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AFit User Guide Abstract Contents - Particle Physics Research Centre

1. 2 24 Voigtian This PDF has a type voigtian when using the AFitPdfFactory Figure 22 shows an example of the Voigtian PDF shape This function is defined as the convolution of a Gaussian and a Breit Wigner distribution It is useful when trying to describe mass peaks with the particle width T is comparable to the detector resolution The functional form of this PDF is 00 1 Plem T 0 N J G 2 0 0 BW z 2 m T dz 31 00 where G is the Gaussian and BW is the Breit Wigner The following code snippet will make a Voigtian PDF shape with a mean of 0 0 a width of 1 0 and ao of 1 0 26 A RooPlot of 0 03 PDF o N oa ME AAA AA ADA tae o N Prejection of Voigtian 2 a 5 4 3 2 1 0 1 2 3 4 5 Figure 22 An example of the Voigtian PDF shape AFitVoigtianShape vbld x pdf RooAbsPdf v vbld getPdf 2 25 Composite Add PDF Composite PDFs can be constructed from their individual components using an AFitAddPdf Each component is added with a relative fraction as follows P x fi f iPDF x foPDF xz saa d fifa In 1 PDFn zx 32 This PDF has a type add x y when using the AFitPdfFactory The comma separated variables after the colon specify the different PDF types to be added together 2 26 Composite Multiply PDF This PDF has a type multiply followed by a colon and a comma separated list of PDF types to i
2. signal default background default The final part of the FitConfiguration block specifies the fit yields and allowed ranges for each of the fit components signalYield 30000 10 000 L 100 1e6 backgroundYield 100 100 000 L 1000 1e4 Each fit component specified has to have the functional form of the PDFs defined The available options are listed in Table 1 For example the signal PDF for this example has an exponential function for the t distribution signal signal_t_type exp The parameters of the exponential function are set by the following signal_t signal_t_constant 2 2 0 1 L 4 0 4 0 signal_t_fitLifetime true where the block name is derived from the component name and variable name this is just lt component name gt lt variable name gt and the PDF parameter names are prefixed by the same string In this particular example the option signal tfitLifetime is set to true so that the functional form of the exponential is e7t constant where constant corresponds to the lifetime see section 2 9 for details The background PDF is defined in an analogous way 6 2 Simple rare B decay search at BaBar or Belle Two kinematic variables can be used to select signal events in an ete Y 45 BB event These are mpg and AE mgs V s 2 pi ps E ph is the beam energy substituted mass and AE Ef s 2 is the difference betgween the B candida
3. toy generateToys macros MuonLifetime txt fpr 1 100 0 The AFitToy instance uses the input configuration file to build the likelihood fit model The second argument fpr determines what is done with the generated data described below Using the likelihood toys with initial random number seeds corresponding to the toy number 1 through 100 the third and fourth arguments to the generateToys member function The final argument is a prototype that can be used when generating toy samples for any likelihoods that depend on conditional variables The options fpr determine the following e f Fit the generated data e p Persist the generated data in a root file The file name is Seed_toydata root and the data are saved as a RooDataSet data e r Persist the results in a root file The file name is Seed_toyresults root and the RooFitResult fit Result is saved to the file along with a TTree resultdata that contains the fit result information In order to inspect the results of an ensemble of toy MC experiments it is easy to chain together the resultdata TTrees obtained using the following TChain chain resultdata toy setOutputDir toy toy chainResults chain 1 100 If you want to make sure that fits converged ok with status 0 then the last line should be re placed by toy chainResults chain 1 100 kTRUE The chainResults member fun
4. Bbg0 default give initial values for the yields of each component signalYield 500 00 10 000 L 100 10000 continuumYield 2000 00 10 000 L 100 10000 Bbg0Yield 50 000 10 000 L 100 10000 define the shapes used for the signal Mpg and AE PDFs signal Signal_bMes_type gaussian Signal_bDeltaE_type landau define the shapes used for background Mpg and AE PDFs continuum continuum_bMes_type argus continuum_bDeltaE_type poly2 define the shapes used for background Mpg and AE PDFs Bbg0 Bbg0_bMes_type argus Bbg0_bDeltaE_type poly2 With this information specified in a configuration file you can generate a configuration that 34 includes PDF parameters by running the following commands AFitMaster master my_model configuration txt RooAbsPdf pdf master getPdf master writeDataCard cout where cout can be replaced by any ostream object for example a text file you ve just opened 4 7 The fitData function This member function is used in order to fit the data to the total PDF model build from the AFitMaster If the fitOptions variable in the configuration file s FitC on figuration block specifies r this function will return a pointer to a RooFitResult that should be non zero 4 8 Blinding yields The fit yields for components are by default unblind However AFit is written so that any of the yields may be blinded using the appropriate
