Physics Computations I
Physics Computations I PHY 102
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This 14 page Class Notes was uploaded by Quinn Larkin on Saturday September 19, 2015. The Class Notes belongs to PHY 102 at Michigan State University taught by Jon Pumplin in Fall. Since its upload, it has received 9 views. For similar materials see /class/207618/phy-102-michigan-state-university in Physics 2 at Michigan State University.
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Date Created: 09/19/15
Data Set Diagonalization in Global fitting Jon Pumplin Michigan State University PDF4LHC meeting CERN 29 May 2009 A Data Set Diagonalization technique lets us measure the compatibility between a subset S of the data eg a data from a single experiment c all the data that use nuclear targets a all the data from low Q where higher twist corrections might play a role and the remainder of the global data set g This technique can also be used to determine which aspects of the global fit are determined by a specific subset of the input data Preliminary results of a study of the internal compatibility of the CT09 data set will be shown Old method 2001 J C Collins and J Pumplin Tests of goodness of fit to multiple data sets hep ph0105207 In addition to the Hypothesis testing criterion AX2 N W use the stronger Parameter fitting criterion AX2 N 1 The parameter being fitted here is the relative weight assigned to subset S We minimize the weighted X2 for a series of values of the weight We can then 0 plot minimum Xg vs X 2 or 0 Plot both as function of Lagrange multiplier u where 1 uX 1 I uX 2 is the quantity minimized We obtain quantitative results by fitting to a model with a single common parameter p 2 ngA gt pOlSIn6 2 ngB 30556 gt pzsiCOSQ These differ by s j 1 ie by 5 standard deviations 2 1 1 1 4n 7 7 7 NMC D2H2 NMC D2H2 30 7 S 2 6 7 307 7 207 207 7 E gt3 10 lt1 7 m7 7 07 q 7 quotquotquot 710 7107 7 7 t t 1 t 1 t t t t 1 t t 1 t i 710 0 1a 20 30 40 291 0 705 00 10 Axnoti u 100 W t t t 7 1 BCDMS D2 7 507 BCDMS D2 7 75 7 4o 7 i f 7 7 gt3 207 7 lt1 7 7 7207 7 m m F 740 720 0 20 40 so 80 sznuti Fits to 8 of the experiments in the CTEQ5 analysis Expt 1 2 3 4 5 6 7 8 s 27 33 33 42 53 76 74 83 tangb 056 054 099 086 071 114 065 039 The approach John and I considered maps X3 as a function of Xg The method has three problems 1 Since traditional Gaussian statistics don t seem to apply to our problem because of unknown systematic errors both in theory and in experiment we don t know how to decide whether a particular X3 vs X 2 curve shows compatibility or incompatibility 2 The method doesn t directly show what parts of the theory are affected by the tension between S and S 3 Discrepancies between experiments don t matter if they are along parameter directions that are well constrained by other experiments A new Data Set Diagonalization method which expands upon the Hessian method appears to solve problems 2 and 3 The new method works in multiple dimensions it finds all of the directions in parameter space that are controlled by the particular experiment that is under study DSD method The quality of the fit of a PDF set to the data is measured by 2 M Dz T239 2 where D and E represent a data point and its uncertainty Ti is the theoretical prediction which depends on shape parameters 01 aN which describe the PDFs Near the Best Fit minimum Taylor series implies that X2 is a quadratic function of the shape parameters N N N X2 f Zgz39azI Z Z hijaz39aj i1 i1j1 Using the eigenvectors of the Hessian matrix h we can make a linear transformation to obtain new shape coordinates such that N 2 2 2 X Xmin 39 i1 Owing to numerical instabilities associated with flat directions it is generally necessary to calculate that transformation by an iterative method The contribution to X2 from subset S of the data can also be expanded to quadratic accuracy N N N X3 0 Z bzzz Z 2 szsz i1 i1j1 The key step of the DSD method occurs now a further orthogonal transformation defined by the eigenvectors of cm makes the matrix c diagonal Thus N X52 04 252 72 221 while preserving Assume for the moment that all of the vi parameters lie between 0 and 1 Then by simple algebra we obtain X2 X32 X 2 const N 2 2 Zz39 TAz39 XS z lt gt 21 Bi N 2 2 Zz39 Oz 1 XS lt Di gt This has a simple interpretation 5 and its complement take the form of independent measurements of the quantities 21 zN The difference between these is AZ 0 j iBZ D2 The incompatibility between S and the rest of the global fit along direction 139 is thus given by lAi OlinBE DB in standard deviations This decomposition answers the question What is measured by data subset S it is those parameters zz39 for which the Bi 3 Di Compatibility between S and its complement only matters for the directions in which the two measurements have comparable errors In practice that is roughly the range 02 lt 7239 lt 08 7239 BiDi 01 3 02 2 05 l 08 12 09 13 Relation between Bi 2 uncertainty from S and Di uncertainty from S for various m In particular directions for which w is outside the range 0 to 1 are irrelevant for the comparison Example E605 DY pair production in pCu z m z from S 2 from S Difference 1 093 040 j 106 378 j 255 418 j 276 151 0 2 042 122j 152 093j 134 215j203 1060 3 009 171 334 015j 101 186j348 0530 4 005 057j454 003j 103 060j465 0130 Parameter 21 is determined almost entirely by this experiment Parameter 22 is determined about equally by E605 and its complement E605 doesn t give any useful information about any of the other parameters Relative X 2 Relative X 2 Relative X 2 Relative X 2 720 710 O 10 20 720 710 O 10 20 X2 for fit to E605 dashed curves and to the rest of the data solid curves along the four leading eigenvector directions in descending order of 7 The overall best fit is at zz39 O in each case These figures confirm the results of the previous table E605 dominates the measurement of Z1 and is mildly in conflict with the other experiments along that direction it plays an important role in determining z2 and it has nothing to say about Z3 Z24 10 Example Tevatron Run II Jet experiments 7i 2i from S 2i from Difference 079 009 113 034210 043239 020 072 119 116 324 192 442224 200 010 041 d 308 005 d 106 045 d 325 01 0 003 616j647 018j 107 634i655 100 wMHs S CDFDO Run II Run Ijet data removed for simplicity The jet experiments dominate along two directions showing a mild incompatibility along one of them Note these zz39 are the result of diagonalizing the gtlltjetgtllt contribution to X2 they are gtlltnotgtllt the same parameters as the zz39 used in the study of E605 11 m o i as o 4 O 4gt O K O K O lllll 0 Relative X 2 Relative X 2 i 0 will X2 for CDF blue DO magenta and rest of the data black along 21 and 22 0 Good agreement between the average of CDF and DO with the rest of the global fit along Z1 but there is clearly some difference between CDF and DO along that direction 0 Also some difference between CDF and DO with regard to z2 but this time DO agrees with the non jet data while CDF has a bit of tension with it 20 12 60 4 O 4gt O K O K 0 Relative X 2 Relative X 2 X2 for fit to CDF blue DO magenta and the rest of the data black along z3 and z4 0 Parameters z3 z4 224 don t matter because the non jet data determine those parameters The apparent incompatibility between CDF and DO along the z4 axis for example is no cause for concern because the non jet data determine that parameter very well as shown by the very narrow black parabola 13 The discrepancy between the two run II jet experiments run I experiments were not used in this study is given by 2i from CDF 2i from DO Difference 1 27sz 165 245i 138 515i215 2400 2 233 135 174i222 407i260 1570 N The Data Set Diagonalization method thus gives a quantitative measure of the mild conflict between the two run II inclusive jet data sets which could only be seen qualitatively using the classical methods employed in the talk I gave this morning 14