센터 2002. 6. 25. CENTER FOR HIGH ENERGY PHYSICS
Fitting Methods Fitting Methods
Moment– Simple, but inefficiency
Maximum likelihood Method– More general case
Least Square Method– In case of statistical error
Example) (xi, yi) 인 n 개의 데이터가 y=ax+b 인 일차식으로 가정하여 fitting 하는 방법
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54고에너지물리연구센터 2002. 6. 25. CENTER FOR HIGH ENERGY PHYSICS
MomentMoment
Method is to calculate the average Simplicity Example
– 일차식
• Parameter a is
ii axy
nx
ya
i
in
i
/)(1
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55고에너지물리연구센터 2002. 6. 25. CENTER FOR HIGH ENERGY PHYSICS
Maximum likelihood Maximum likelihood MethodMethod
The likelihood L
Where is the parameter to find yi is the function given variable xi
To find maximize L To maximize l= log L Normalization is essential. Ex) 일차식
n
iiyL
1
)()(
baxy ii
n
ii baybaL
1
),(),(
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56고에너지물리연구센터 2002. 6. 25. CENTER FOR HIGH ENERGY PHYSICS
Maximum likelihood Maximum likelihood Method (cont’d)Method (cont’d)
The most powerful one for finding the values of unknown parameters
No histogram needed (event by event)
Efficient Method -> Most case works
We can transform one variable to anotherEx)
00 /1
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57고에너지물리연구센터 2002. 6. 25. CENTER FOR HIGH ENERGY PHYSICS
Least Square Method ILeast Square Method I(( 최소자승법 최소자승법 - - 직선회귀법직선회귀법 ))
(xi, yi) 인 n 개의 데이터가 y=ax+b 인 일차식으로 가정하여 fitting 하는 방법
가정하는 직선과 데이터가 최소가 되도록 a 와 b 를 구한다 . 즉 Q를 아래처럼 두면
다음식을 만족하는 a 와 b 를 구한다 .
i
ii ybxaQ 2)(
0 & 0
b
Q
a
Q
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58고에너지물리연구센터 2002. 6. 25. CENTER FOR HIGH ENERGY PHYSICS
Least Square Method IILeast Square Method II(( 최소자승법최소자승법 -- 일반적 경우일반적 경우 ))
선형 최소자승법– m 개의 미지변수 (a1, a2, a3,… am)
– F(x)=a1f1(x)+a2f2(x)+ + am fm(x)– 직선 최소 자승법과 같음– m 개의 연립방정식의 해
비선형 최소자승법– Taylor 시리즈로 전개하여 선형으로 바꾸어서 계산
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59고에너지물리연구센터 2002. 6. 25. CENTER FOR HIGH ENERGY PHYSICS
Least Square Method Least Square Method (Example)(Example)
Mn_fit used Least Square Method
Signal is gaussian. Background is Chebyshe
v polynomial.
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60고에너지물리연구센터 2002. 6. 25. CENTER FOR HIGH ENERGY PHYSICS
Maximum Likelihood vs.Maximum Likelihood vs. Least Square Method Least Square Method
Maximum like. Least Square
How easy Normalization and maximization can be
messy
Needs minimization
Efficiency Usually most efficient Sometime equivalent to max.
like.Input data Individual events HistogramsEstimate of
goodness of fitVery difficult Easy
Zero event Cover well Troublesome
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61고에너지물리연구센터 2002. 6. 25. CENTER FOR HIGH ENERGY PHYSICS
X-Y plane Errors in y-direction are Gaussian X-values are precisely determined
The maximum likelihood and the least square methods are equivalent.
Example) Mass distributions
Maximum Likelihood =Maximum Likelihood = Least Square Method Least Square Method
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62고에너지물리연구센터 2002. 6. 25. CENTER FOR HIGH ENERGY PHYSICS
Fitting PackageFitting Package
PAW
Mn_fit
Root
……
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63고에너지물리연구센터 2002. 6. 25. CENTER FOR HIGH ENERGY PHYSICS
PAW PAW
Physics Analysis Workstation Inside of CERN library Ntuple – n dimensional variables Good to make histogram Include some fitting
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64고에너지물리연구센터 2002. 6. 25. CENTER FOR HIGH ENERGY PHYSICS
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65고에너지물리연구센터 2002. 6. 25. CENTER FOR HIGH ENERGY PHYSICS
Mn_fitMn_fit
Using fitting program in minuit at CERN library Powerful for fitting Easily check the results whether the fitting resul
ts are good or not.
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66고에너지물리연구센터 2002. 6. 25. CENTER FOR HIGH ENERGY PHYSICS
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67고에너지물리연구센터 2002. 6. 25. CENTER FOR HIGH ENERGY PHYSICS
mn_fit (example)mn_fit (example)
Signal is Gaussian
Maximum likelihood is same as least square method
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68고에너지물리연구센터 2002. 6. 25. CENTER FOR HIGH ENERGY PHYSICS
ROOT ROOT
To Handle large data
An object oriented HEP analysis Framework
ROOT was created by Rene Brun and Fons Rademakers in CERN
The ROOT system website is at http://root.cern.ch/
KimJieun
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69고에너지물리연구센터 2002. 6. 25. CENTER FOR HIGH ENERGY PHYSICS
Differences from PAW Differences from PAW
Regular grammar (C++) on command line Single language (compiled and interpreted) Object Oriented (use your class in the interpreter) Advanced Interactive User Interface Well Documented code. HTML class descriptions for every
class. Object I/O including Schema Evolution 3-d interfaces with OpenGL and X3D.
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70고에너지물리연구센터 2002. 6. 25. CENTER FOR HIGH ENERGY PHYSICS
ROOT exampleROOT example
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71고에너지물리연구센터 2002. 6. 25. CENTER FOR HIGH ENERGY PHYSICS
결론
자료처리 방법론은 중요 물리학에서 중요
Welcome to KNU! Go Final!
Ref.– Louis Lyons, Statistics for nuclear and particle physicists (Cambridge Press)
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72고에너지물리연구센터 2002. 6. 25. CENTER FOR HIGH ENERGY PHYSICS
도움말
PAW 와 ROOT 는 아래에서 프로그램 , 매뉴얼 그리고 예제를 배포하고 있읍니다 .
PAW : http://wwwinfo.cern.ch/asd/paw/ ROOT : http://root.cern.ch