5. 2 10 Flatte Function An example of the Flatte function distribution is shown in Fig 9 More details will be added in due course 2 Entries 0 01 Figure 9 An example of the Flatte PDF shape The following code snippet is an illustration of how to construct an AFitFlatte pdf RooRealVar x x 0 10 0 AFitFlatte flatteshape x mypdfname RooAbsPdf flattePDF flatteshape getPdf 14 2 11 Gaussian This is defined by a mean and width u and 0 and is given by 1 P x u o ex x 2 20 17 03 0 5 exp Ela 10 20 17 The Gaussian function implemented in AFit allows you to choose to have different widths above and below the mean value This PDF has a type gaussian when using the AFitPdfFactory and Figure 10 shows an example of the Gaussian PDF shape A RooPlot of Gaussian PDF o SF o e N o o o a o a gt Projection of o N Figure 10 An example of the Gaussian PDF shape The following code snippet will make a Gaussian PDF with a mean of 0 0 and a width of 1 0 RooRealVar x x 4 0 4 0 The Gaussian PDF AFitGaussian gausshape x mypdfname RooAbsPdf gausPDF gausshape getPdf 2 12 Generic PDF with functional form f x This PDF has type generic when using the AFitPdfFactory The functional form of a Generic PDF is defined by a string input into the cons
6. 30 3 3 Composite component model composite 0 o o 31 Building a Fit model 31 Al The makePdf fiunction omar ds a a da 32 4 2 The makeSimPdf function 200 000 0022 eee 32 43 The makeConditionalPdf function o o o 32 4 4 The fitParameters function 0 0000 eee ee 32 4 5 The persist function 33 4 6 The writeDataCard function e 33 4 7 The fitData function 35 48 Blinding yields pita lo a each te BSS GoW ox A ta 35 4 9 Replacing Variables In A PDF 2 2 2 0 0 2000 35 4 10 Computing systematic uncertainties 2 20 20 0020 2 002 ee eee 36 5 Utilities 36 5 1 Statistical analysis ea 2 ee 36 5 1 1 Pearson Correlation Coefficients 2 2 02004 36 5 1 2 Spearmans Rank Correlation Coefficients 28 4 37 5 1 3 Miscellaneous e tiles Be ea a a ha 38 5 2 Projection Plots s scoa saie 2 aye ee ha a Ee ed eae Se a 38 5 2 1 Example Enhancing the signal for a 2D fit model 38 5 2 2 Example Plotting an asymmetry s ooa a 39 S237 Pull Plots s 2 94 aeei a ee ead a Ee a e a 40 5 2 4 Likelihood Projection Plots ooa aa e o o 41 5 3 Likelihood Ratio Plots ss cs secar sue o noisa a p 0000 B Ei eee 43 54 TMVA Interface cas so Re a a ede ae es 44 5 5 Toy Monte Carlo Validation of the likelihood 45 5 6 Toy Monte Carlo Validation Embedded Toys o o 47 6 Exam
7. 76 775 305 881 systematic error shifts 0 0 251594 continuumYield 915 797 171 32 systematic error shifts 0 0 192007 signalYield 161 237 14 9639 systematic error shifts 0 12581 0 12946 from which one can read off the systematic uncertainty of interest from the last two columns of data If the systematic shift is one sided as in the case of BogOYield and continuumYield above then the larger of the two shifts is taken as the systematic uncertainty otherwise the asymmetric error is reported The output file is written in such a way that it can easily be parsed by scripts 5 Utilities 5 1 Statistical analysis 5 1 1 Pearson Correlation Coefficients The Pearson correlation coefficient p 0 y 0 0y between variables in a TTree can be computed using the correlation member function of the AFitStatTools class This function requires that 36 the branches of the TTree are of type Double_t and the correlation coefficients are calculated for all possible combinations of a comma separated list of variables For example given a pointer to the TTree tree one can compute the correlations between the variables mass and energy using AFitStatTools st st correlation tree mass energy or AFitStatTools st st pearsons_correlation tree mass energy the output of this will look something like Results from AFitStatTools correlation for the variables mass
