GammaCombo User Manual
Contents
1. 12 bin tutorial c 6 var a_gaus var b gaus a plot i c 4 combine the circular constraint with the two dimensional Gaussian bin tutorial c 7 var a_gaus var b_gaus plot all three into the same plot bin tutorial c 6 var a_gaus var b_gaus a plot i c 4 c 7 4 j T T 4 T 7 T l Circle E 2D Gaus Deaus Beaus Figure 3 Result of the tutorial 6 a circular constraint Sec 4 4 Left the circular constraint alone Right added the two dimensional Gaussian from the previous example Desig 2 1 0 1 2 Figure 4 Result of the tutorial 6 a circular constraint Sec 4 4 Combining the circular constraint with the two dimensional Gaussian measurement 4 5 Changing combinations on the fly It is possible to change existing combinations on the fly from the command line by adding and or removing measurements The measurements are refered to by their internal 13 numbers which can be obtained from the usage printout u Example 1 add a measurement Lets add a measurement on the fly to one of the tutorial combinations The resulting plot is shown in Fig Lets start by running the unmodified combination 1 just consisting of measurement 1 bin tutorial i var a_gaus c 1 Now print out the usage message to find a measurement we can add bin tutorial u AVAILABLE MEASUREMENTS 1 1D Gaussian a_ obs 0 5 2 1D Gaussian2. c combination ID This refers to pre defined combinations in this case nothing is combined really it is just the simple Gaussian A list of available IDs can be obtained by running with the usage flag bin tutorial u i interactive mode Any plot will be shown directly in the familiar ROOT canvas Exit with Ctrl c var the variable name of the theory parameter to be scanned ps 1 print solution on the plot The given value configures the position of the numerical value on the plot See the help bin tutorial h for options The resulting plot is shown in Fig Gaus 1 m J i J Q 127 Gaus 1 J Q 1 21 Gaus 2 J 5 J baa w Gaus 1 amp Gaus 2 J 1 i rm T 0 86 GammaCombo 0 66 4 0 4 4 0 2 4 2 1 0 1 2 AGaus Agaus Figure 1 Left 1 CL plot resulting from tutorial measurement 1 a simple Gaussian Right Curves from tutorial measurements 1 and 2 together with the result of combining both Sec 4 2 4 2 Combining two Gaussian measurmeents In the tutorial executable there is also a second Gaussian defined measurement 2 which constrains dp to a different value debe 1 04 0 5 stat 0 15 syst 18 The combination of Eq 17 and 18 should reproduce the usual weighted average _ W10 W202 Z ei as 19 a wy We ee 1 ih 20 ywi w2 1 wi z gt 21 oO where the a refer to Eq I7 and 18 and the statistical and sytematic uncertainty o
3. Circle 5 5 FE 2D Gaus amp Circle Prob 5 E Eo 3 H 2D Gaus amp Circle Plugin 4 contours hold 68 95 CL fi 2 2 I 0 2 Agaus Figure 9 Result of running the PLUGIN method in two dimensions on combination 7 Top left confidence contours for the PROB and PLUGIN methods when not using the 2dcl option causing inconsistent CL contents in the plot see text Top right Using the 2dcl1 option Bottom left Using the 2D smoothing option smooth2d Bottom right The raw p value histogram obtained from the PLUGIN method with a toy statistics of 300 toys per grid point 20 5 Create a new combiner module In this section we describe how to add a new combiner module to GAMMACOMBO that will contain everything that is needed for a new combination project all measurement classes containing the observed central values uncertainties correlations as well as the PDF parameter definitions and a set of predefined combinations We will take a real life example based on a measurement of the CKM angle y as reported in Ref 6 The observables are cartesian coordinates x and y which are linked to the parameter of interest y and two nuisance parameters 62 and r2 through the relations r4 rBE cos 6b 7 30 ys rBE sin 62 7 31 The observables are measured to be zr _ 2525 1 1 x 107 32 y 7 5 2 91 5 x 107 33 4 7 7 2 44 1 1 x 107 34 yy 2
4. a Gaus AGaus 2 Figure 5 Result of Examples 1 and 2 Left Add a measurement to an existing combination from the command line Right Delete a measurement from an existing combination 4 6 Running the PLUGIN method Running the pseudo experiment based PLUGIN method for a particular combination consists of two steps 1 generate and fit the pseudo experiments possibly on a batch farm 2 read in the fit results and compute the CL intervals Running the PLUGIN method requires that a profile likelihood scan is already set up because each batch job will recompute the profile likelihood In practice this means one should have successfully made the PROB plot before whithout running into problems For the tutorial this is the case Lets take as an example combination number 7 2D Gaus amp Circle and scan for parameter a The PLUGIN result is shown in Fig 6 Run three jobs to generate and fit toy experiments possibly on different cores or batch nodes bin tutorial c 7 var a_gaus a pluginbatch ntoys 100 nrun 1 bin tutorial c 7 var a_gaus a pluginbatch ntoys 200 nrun 2 bin tutorial c 7 var a_gaus a pluginbatch ntoys 400 nrun 3 Compute the plugin intervals from all toys bin tutorial c 7 var a_gaus a plugin j 1 3 i Only plot the plugin curve bin tutorial c 7 var a_gaus a plugin j 1 3 i po The arguments mean a pluginbatch batch mode This will run a number of PLUGIN to
5. a_ obs 1 5 3 2D Gaussian a_ obs b_f obs 4 circle a_fobs b_f obs Lets add number 2 to the combination bin tutorial i var a_gaus c 1 2 For comparison lets add the unmodified combination to the same plot bin tutorial i var a_gaus c 1 2 c 1 Example 2 delete a measurement Lets delete a measurement on the fly from one of the tutorial combinations The resulting plot is shown in Fig This is the unmodified tutorial combination 8 consisting of three measurements bin tutorial i var a_gaus c 8 Combiner Configuration Gaus 1 amp Gaus 2 amp 2D Gaus 1 measurement 1 1D Gaussian A 2 measurement 2 1D Gaussian B 3 measurement 3 2D Gaussian A B Lets delete measurement 1 from the combination bin tutorial i var a_gaus c 8 1 c 8 Example 3 a new combiner from scratch An easy trick to make completely new combinations from the command line is to define an empty combination in the main file In the tutorial this combination has index 0 Then one can add whatever one wants Adding several measurements to the empty combination to reproduce combination 8 bin tutorial i var a_gaus c 0 1 2 3 14 Gaus 1 Meas 2 Gaus 1 amp Gaus 2 amp 2D Gaus w o Meas 1 1 CL i TTT T T LI I L 1 CL E Gaus 1 Gaus 1 amp Gaus 2 amp 2D Gaus m 0 8 0 8 0 6 0 6 0 4 0 4 0 2 0 2 2
6. be an arbitrary string however setting it to Rad will tell GAMMACOMBO that this parameter is an angular parameter During minimization angular parameters will be kept within the range 0 27 The start value is being used in the first minimization of the y function Some care needs to be taken here because it is very easy to pick starting values so far away from the sensible region that the x function reaches too large values that prevent the fit from converging The default scan range applies to both one and two dimensional scans if not being overwritten from the command line using the options scanrange horizontal axis and scanrangey vertical axis The physically allowed range can be used to impose hard limits on the value of a parameter which get activated through the pr option Create the measurement class holding the measurement of the cartesian observables as well as the definition of their Gaussian PDF At first create the the header file in include PDF_Cartesian h ifndef PDF_Cartesian_h define PDF_Cartesian_h include PDF_Abs h include ParametersCartesian h using namespace RooFit using namespace std using namespace Utils class PDF_Cartesian public PDF_Abs 23 public PDF_Cartesian TString cObs TString cErr TString cCor PDF_Cartesian void buildPdf void initObservables virtual void initParameters virtual void initRelations void setCorrelations TString c
