Sept 2003 PHYSTAT 21Our treatment of systematic errorsSept 2003 PHYSTAT 22What 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. BarlowStatisticsSept 2003 PHYSTAT 23Imagine that two experimental groups have measured a quantity θ, with the results shown.OK, what is the value of θ?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 24We use χ2minimization with fitting of systematic errors.Minimize χ2w. r. t. {aµ} ⇒ optimal parameter values {a0µ}.All this would be based on the assumption thatDi= Ti(a0) + σiridrredP 22/2π−=For statistical errors define()−=∑=error lstatistica : valueal theoretic: valuedata : 1222iiiNiiiiTDTDσσχTi = Ti(a1, a2, ..,, ad) a function of d theory parameters(S. D.)Sept 2003 PHYSTAT 25Treatment of the normalization error()∑=−+−=NiiiiTDfffa122N2normNN21),(σσχIn scattering experiments there is an overall normalization uncertainty from uncertainty of the luminosity. We definewhere fN= overall normalization factorMinimize χ2w. r. t. both {aµ} and fN.Sept 2003 PHYSTAT 26A method for general systematic errorsjKjijiiiirraTDˆ)(10∑=++=βα()∑∑∑==+−−=′KjjNiijijijisTsD121222αβχMinimize χ′2with respect to both shape parameters {aµ} and optimized systematic shifts {sj}.quadratic penalty termαi: statistical error of Diβij: set of systematic errors (j=1…K) of DiDefineSept 2003 PHYSTAT 27K x K matrix K x 1 matrix()mKmjmjjjBAasss∑=−===∂′∂11)0(2)( implies 0χ()∑∑==−=+=NiiijiijNiiikijjkjkTDBA1212 and αβαββδwhereBecause χ′2 depends quadratically on {sj} we can solve for the systematic shifts analyticallySept 2003 PHYSTAT 28Now let()∑′=sexperiment022global)(, )( asaaχχand minimize w.r.t {aµ}.The systematic shifts {sj} are continually optimized[ s → s0(a) ]Sept 2003 PHYSTAT 29So, 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 factorsDefault : weand wN,e = 1The 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!)∑∑−+=eeeeeeefwfawfa2N,2,NN2N2global)1(}){},({}){},({σχχSept 2003 PHYSTAT 30How well does this fitting procedure work?QualitySept 2003 PHYSTAT 31Comparison 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 32Comparison 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 33Comparison of the CTEQ6M fit to the BCDMS and NMC data on µp scattering. The data points include optimized shifts for systematic errors. The error bars are statistical only.Sept 2003 PHYSTAT 34Comparison of the CTEQ6M fit to the inclusive jet data. (a) D0 cross section versus pTfor 5 rapidity bins; (b) CDF cross section for central rapidity.Sept 2003 PHYSTAT 35Closer comparison between CTEQ6M and the D0 jet data as fractional differences.Sept 2003 PHYSTAT 36How large are the optimized normalization factors?1.004CDF0.974D00.950E6051.020CCFR1.011NMC0.997ZEUS0.988H1 (b)1.010H1 (a)0.976BCDMSf NExptSept 2003 PHYSTAT 37We must always check that the systematic shifts are not unreasonably large.0.2110-0.4090.6281.287-1.0760.005-0.444-1.253-0.6721.671sjj0.22110.26100.6191.058-0.3170.7060.0550.254-0.353-0.8120.671sjj10 systematic shifts NMC data11 systematic shifts ZEUS dataSept 2003 PHYSTAT 38Comparison to NMC F2iiiiiTDα/)( −⋅−=∆ βswithout systematic shiftsSept 2003 PHYSTAT 39Comparison to ZEUS F2iiiiiTDα/)( −⋅−=∆ βswithout systematic