sept 2003phystat21 our treatment of systematic errors
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Sept 2003 PHYSTAT 1
Our treatment of systematic errors
Sept 2003 PHYSTAT 2
What is a systematic error?
“This is why people are so frightened of systematic errors, and most other textbooks avoid the subject altogether. You never know whether you have got them and can never be sure that you have not – like an insidious disease…
The good news, however, is that despite popular prejudices and superstitions, once you know what your systematic errors are, they can be handled with standard statistical methods.”
R. J. BarlowStatistics
Sept 2003 PHYSTAT 3
Imagine that two experimental groups have measured a quantity , with the results shown.
OK, what is the value of ?
This is very analogous to what happens in global analysis of PDF’s. But in the case of PDF’s the systematic differences are only visible through the PDF’s.
Sept 2003 PHYSTAT 4
We use 2 minimization with fitting of systematic errors.
Minimize 2 w. r. t. {a} optimal parameter values {a0}.
All this would be based on the assumption that
Di = Ti(a0) + i ri
drre
dP 2
2/2
For statistical errors define
error lstatistica :
valueal theoretic:
valuedata :
1
2
22
i
i
iN
i i
ii T
DTD
Ti = Ti(a1, a2, ..,, ad) a function of d theory parameters
(S. D.)
Sept 2003 PHYSTAT 5
Treatment of the normalization error
N
i i
ii TDfffa
12
2N
2
norm
NN
2 1),(
In scattering experiments there is an overall normalization uncertainty from uncertainty of the luminosity. We define
where fN = overall normalization factor
Minimize 2 w. r. t. both {a} and fN.
Sept 2003 PHYSTAT 6
A method for general systematic errors
j
K
jijiiii rraTD ˆ)(
10
K
jj
N
i i
j ijijis
TsD
1
2
12
2
2
Minimize 2 with respect to both shape parameters {a} and optimized systematic shifts {sj}.
quadratic penalty term
i : statistical error of Di
ij : set of systematic errors (j=1…K) of Di
Define
Sept 2003 PHYSTAT 7
K x K matrix K x 1 matrix
m
K
mjmjj
j
BAasss
1
1)0(2
)( implies 0
N
i i
ijiij
N
i i
ikijjkjk
TDBA
12
12
and
where
Because 2 depends quadratically on {sj} we can solve for the systematic shifts analytically
Sept 2003 PHYSTAT 8
Now let
sexperiment
022
global )(, )( asaa
and minimize w.r.t {a}.
The systematic shifts {sj} are continually optimized[ s s0(a) ]
Sept 2003 PHYSTAT 9
So, we have accounted for …• Statistical errors• Overall normalization uncertainty (by fitting {fN,e})• Other systematic errors (analytically)
We may make further refinements of the fit with weighting factors
Default : we and wN,e = 1
The spirit of global analysis is compromise – the PDF’s should fit all data sets satisfactorily.If the default leaves some experiments unsatisfied, we may be willing to reduce the quality of fit to some experiments in order to fit better another experiment. (We use this sparingly!)
e e
ee
eee
fwfawfa
2N,
2
,NN2
N2global
)1(}){},({}){},({
Sept 2003 PHYSTAT 10
How well does this fitting procedure work?
Quality
Sept 2003 PHYSTAT 11
Comparison of the CTEQ6M fit to the H1 data in separate x bins. The data points include optimized shifts for systematic errors. The error bars are statistical only.
Sept 2003 PHYSTAT 12
Comparison of the CTEQ6M fit to the ZEUS data in separate x bins. The data points include optimized shifts for systematic errors. The error bars are statistical only.
Sept 2003 PHYSTAT 13
Comparison of the CTEQ6M fit to the BCDMS and NMC data on p scattering. The data points include optimized shifts for systematic errors. The error bars are statistical only.
Sept 2003 PHYSTAT 14
Comparison of the CTEQ6M fit to the inclusive jet data. (a) D0 cross section versus pT for 5 rapidity bins; (b) CDF cross section for central rapidity.
Sept 2003 PHYSTAT 15
Closer comparison between CTEQ6M and the D0 jet data as fractional differences.
Sept 2003 PHYSTAT 16
How large are the optimized normalization factors?
Expt f N
BCDMS 0.976H1 (a) 1.010H1 (b) 0.988ZEUS 0.997NMC 1.011CCFR 1.020E605 0.950D0 0.974CDF 1.004
Sept 2003 PHYSTAT 17
We must always check that the systematic shifts are not unreasonably large.
j sj1 1.672 -0.673 -1.254 -0.445 0.006 -1.077 1.288 0.629 -0.40
10 0.21
j sj1 0.672 -0.813 -0.354 0.255 0.056 0.707 -0.318 1.059 0.61
10 0.2611 0.22
10 systematic shifts NMC data
11 systematic shifts ZEUS data
Sept 2003 PHYSTAT 18
Comparison to NMC F2iiiii TD /)( βs
without systematic shifts
Sept 2003 PHYSTAT 19
Comparison to ZEUS F2
iiiii TD /)( βs
without systematic shifts