7. confidence level The default is to plot contours such that their projection to the axes coincides with the one dimensional scans In this case the resulting innermost contour corresponds to 39 CL When the option is given the innermost contour is enlarged to hold 68 CL Sect asimov run an Asimov toy Sect color changes the plot color of the plotted combination There are six different colors for one dimensional plots defined in GammaComboEngine defineColors that can be specified through numbers 0 5 For two dimensional plots there are four different colors defined in OneMinusC1Plot2d OneMinusC1Plot2d that can be specified through the numbers 0 3 controlplots Make control plot for the PLUGIN toys Sect covCorrectPoint Define the point for the coverage correction experimental Sect covCorrect Activate a coverage correction experimental Sect digits s Set the number of printed digits right of the decimal point Default is automatic evol e Plot the profile likelihood parameter evolution of a PROB scan Sect fix g 1 7 r_dk 0 09 Fix given scan parameters to the given values group LHCb Change the GAMMACOMBO logo to LHCb id 57 When making controlplots controlplots only plot toys generated at a specific scan point e g 57 The id number refers to the scan step and therefor to a specific value of the parameter of interest importance Activate importance sampling for PLUGIN toys Th
8. constraints on CKM parameters V and Ve 4 Tutorial The tutorial consists of three parts a simple Gaussian a two dimensional Gaussian and a circular PDF that puts a constraint on a radius parameter Lets run the tutorial executable to get an overview of what measurements and combinations are defined in the tutorial main tutorial cpp cd tutorial bin tutorial u u always prints the usage and exits AVAILABLE MEASUREMENTS 1 1D Gaussian a_ obs 0 5 2 1D Gaussian a_fobs 1 5 3 2D Gaussian a_fobs b_fobs 4 circle a_ fobs b_ obs AVAILABLE COMBINATIONS 0 empty 1 Gaus 1 2 Gaus 2 3 Gaus 1 amp Gaus 2 4 2D Gaus 5 2D Gaus amp Gaus 1 6 Circle 7 2D Gaus amp Circle 8 Gaus 1 amp Gaus 2 amp 2D Gaus 4 1 Simple Gaussian measurement We start with the probably most simple case of all a theory parameter being constrained through a single measurement that follows a simple Gaussian PDF To find the best estimate for the theory parameter a the following likelihood function is maximized 1 obs tn SC eC ae ee 15 44 Glam Par en a exp 15 Qobs Here the result of the measurement is denoted as Qobs E Faors gt 16 and the numerical values used in the tutorial s measurement 1 are mem EL 17 The resulting confidence intervals for aj are computed with bin tutorial c 1 i var a_gaus ps 1 j where the options mean
9. for each axis nrun 1 run job number 1 This number will be added to the filenames of the produced PLUGIN toy files Files will be overwritten if the same number is given ntoys 100 produce 100 PLUGIN toys per scan point parfile Give a specific parameter file to provide for example start parameters Sect plotid Make the control plot with given ID to save time controlplots Sect pluginplotrange Restrict the PLUGIN plot to a given range to rejcet low statistics outliers po make a PLUGIN only plot prelim Plot Preliminiary into the plots See also unoff printcor Print correlation matrix of each solution found probforce Use a stronger minimum finding algorithm for the PROB method Sect 6 35 pr physical range This will enforce the physical parameter ranges defined in src ParametersTutorial cpp Sect ps 1 print a marker at all local minima If ps 2 is given only the best fit value is indicated ps 1 print solution on the plot The given value configures the position of the numerical value on the plot See the help bin tutorial h for options pulls Make a pull plot illustrating the consistency of the best solution with the observables Sect 6 4 qh A list of quick hacks that can be enabled through the command line that modify very specific areas of the code These areas can be found by searching for calls to the isQuickhack function scanforce f Use a stronger min
10. in obsValSource is meant for information only it will be printed in the verbose output v The truth and toy entries should not be touched void PDF_Cartesian setObservables TString c i if c EqualTo truth setObservablesTruth else if c EqualTo toy setObservablesToy else if c EqualTo year2014 obsValSource arxiv 1408 2748 setObservable xm_dk_obs 2 5e 2 setObservable ym_dk_obs 7 5e 2 25 setObservable xp_dk_obs 7 7e 2 setObservable yp_dk_obs 2 2e 2 J else cout lt lt PDF_Cartesian setObservables ERROR config not found lt lt c lt lt endl exit 1 9 Implement the measured uncertainties of the observables It is important to match the order that of the observables The systematic error can be set to zero without problems but the statistical error should always be finite The obsErrSource string is again solely for informational purposes void PDF_Cartesian setUncertainties TString c if c EqualTo year2014 obsErrSource arxiv 1408 2748 StatErr 0 0 025 xm StatErr 1 0 029 ym StatErr 2 0 024 xp StatErr 3 0 025 yp SystErr 0 0 011 xm SystErr 1 0 015 ym SystErr 2 0 011 xp SystErr 3 0 011 yp else cout lt lt PDF_Cartesian setUncertainties ERROR config not found lt lt c lt lt endl exit 1 10 Implement the
11. measured correlations of the observables Here the order needs to match that of the observables The function resetCorrelations sets all correlations to zero void PDF_Cartesian setCorrelations TString c resetCorrelations if c EqualTo year2014 corSource arxiv 1408 2748 double dataStat xm ym xp yp i 0 247 0 038 0 003 xm 0 247 1 0 011 0 012 ym 0 038 0 011 1 0 002 xp 0 003 0 012 0 002 1 yp 26 J corStatMatrix TMatrixDSym n0bs dataStat double dataSyst xm ym xp yp i 0200525 0 025 0 070 xm 0 005 1 0 009 0 141 ya 0 025 0 009 1 0 008 xp 0 070 0 141 0 008 1 yp bg corSystMatrix TMatrixDSym n0bs dataSyst else cout lt lt PDF_Cartesian setCorrelations ERROR config not found lt lt c lt lt endl exit 1 11 Implement the Gaussian PDF It is possible to use any other functional form for the PDF as will be described in Sec but the Gaussian is probably the most common choice Don t change the name of this PDF object void PDF_Cartesian buildPdf pdf new RooMultiVarGaussian pdf_ tname pdf_ name RooArgSet observables RooArgSet theory covMatrix 12 Implement the second PDF class for the measurement of r2 We will just adapt the existing class for a simple Gaussian measurement from the tutorial PDF_Gaus e copy the header file include
