1. Sept 2003PHYSTAT21 Our treatment of systematic errors

2. Sept 2003PHYSTAT22 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. Barlow Statistics

3. Sept 2003PHYSTAT23 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.

4. Sept 2003PHYSTAT24 We use  2 minimization with fitting of systematic errors. Minimize  2 w. r. t. {a  }  optimal parameter values {a 0  }. All this would be based on the assumption that D i = T i (a 0 ) +  i r i For statistical errors define T i = T i (a 1, a 2,..,, a d ) a function of d theory parameters (S. D.)

5. Sept 2003PHYSTAT25 Treatment of the normalization error In scattering experiments there is an overall normalization uncertainty from uncertainty of the luminosity. We define where f N = overall normalization factor Minimize  2 w. r. t. both {a  } and f N.

6. Sept 2003PHYSTAT26 A method for general systematic errors Minimize  2 with respect to both shape parameters {a  } and optimized systematic shifts {s j }. quadratic penalty term  i : statistical error of D i  ij : set of systematic errors (j=1…K) of D i Define

7. Sept 2003PHYSTAT27 K x K matrixK x 1 matrix where Because  2 depends quadratically on {s j } we can solve for the systematic shifts analytically

8. Sept 2003PHYSTAT28 Now let and minimize w.r.t {a  }. The systematic shifts {s j } are continually optimized [ s  s 0 (a) ]

9. Sept 2003PHYSTAT29 So, we have accounted for … Statistical errors Overall normalization uncertainty (by fitting {f N,e }) Other systematic errors (analytically) We may make further refinements of the fit with weighting factors Default : w e and w N,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!)

10. Sept 2003PHYSTAT30 How well does this fitting procedure work? Quality

11. Sept 2003PHYSTAT31 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.

12. Sept 2003PHYSTAT32 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.

13. Sept 2003PHYSTAT33 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.

14. Sept 2003PHYSTAT34 Comparison of the CTEQ6M fit to the inclusive jet data. (a) D0 cross section versus p T for 5 rapidity bins; (b) CDF cross section for central rapidity.

15. Sept 2003PHYSTAT35 Closer comparison between CTEQ6M and the D0 jet data as fractional differences.

16. Sept 2003PHYSTAT36 How large are the optimized normalization factors? Exptf N BCDMS0.976 H1 (a)1.010 H1 (b)0.988 ZEUS0.997 NMC1.011 CCFR1.020 E6050.950 D00.974 CDF1.004

17. Sept 2003PHYSTAT37 We must always check that the systematic shifts are not unreasonably large. jsj 11.67 2-0.67 3-1.25 4-0.44 50.00 6-1.07 71.28 80.62 9-0.40 100.21 jsj 10.67 2-0.81 3-0.35 40.25 50.05 60.70 7-0.31 81.05 90.61 100.26 110.22 10 systematic shifts NMC data 11 systematic shifts ZEUS data

18. Sept 2003PHYSTAT38 Comparison to NMC F 2 without systematic shifts

19. Sept 2003PHYSTAT39 Comparison to ZEUS F 2 without systematic shifts