12. o aa a Se 2 4 oe Ree Oe es 6 5 Run an Asimov TOY ooa 2 4 6 eee eee eee SS 6 6 Predict observables a 0 a a a a a 6 7 Attempt a coverage correction Use cases hints and advice 8 Command line options 9 Acknowledgements 10 12 13 15 16 18 21 29 29 30 30 31 32 32 32 32 33 34 34 38 38 1 Introduction The interpretation of observables in terms of parameters of the underlying theory is a common problem in physics Well established methods exist on how to do that but in practice their application is often found to be cumbersome GAMMACOMBO is a framework that hopes to turn this process into a more pleasurable experience The basic principle is to assume a probablility density function PDF for the observables and form a likelihood function From these best fit values as well as confidence intervals or upper limits can be computed following two staticical methods the profile likelihood construction and the PLUGIN method Both methods are located at the more frequentist like side of the spectrum of statistical schools The strength of GAMMACOMBO is to provide tools helping in the following areas e provide a framework to construct the likelihood function e implement minimization strategies to aid the profiling of nuisance parameters provide debug tools helping to identify problems during profiling make publication quality plots in both one and two dimensions e provide a wa
13. 2 2 5 1 1 x 107 35 where the first uncertainty is statistical and the second is systematic The statistical and systematic correlations are given in Tables 3 and Table 3 Statistical cartesian correlations z y T4 Y4 x 1 0 247 0 038 0 003 Y 1 0 011 0 012 A 1 0 002 Y 1 Table 4 Systematic cartesian correlations m y U4 z 1 0 005 0 025 0 070 ae 1 0 009 0 141 7 1 0 008 Y 1 In addition to this real life cartesian example we will add a mock up measurement of the nuisance parameter rB which we can then combine with the cartesian example rBK 0 10 0 01 36 21 1 The following steps describe how to create the source files needed for a new combiner The complete solution is contained in the directory tutorial cartesian so the quick way to get it run is to just copy this directory one level up to where the other combiner directories live cp r tutorial cartesian For a more step by step like experience we will copy the existing tutorial directory and modify the source files to make the new cartesian combiner from scratch cp r tutorial cartesian rm r cartesian cartesian delete the solution 2 Tell the GAMMACOMBO build system about the new combiner vi cmake combiners cmake edit the file to look like SET COMBINER_MODULES tutorial cartesian vi cartesian CMakeLists txt change the name of the project to look like this SET COMB
14. INER_NAME cartesian 3 Define the project parameters These will be y r85 and 62 All the parameters are defined in one single class Here their names are defined their titles unit the default start value for the fit the default scan range and the physically allowed region rename the parameter class header file to match the new project cd cartesian include mv ParametersTutorial h ParametersCartesian h vi ParametersCartesian h edit to change Tutorial for Cartesian also rename the implementation file cd cartesian src mv ParametersTutorial cpp ParametersCartesian cpp vi ParametersCartesian cpp Add the parameters to the ParametersCartesian cpp file void ParametersCartesian defineParameters Parameter p 0 p newParameter g 22 p gt title gamma p gt startvalue DegToRad 70 p gt unit Rad p gt scan range DegToRad 0 DegToRad 180 p gt phys range 7 7 p newParameter d_dk p gt title delta_ B Dk p gt startvalue DegToRad 127 p gt unit Rad p gt scan range DegToRad 0 DegToRad 180 p gt phys range 7 7 p newParameter r_dk p gt title r_ B 7 DK p gt startvalue 0 09 p gt unit p gt scan range 0 02 0 2 p gt phys range 0 1e4 The name is being used on the command line while the title of a parameter will be used in the plots The unit of a parameter can
15. June 8 2015 version 1 0 GAMMACOMBO USER MANUAL Till Moritz Karbach Matthew William Kenzie 1 European Organization for Nuclear Research CERN Geneva Switzerland GAMMACOMBO is a framework to combine measurements in order to compute confidence intervals on parameters of interest A global likelihood function is constructed from the probability densitiy functions of the input observables which is used to derive likelihood based intervals and frequentist intervals based on pseudoexperiments following the PLUGIN method to handle nuisance parameters Contents 2 1 Basic Principle 2 2 The profile likelihood method PROB 2 3 The pseudo experiment based frequentist method PLUGIN 3 Installation 3 1 Installation of subpackages 04 4 Tutorial 4 1 Simple Gaussian measurement merererecre 4 3 Two dimensional Gaussian measurement 4 4 A more advanced example circular constraint 4 5 Changing combinations on the fly 0 2 4 6 Running the PLUGIN method Revie tenes 4 OTOR 5 Create a new combiner module 5 1 Advanced C truth relations 00 5 2 Advanced Non Gaussian observables 0 00 aa aaa 6 Advanced Topics 6 1 Numerical results 0a a a a a a a a a 6 2 Defining a custom scan strategy and starting points 6 3 Plotting an overview of a combination s structure 6 4 Control Plots
16. N 0 20063 EDM 1 58985e 07 COV quality 3 status 0 confirmed yes a_gaus 0 411559 0 758865 0 758865 b_gaus 1 932237 0 265757 0 265757 Here errLow and errHigh are obtained from the parabolic HESSE errors so will always be the same It is possible to print out the correlation matrices of each solution This is triggered by the command line option printcor 6 2 Defining a custom scan strategy and starting points In more complex combinations it can happen that the x function has multiple local minima that are well separated from each other As the scan progresses different values of nuisance parameters can shrink the x hill separating the minima and it may happen that the fit scanner jumps into the second mininum and stays there This will usually be visible as a jump in the p value curve One can be however unlucky and completely miss a second mininum A good way to alleviate this problem is to redo the scan using another starting point and to accept points with a lower x value a higher p value into the final curve if any are found In GAMMACOMBO the default scans are done in drag mode where the fit at each scan step uses as start parameters the fit result from the previous scan step Therefore it matters what the start parameters of the first scan were The default is to use the parameter values defined in the parameter class ParametersTutorial cpp in the tutorial It is possible to override thi
17. PDF_Gaus h into include PDF_rb h e edit the new header file to adjust the class name PDF_rb and to in clude the header file of the parameter class of the cartesian project ParametersCartesian h e copy the implementation file src PDF_Gaus cpp into src PDF_rb cpp e edit the class name change to the parameter class of the cartesian project e include the paramter r_dk rather than a_gaus e change the name of the r2 observable to e g r_dk_obs similar for the theory value e set the observed central values for the identifier string year2013 to 0 1 and the statistical error to 0 01 27 13 14 15 If you created the folder for the cartesian project as suggested by copying the tutorial folder you should remove now the non needed classes PDF_Circle h PDF_Gaus h PDF_Gaus2d h to not confuse the build system Also remove the corresponding implementation files from the src directory Now we need to create the new main file Rename the one from the tutorial to main cartesian cpp and change it to look like this include lt stdlib h gt include GammaComboEngine h include PDF_rb h include PDF_Cartesian h using namespace std using namespace RooFit using namespace Utils int main int argc char argv i GammaComboEngine gc cartesian argc argv define PDFs gc addPdf 1 new PDF_Cartesian year2014 year2014 year2014 Cartesian gc addPdf 2 new PDF_rb year2013 year2013 y
18. a plugin j 1 pr scanrange 1 5 2 5 i a_gaus 1 24 0 51 0 50 0 74 1 01 0 68CL Plugin a_gaus 1 5 1 4 0 5 1 0 1 9 0 95CL Plugin Gaus 1 1 CL P Gaus 1 GammaCombo_ l 0 1 2 AGaus Figure 8 Applying a physical range to combination 1 and running the PLUGIN method In this case the PLUGIN method is equivalent to the method by Feldman Cousins The difference to the profile likelihood method is most apparent close to the physical boundary at a 1 5 4 8 Running the PLUGIN method in 2D It is also possible to run the PLUGIN method in two dimensions The commands follow the previos lines however a few additional complications arise Mostly the 2D scan becomes computationally demanding quickly due to the vast number of points needed for smooth 2D contours Lets take as an example combination 7 2D Gaus amp Circle which is the pink contour in Fig 4 We will restrict the 2D grid to a size of 30 x 30 bins The result is shown in Fig P A first attempt just plots the PLUGIN contours along with the PROB contours without specifying the contours should be true 2D confidence contours and should therefore 18 cover the true value in e g 68 of the cases in two dimensions Since one would need additional assumptions on how to translate the p value which is computed directly by the PLUGIN method into an 1D equivalent this is not done in GAMMACOMBO There
19. ar b_gaus i redo the scan to print 2D confidence contours and a marker at the best fit value bin tutorial c 4 var a_gaus var b_gaus i npoints 70 2dcl ps 1 just remake the plot without redoing the scan and activate magnetic plot boundaries bin tutorial c 4 var a_gaus var b_gaus i 2dcl ps 1 magnetic a plot The command line options mean var a_gaus var b_gaus specify the two variables to perform the scan for The variable given first will be plotted on the horizontal axis the second one on the vertical axis 2dcl plot contours corresponding to two dimensional confidence level The default is to plot contours such that their projection to the axes coincides with the one dimensional scans In this case the resulting innermost contour corresponds to 39 CL When the option is given the innermost contour is enlarged to hold 68 CL npoints 70 use 70 scan points in each direction so 70 4900 in total ps 1 print a marker at all local minima If ps 2 is given only the best fit value is indicated magnetic switches on magnetic plot boundaries that will drag the contours towards them In many cases this results in nicer contours but not always a plot activates the plot action Whenever a scan is done the result is saved into a file in plots scanner This can be read in to remake the plot without having to rerun the entire scan With more complex combinations this s
20. aus AGaus Figure 2 Result of tutorial 4 a two dimensional Gaussian ellipse Sec 4 3 Left with 1D confidence regions Right with 2D confidence regions and a marker at the best fit value 4 4 A more advanced example circular constraint The next measurement of the tutorial is a circular constraint for the parameters a and b This happens if we have a measurement of a radius like parameter In measurement 4 of the tutorial a measurement of r is implemented r 2 0 0 25 27 which is related to the parameters a and b through the truth relation r va b2 28 The constraint is shown in Fig 3 left Note that when computing it code below it is likely that several best fit points are found This is because all points on the circle have the same likelihood and due to the numerical accuracy some get picked up In the default configuration a rescan is done for each local maximum of the likelihood is found this might take a while in this case Let s combine the circular constraint with the two dimensional Gaussian measurement from the previous example At first both could be added to the same plot Fig 3 right Then we could combine the circular constraint with the two dimensional Gaussian measurement and plot all three combinations together This is shown in Fig scan the circular constraint of the tutorial bin tutorial c 6 var a gaus var b gaus i add the previous 2D Gaussian to the same plot
21. aves time While a two dimensional scan is running in interactive mode i a window opens illustrating the scan progress The scan is done in a trajectory spiralling away from the start points In this plot the current Ay is plotted Any subsequent scan opens one more of these plots If in a subsequent scan lower Ay values are encountered this will show up At the end of the scan one more window is opened showing the final Ay from which the confidence contours are computed Updating these control plots can consume a significant computing time for simple combinations like the tutorial In these cases the program runs much faster non interactively without i For more complex combinations two dimensional scans can take a long time especially with a high granularity For these cases the a plot option is very useful which allows the plot to be redone without having to rerun the scan This way the look and feel can be changed plot markers at best fit points change titles change colors change the number of contours activate magnetic boundaries etc When running with the a plot option 11 the only other important options are c 6 var a_gaus var b_gaus all other ones are not needed to identify the correct file and can be omitted 4 7 T 7 j 7 T 2 3L 2D Gaus o 2 3H 2D Gaus zamma Combo 21 5 2 1 1 OF 5 0 IF 1 J rn A i i j i _ L iae a i 1 i 2 2 1 0 1 2 2 2 1 0 1 2 AG
22. bination 7 see Sec is SOLUTION 0 combination tutorial7 title 2D Gaus amp Circle date Sun Feb 22 23 01 27 2015 FCN 0 20063 EDM 1 58985e 07 COV quality 3 status 0 confirmed yes Parameter FinalValue Error HESSE a_gaus 0 411559 0 758865 1 b_gaus 1 93224 0 265757 The several components have the following meaning SOLUTION 0 is the running index of the displayed solution combination the name given to the combination in the main file title the title given to the combination in the main file FCN the value of the x function EDM estimated distance to minimum as provided by MINUIT COV quality result of RooFitResult covQual status result of RooFitResult status confirmed solutions are first found as maxima of the p value curve which is obtained while the scan parameter is fixed to the scanned point Before printout the y function gets reminimized with the scan parameter floating too Only if the reminimization agrees reasonably with the result from the scan the solution is regarded as confirmed Error HESSE the parabolic error obtained by HESSE All solutions get stored to disk into parameter files into the plots par directory They have the following format 30 cat plots par tutorial_tutorial7_a_gaus dat auto generated by ParameterCache printed on Sun Feb 22 23 01 27 2015 ParameterName value errLow errHigh o SOLUTION 0 FC
23. do so at this page Please also sign up to our mailing list here so we can keep you informed of updates downtime changes etc 3 You should be able to browse subpackages on the web in the git private area of GAMMACOMBO on hepforge which can be found To install a subpackage on your system you can do the following cd gammacombo the location you installed the main package above set useful environment variable HEPFORGE lt hepforge_user_name gt login hepforge org clone the package git clone HEPFORGE hepforge git gammacombo private lt sub_package_name gt edit cmake combiners cmake file to include the sub_package_name cd build cmake make can use j4 for 4 CPU cores make install To run the subpackage follow steps as per the tutorial subpackage in the rest of this document For example to print the usage to check that it runs ok cd lt sub_package_name gt bin lt sub_package_name gt u print the usage Table 2 A list of GAMMACOMBO subpackages and their uses Subpackage name Details gammacombo Combination of LHCb analyses which measure or con strain the CKM angle y Note that this has the same name as the main package but only for historical reasons biggammacombo Combination of analyses which measure or constrain B mixing parameters ls ATs hfag Combination of charm mixing averages belle2_Vub Combination of constraints on CKM parameter Vap Vub_Vcb Combination of
24. ear2013 rb Define combinations gc newCombiner 0 empty empty gc newCombiner 1 cartesiani Cartesian 1 gc newCombiner 2 cartesian2 rb Die gc newCombiner 3 cartesian3 Cartesian amp rb 1 2 Run gc run Change to the build directory and build the new project cd build make install j4 16 Run the new combiner The results are shown in Fig cd cartesian bin cartesian u bin cartesian c 1 c 3 var g i bin cartesian c 1 c 2 c 3 var r_dk i 28 Cartesian rb Cartesian amp rb T Cartesian Cartesian amp rb 1 CL x 1 CL i a MP call x an i GammaComb rescued he GammaComb gt oo S a 2 NK Ope eo gt N Jit tut Lit ial c S y 5 P 150 0 2 yI rp Figure 10 Results of the new combiner module cartesian Left curves for parameter y Right curves for nuisance parameter r2 5 1 Advanced C truth relations To be filled 5 2 Advanced Non Gaussian observables To be filled 29 6 Advanced Topics Most of this section is yet to be filled 6 1 Numerical results The printout of GAMMACOMBO contains the parameter values of each local minimum of the xy function that was found during the scan These solutions are ordered by their x value so the global minimum is displayed first An example from tutorial com
25. ed on your system It also requires an installation of Root compiled with the RooFit library and a C compiler recent enough to support C 11 See a full list of dependencies in Tab 1 The source code is hosted on github com The following steps install GAMMACOMBO on your system get and build gammacombo git clone https github com gammacombo gammacombo git cd gammacombo if you want to work with the most recent version of the code switch to the development branch git checkout development mkdir build cd build cmake make can use j4 to use 4 CPU cores make install In addition to the GAMMACOMBO core library this also builds an example combiner called tutorial It is also possible to compile with debug symbols cmake DCMAKE_BUILD_TYPE Debug To run a first combination try the following run the tutorial cd tutorial bin tutorial u usage bin tutorial c 1 var a_gaus i run combination 1 ls 1 plots pdf tutorial_tutoriall_a_gaus pdf the plot that was created More on the tutorial is described later in Sec The GAMMACOMBO source code is documented with doxygen You can build the doxygen web page documentation if you have doxygen installed and probably also graphviz s dot command The documentation is created in the doc subdirectory of the top level directory build the doxygen documentation still in the build directory make doc open the created webpages
26. ee f f AlB 2 The likelihood is obtained by fixing the observables to the measured values If multiple measurements of the same kind were performed the likelihood is the product of as many PDF terms But in the context of the combination each measurement was performed exactly once thus the distinction is subtle gt gt L FAD 3 A Apps To give an example lets consider the combination of two measurements The first is a cartesian measurement which measures four cartesian observables that are related to polar coordinates through ta rBE cos 68 7 4 y p sin g 7 5 The second measurement is a direct measurement of the parameter r2 robs tR 6 Therefore the vector of parameters has the following components y rg B V 7 where T means transpose The purpose of the combination is to find the best values of the components of a defined as those that maximize the combined likelihood and to compute well defined error intervals The vector of observables A has the following form gt A Sages Y Ge dee s 8 The components of A are related to those of amp by the truth relations Eqns 4 and lel We can now construct the combined likelihood as L a Asl A amp 9 where Avis holds the measured results and A q are the truth relations In case of uncorrelated observables this factors into L a LIAA ons Aa f 10 To obtain confidence intervals a first a
27. eory gt add new RooFormulaVar ym_dk_th ym_dk_th r_dk sin d_dk g p theory gt add new RooFormulaVar xp_dk_th xp_dk_th r_dk cos d_dk g p theory gt add new RooFormulaVar yp_dk_th yp_dk_th r_dk sin d_dk g p 7 Implement the definition of the observables The names of the observables need to match those of the predicted ones with the only difference that they end on _obs Their titles will show up in pull plots The configured value is not important but the ranges should be far away from any boundary there might be as they will affect the toy generation within the PLUGIN method void PDF_Cartesian initObservables observables new RooArgList observables The order of this list must match that of the COR matrix observables gt add new RooRealVar xm_dk_obs x 0 1 1 observables gt add new RooRealVar ym_dk_obs y O 1 1 observables gt add new RooRealVar xp_dk_obs x 0 1 1 observables gt add new RooRealVar yp_dk_obs y 0 1 1 8 Implement the measured values of the observables Based on a config string dif ferent values can be loaded This is very useful if different versions of a given measurement exist for example a first measurement from the year 2010 and an update from 2012 These config strings are being used inside the main program file main cartesian cpp see below The string stored
28. f Eq was combined in quadrature The numerical result of Eq 19 is an 1 07 0 46 22 Lets put them both into the same plot and then also plot the combination of both The resulting plot is shown in Fig multiple instances of c will be added to the same plot bin tutorial c 1 c 2 i var a_gaus combination number 3 combines both Gaussians bin tutorial c 1 c 2 c 3 i var a gaus ps 1 Options used c 1 c 2 c 3 When the combination argument is given multiple times all combinations will be computed and added to the same plot The order controls the order in the plot where the last one is in the foreground 4 3 Two dimensional Gaussian measurement The next measurement of the tutorial measurement 3 contains a two dimensional Gaussian PDF in the previous variable a and another one b Now the likelihood function is given by L a bin G aobs Oaons Aths bobs O bobs bin 1 l asyr a exp Zobs Lin V Zobs Tth 23 Vdet V V27 7 ai a where Z dobs bobs is the vector of observables n is the number of observables and V is the covariance matrix Combination number 4 contains only this two dimensional measurement the numerical values of the observations are een Ween On 24 bie ates ELD 25 pla b 0 6 26 The resulting plots are shown in Fig 10 the 2D scan is triggered by giving a second var argument bin tutorial c 4 var a_gaus v
29. filled curve without the PROB curve 4 7 The standard Feldman Cousins example The presence of a physically forbidden region for a parameter complicates the statistical treatment In particular the profile likelihood construction is easily spoiled when the boundary is close to the region of interest In this situation the method by Feldman and Cousins 3 provides an accepted frequentist alternative In the absence of nuisance parameters the PLUGIN method is equivalent to the method by Feldman and Cousins 16 Lets take the example of the simple Gaussian measurement given in Sec The parameter a has a physical limit configured see src ParametersTutorial cpp a gt 1s 29 which has not been enforced so far It can be activated through the physical range option pr The resulting plot is shown in Fig 7 left Note that we are trying to scan in the non physical region which is currently not well supported The drop at a 1 5 is not vertical due to the finite binning and as a result the interpolated confidence interval is not accurate This can be overcome by restricting the scan range to the physical region Fig right activate physical parameter limits bin tutorial c 1 i var a_gaus ps 1 pr restrict scan range to the physically allowed region bin tutorial c 1 i var a gaus ps 1 pr scanrange 1 5 2 5 The arguments mean pr physical range This will enforce the physical parameter ra
30. fore the contours don t match until one give the 2dcl option Both the PLUGIN and PROB contours seem to match quite well in this example too This is not trivial but perhaps not too surprising given that they already matched in the one dimensional case of Fig To make the best out of scarce toy statistics in the 2D case a smoothing option was implemented for 2D contours However it needs to be used with care in this example Fig 9 it seems to artificially widen the contour over what appears to be reasonable The 2D PLUGIN method also produces a p value histogram that allows onen to directly judge the status of toy statistics This is also shown in Fig 9 run 2D plugin toys for combination 7 2D Gaus amp Circle bin tutorial c 7 var a_gaus var b_gaus a pluginbatch ntoys 100 npoints 30 npointstoy 30 nrun 1 bin tutorial c 7 var a_gaus var b_gaus a pluginbatch ntoys 100 npoints 30 npointstoy 30 nrun 2 bin tutorial c 7 var a_gaus var b_gaus a pluginbatch ntoys 100 npoints 30 npointstoy 30 nrun 3 compute the plugin contours from all toys This will produce non matching contours bin tutorial c 7 var a_gaus var b_gaus a plugin j 1 3 i group off legsize 0 38 0 10 leg 0 17 0 80 with the 2dcl option the contours match bin tutorial c 7 var a_gaus var b_gaus a plugin j 1 3 i group off legsize 0 38 0 10 leg 0 17 0 80 2dcl lets try to use the
31. ich observables deviate the most from the prediction It will also show the value of the x function at the best fit value and the corresponding naive fit probability e Giving the option e will produce a plot of the parameter evolution of a PROB scan e Many control plots exist to judge the quality of the PLUGIN toys 6 5 Run an Asimov toy 6 6 Predict observables e Observable names are unique in the Combiner so multiple PDFs with observables of the same name can be combined For this purpose each observable has a unique ID appended for example UID2 for ID 2 The number corresponds to the number that the PDF which provides this observable has in the Combiner e One can make predictions for observables by simply scanning for them When doing this the corresponding x constraint usually needs to be tightened This effectively replaces the inversion of the truth equation by the minimization A plot is being produced showing the deviation of the actual value of the observable and the predicted one as a function of the scan step This should be a flat line at zero and allows to judge if the constrained was tightened enough 32 6 7 Attempt a coverage correction 33 7 Use cases hints and advice to be filled 8 Command line options Here is a list of available command line options The full list can always be obtained by running bin tutorial h 2dcl plot contours corresponding to two dimensional
32. imum finding method for the PLUGIN method scanrange 1 5 2 5 Adjust the scan range to the given range The default range is defined in src ParametersTutorial cpp Sect scanrangey 1 5 2 5 For 2D plots Adjust the scan range of the vertical axis to the given range The default range is defined in src ParametersTutorial cpp Sect sn2d 2D version of sn sn Save nuisances to a parameter cache file at certain scan point Sect title Override the title of a combination unoff Plot Unofficial into the plots See also prelim var a_gaus var b_gauss specify the two variables to perform the scan for The variable given first will be plotted on the horizontal axis the second one on the vertical axis var the variable name of the theory parameter to be scanned a plot activates the plot action Whenever a scan is done the result is saved into a file in plots scanner This can be read in to remake the plot without having to rerun the entire scan With more complex combinations this saves time a pluginbatch activates the PLUGIN batch mode This will run a number of PLUGIN toys and save them into an output file Many of these jobs can be run in parallel for example on a batch farm a plugin activates the PLUGIN mode This will read in a number of PLUGIN toy files that where previously produced These files need to be specified using the j option c 1 2 modifying combination number 1 by adding PDF number 2 Sec
33. is reduces the number of toys generated in a region with large expected p value therefore saving computing cycles that can better be invested to get a better p value accuracy in the tails 34 intprob Use the internal PROB x histogram instead of the x from the toy files to evaluate 1 CL of the PLUGIN method Sect largest Report largest CL interval from the lowest boundary of all intervals to the highest boundary of all intervals Useful if two intervals are very close together leg Give print options for the legend such as its position or to turn it off lightfiles Produce only light weight PLUGIN toy files They cannot be used for control plots but save disk space log make logarithmic one dimensional 1 CL plots magnetic switches on magnetic plot boundaries that will drag the contours towards them In many cases this results in nicer contours but not always Sect ncontours Plot this many sigma contours in 2D plots max 5 ndivy Set the number of axis divisions y axis in 1D and 2D plots ndiv Set the number of axis divisions x axis in 1D and 2D plots nosyst Ignore systematic uncertainties npoints2dx Configure the number of scan points on x axis in 2D scans npoints2dy Configure the number of scan points on y axis in 2D scans npointstoy Number of scan points used by the plugin method npoints Configure the number of scan points used in 1D and 2D scans for 2D use same number of points
34. le described in the PDG booklet 2 It is equivalent to the profile likelihood method implemented e g in the MINOS method of MINUIT The confidence intervals are obtained from the 1 CL curves by finding the point where it intersects the 1 CL 1 0 68 and 1 0 95 levels These points are found by interpolating between the four nearest scan points The PROB method is known to undercover in many situations that is the reported intervals contain the true value in a lesser number of times than claimed However it can also overcover in which case the reported intervals are too large 2 3 The pseudo experiment based frequentist method PLUGIN A more accurate approach is to not rely on the linearity of the truth relations that relate the observables to the parameters for example Eqns 4 and 6 Instead on assuming that the test statistics Ax is distributed as a x7 one should compute its distribution using pseudo experiments This method is based on the approach by Feldman and Cousins 3 and extends it by introducing the concept of nuisance parameters In the PLUGIN method sometimes also ji method the nuisance parameters are kept at all times at their best fit values determined by the observed data that is they are kept at their profile likelihood point The algorithm of the method is equivalent to the algorithm propsed by Feldman and Cousins in the absence of nuisance parameters 5 It is briefly described in the f
35. nd simple approach is to inspect the profile likelihood curve which is described in Section 2 2 This has the advantage of being computationally cheap but in many examples comes at the price of bad frequentist coverage A better method is to rely on a method based on pseudo experiments the PLUGIN method which is described in Section 2 3 This method is much more computing expensive It builds upon the profile likelihood method so in any case it does make sense to start with the easier PROB method 2 2 The profile likelihood method PROB We define a y function as xX a 2In L G 11 In general there are many equivalent global minima reflecting the fact that there are multiple solutions possible At each of these minima the x has a value y2 To evaluate the confidence level of a certain truth parameter for example y at a certain value yo we consider the value of the y function at the new minimum x2 where the y component of a is fixed to 7 yo The new minimum satisfies Ay x2 x2 gt 0 In a purely Gaussian situation for the truth parameters the p value p 1 CL is given by the probability that Ay is exceeded for a y distribution with one degree of freedom 1 CO 1 CL _ es ars 2 12 V2T 1 2 i 12 Because Eq can be evaluated using the TMath Prob function 1 CL Prob A xX nao 1 13 this method is sometimes called the PROB method for examp
36. nges defined in src ParametersTutorial cpp scanrange 1 5 2 5 Adjust the scan range to the given range The default range is defined in src ParametersTutorial cpp 4 a 4 c J Q 1 2F Gaus 1 q Q 1 27 Gaus 1 J 4 E J Ir m IF 0 8 F GammaCombo 0 8 GammaCombo 0 6 ee 0 6 J 0 4 0 46 2 AGaus AGaus Figure 7 Applying a physical range to combination 1 Left We tried to scan in the now unphysical region which caues some inaccurracies see text Right We restricted the scan range to the physical region resulting in accurately deterimined but statistically questionable intervals Lets now run the PLUGIN toys for this example The result is shown in Fig The difference to the profile likelihood method is most apparent close to the physical boundary the boundary seems to push away the lower boundary of the confidence interval The 17 observed interval boundaries reproduce the nominal ones taken from the original Feldman Cousins paper 8 The values to compare to are given in Tab X of Ref at zo 1 0 because our Gaussian is shifted with respect to theirs by 1 5 units run plugin toys for combination 1 simple Gaussian with physical boundary try to run mulitple jobs to generate more toys bin tutorial c 1 var a_gaus a pluginbatch ntoys 400 pr scanrange 52o nrun compute the plugin intervals from all toys bin tutorial c 1 var a gaus
37. ollaboration R Aaij et al Measurement of the CKM angle y using Bt gt DKF with D K atn K8K K decays arXiv 1408 2748 38
38. ollowing For a certain value of interest yo we 1 calculate Ay y2 Xin as before 2 generate a toy result ve using Eq 3 with parameters set to amp as the PDF 3 calculate Ax of the toy result as in the first step by replacing An Ae i e minimize again with respect to once with the scan parameter floating and once with the scan parameter fixed 4 calculate 1 CL as the fraction of toy results which perform worse than the measured data i e 1 CL N Ax lt Ay Mioy 14 It has better coverage properties than the PROB method but also not necessarily a perfect coverage The reason is that at each scan step the nuisance parameters components of a other than the scan parameter are plugged in at their best fit values for this step as opposed to computing an n dimensional Neyman confidence belt In our simple example we have already n 3 but situations with many more parameters are very common This would render a fine grained scan of the parameter space computationally very expensive In our experience the intervals obtained using the PLUGIN method are usually within 10 20 of those obtained using the PROB method This means there are only few cases where qualitative statements obtained using the PROB method become obsolete when considering the PLUGIN results 3 Installation The installation of GAMMACOMBO requires the version management system git and the build system cmake to be install
39. on Mac cd open doc html index html Table 1 Dependencies of GAMMACOMBO on libraries and software packages It is not excluded that it can run with older versions as well however these are the versions GAMMACOMBO has been tested with dependency minimum version details ROOT 5 34 23 GAMMACOMBO is able to run also with previ ous versions but it has been thorougly tested with this version ROOT needs to be com piled to include RooFit for example using the enable roofit option in the configure step RooFit 3 60 see ROOT boost 1 57 0 Doxygen 1 8 2 optional 3 1 Installation of subpackages Each separate combination requires a new subpackage which exists in its own directory The example given in this manual is a subpackage called tutorial There are several combinations implemented by the GAMMACOMBO code which each have their own subpackage These are listed in Table 2 Some of these contain private data which has not yet been published or is not intended to be public Consequently each subpackage is held in a private location on a hepforge server and only certain users are allowed access to them If you require access to the subpackages then you will need to follow these steps 1 Inform us you require access by emailing us gammacombo owner projects hepforge org with your name and a justification 2 Register as a hepforge user citing your reasons for joing as a GAMMACOMBO user developer You can
40. s default to explicitly start at a certain point For this the parameter files see Sec 6 1 are used In order to make a start parameter file one simply copies one parameter file adding _start to the name For example cp plots par tutorial_tutorial7_a_gaus dat plots par tutorial_tutorial7_a_gaus_start dat Start parameter files can contain any number of parameter points labelled SOLUTION inside the file For each point found one rescan will be performed 31 Parameter files can also be used from other combinations If it contains values for parameters that are not in the current combination they will simply be ignored Instead of creating a start parameter file one can also just provide a dat file through the command line option parfile A good strategy to obtain flawless p value curves is to rescan once per local minimum of the y function This requires that one knows already more or less where the minima are more about strategies how to find them in Sec 7 For this reason the default scan strategy in GAMMACOMBO is to first run an initial scan and then to rescan for each local minimum encountered 6 3 Plotting an overview of a combination s structure GAMMACOMBO produces a graphical representation of the content of a combination in the dot format that can be interpreted by the dot command of the graphviz package 6 4 Control Plots e Giving the option pulls will produce a pull plot showing wh
41. smoothing option for the contours in this case it doesn t seem to do any good bin tutorial c 7 var a_gaus var b_gaus a plugin j 1 3 i group off legsize 0 38 0 10 leg 0 17 0 80 2dcl smooth2d The arguments mean npoints 30 use 30 x 30 bins for the 2D PROB scan which is done prior to the PLUGIN toys npointstoy 30 use 30 x 30 bins also for the 3D PLUGIN grid In principle they don t have to use the same granularity You will receive warnings if as a consequence of unequal bin sizes the PLUGIN scan point and the PROB point where the values of the nuisance parameters is taken from differs too much group off turns off the group label else in this case the large GAMMACOMBO string would cover the legend 19 legsize 0 38 0 10 legend size we use it to reduce the size of the legend leg 0 17 0 80 legend position we move the legend up a little to not cover 2dcl use two dimensional confidence contours for the PROB method smooth2d apply a smoothing algorithm to the 2D contours to be used with care see text 2 4 i r 1 r 1 r r 1 2 4 i r 1 r 1 r 5 i 2D Gaus amp Circle Prob i 5 pi 2D Gaus amp Circle Prob 7 ej 3 H 2D Gaus amp Circle Plugin co 3 HE 2D Gaus amp Circle Plugin 4 2 2h Ib Ib Ob o 1 IF contours have meone ient Clicontent i l F A 1 0 1 2 AGaus 2 4 i r 1 r i r i p value for 2D Gaus amp
42. t 36 u V 5 combination number 5 This refers to pre defined combinations A list of available IDs can be obtained by running with the usage flag bin tutorial u When the combination argument is given multiple times all combinations will be computed and added to the same plot The order controls the order in the plot where the last one is in the foreground Sect Activate debug level output interactive mode Any plot will be shown directly in the familiar ROOT canvas Exit with Ctrl c 1 3 read in PLUGIN toy files number 1 through 3 Prints usage information and exits Activate verbose output 37 9 Acknowledgements The author would like to thank Max Schlupp Marco Gersabeck Omer Tzuk Matt Kenzie and Florian Bernlochner for valuable discussions and contributions to GAMMACOMBO References 1 LHCb collaboration R Aaij et al Measurement of the CKM angle y from a combina tion of B Dh analyses Phys Lett B726 2013 151 arXiv 1305 2050 2 Particle Data Group J Beringer et al Review of Particle Physics RPP Phys Rev D86 2012 010001 3 G J Feldman and R D Cousins A Unified approach to the classical statistical analysis of small signals Phys Rev D57 1998 3873 4 B Sen M Walker and M Woodroofe On the unified method with nuisance parameters Statistica Sinica 19 2009 301 5 T M Karbach Feldman Cousins Confidence Levels Toy MC Method arXiv 1109 0714 6 LHCb c
43. void setObservables TString c void setUncertainties TString c 5 Constructor implementation d I PDF_Cartesian PDF_Cartesian TString cObs TString cErr TString cCor PDF_Abs 4 lt configure the number of observables name cartesian lt configure the PDF name should be unique initParameters initRelations initObservables setObservables cObs setUncertainties cErr setCorrelations cCor buildCov buildPdf PDF_Cartesian PDF_Cartesian 6 Implement the parameter initialization and the truth relations The truth relations are implemented through RooForlumaVar objects It is also possible to implement them through customized objects that inherit from RooAbsReal This way relations of arbitrary complexity can be added This will be discussed in Sec The names of the predicted observables need to end on _th d void PDF_Cartesian initParameters ParametersCartesian p lt use the project s parameter class parameters new RooArgList parameters parameters gt add p get r_dk parameters gt add p get d_dk parameters gt add p get g void PDF_Cartesian initRelations 24 theory new RooArgList theory RooArgSet p RooArgSet parameters The order of this list must match that of the COR matrix theory gt add new RooFormulaVar xm_dk_th xm_dk_th r_dk cos d_dk g p th
44. y to run large scale toy experiments on a batch system provide debug tools helping to judge the quality of frequentist toys combine likelihood functions corresponding to different measurements While being able to save the user a lot of tedious work one of the design choices is to not invent a black box that is just too dark After all applying statistical methods is in most cases not trivial and requires a detailed understanding of what is going on On the downside this requires the user to code in C GAMMACOMBO is able to run state of the art sized combinations of measurements An example of such combinations is LHCb s combination of measurements that are sensitive to the CKM angle y i 2 Statistical methods 2 1 Basic Principle The strategy to combine the input measurements described in the previous sections is to build a combined likelihood function This combined likelihood is the product of the individual likelihoods Denoting the observables by capital Latin letters and parameters by Greek letters the combined likelihood is c a J E 1 where the product runs over all input measurements the vectors B hold the parameters of the input measurements the vector holds all parameters of the B and the are the input likelihoods In this context the likelihood and the probability density function PDF are very similar The PDF say f is a function not only of the parameters B but also of the observables A s
45. ys and save them into an output file Many of these jobs can be run in parallel for example on a batch farm 15 ntoys 100 produce 100 PLUGIN toys per scan point nrun 1 run job number 1 This number will be added to the filenames of the produced PLUGIN toy files Files will be overwritten if the same number is given j 1 3 read in PLUGIN toy files number 1 through 3 po make a PLUGIN only plot The results in Fig 6 show that first of all the PLUGIN method agrees quite nicely with the profile likelihood method Probably it will yield slightly larger intervals but in order to be conclusive one would have to run more toys Also the result is visibly non Gaussian as it develops an asymmetric tail on the left This was already apparent from the two dimensional Fig 4 In this case however the non Gaussianity likely doesn t affect the statistical coverage as both methods return quite similar results E 2D Gaus amp Circle 1 CL F 2D Gaus amp Circle 2D Gaus amp Circle 3 1 25 j i TTT 0 8 0 6L 0 4 0 2 ilia dia es Pi tiii 2 A Gaus A Gaus Figure 6 Result of running the PLUGIN method with combination number 7 a 2D Gaussian with a circular constraint Left the points are the result from the toys while the filled curve corresponds to the PROB method In this case there is good agreement between both Right Plotting the result of the PLUGIN as a
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