8. e o 12 2 8 Disappearance PDE 200 ir Sea o a A bees 12 14 j bevan qmul ac uk Queen Mary University of London fergus wilson stfc ac uk Rutherford Appleton Laboratory 2 9 Exponential ds A Sate in alk A got Bee ek tes 12 2 10 Flatte Function 10 sce eee a Rae a Be SE CE Bea ee ee UA y 14 21L Gaussian it ety hated ol Se Raid he ee Be A gic ee AA GS ets e 15 2 12 Generic PDF with functional form Pla age ese Ae ae ATA ees 15 2 13 Gounaris Sakurai lineshape PDF 2 200000004 16 2 14 Helicity PDE vergin e Hae ad ne WA BE Sl te eB ee a 17 2 15 Histogram non parametric PDF 2 4 2195 wie cn SS OG AA 18 2 16 KEYS non parametric PDF 4 24 5 aad il sv Ye en be BUA G 19 ZA Landau tt ea et Ra a ke SP a GS ee 20 2 18 NOVOSIDISK s rary exes p AP tek Re AAR A Be pA el oe Be p 20 2 19 Polynomlal osc fey ek See tod Rw A a NA i Pa RO et e 21 2 20 Parametric Step Function oaoa aa e ee 22 2 21 Resol ia aod Rw a be EN A ge ee A a 24 222 SIGINOIG RA ae GR BACH a a tate Ae eS 25 223 Step FUNCOM noa A SB ee a lel a AME Td oy Bia a AE a ae 26 22A VOIE tIAT SS Moe oh Be Ree eet e ee ee Sg an So iad eee ee TA 26 2 25 Composite Add EDEN A Rs he Se SEE as 27 2 26 Composite Multiply PDF 2 2 0 0 0 202000 27 2 27 Multi dimensional PDFs e 28 223 EDE SUMMAry ii A A a eS 28 Fit Components 30 3 1 The default fit component 30 3 2 Scalar to Vector Vector decays vvpolarisation o
9. gt gt pis 27 i 1 where the p are coefficients This PDF has a type polynomial when using the AFitPdfFactory and Figure 17 shows an example of the polynomial PDF shape The following code snippet will construct a cubic polynomial function this is specified by the iOrder argument By default the polynomial parameters are set to 0 0 so you need to modify the parameters either via a configuration file or by accessing the RooRealVar s directly in order to make a non trivial shape 21 A RooPlot of 0 01 o o o o o o o o A EE a Oe Peojectiorzof Polyaomial PDF 0 002 Figure 17 An example of the polynomial PDF shape Int_t iOrder 3 AFitPoly polybld x pdf iOrder RooAbsPdf poly polybld getPdf 2 20 Parametric Step Function The Parametric Step Function PSF is a PDF where data are binned in one dimension and the parameters of the PDF are the fractions of probability in each bin As binning can be non uniform by definition the fitted fractions are in general not the heights of each bin when plotted but also depend on the bin width Figure 18 shows an example of using the PSF pdf The PSF pdf has N bins so there are N coefficients pdfname_PSFcoef 1 and N 1 bin bound aries pdfname PSFlimit_1 that have to be set in order to make this PDF The following code snippet will construct a parametric step function PDF similar to the o
10. Rev Nucl Part Sci 30 1980 253 The TMVA web page is http tmva sourceforge net The apparatus referred to here is described at http www matphys com and references therein 57
11. finished fitting the a new configuration file will be written to the out stream specified in the fitParameters member function call The following code snippet illustrates how to use this function AFitMaster master mydatacard txt RooAbsPdf pdf master getPdf ofstream out outputdatacard txt master fitParameters out out close Note in order to use this function the data card used in the AFitMaster constructor should fully specify the PDF taking note of which shape parameters need to be varied and which are to be held constant If you need to create a data card before fitting the data you can use the writeDataCard function described in Section 4 6 to obtain a template for modification 4 5 The persist function This function will Write out the following e A list of the discriminating variables used in the fit model A list of the component types used in the fit model A list of the component coefficients used in the fit model A list of the options specified in the FitConfiguration block The fit results obtained when fitting the PDF shapes these will be present only if the fitOptions specified in the FitConfiguration block used contains r to the file specified by ofileName If this file exists it will be overwritten otherwise a new file will be created There is only one argument to this member function and it is used as follows master persist myfile root
12. j 4 6 The writeDataCard function When setting up a fit model there can be a very large number of fit parameters that need to be defined in the configuration file or data card that describes your fit The writeDataCard function aims to simplify the process of writing a skeleton data card If you are able to define the variables that go into your fit as well as the fit components signal and backgrounds the component types the yields and the functional forms of the distributions used for the Note that each category yield can be blinded using an independent blinding string and scale factor 33 component PDFs then one can write out a new data card that has all of the aforementioned information included in addition to the the PDF parameters that will need to be defined by the fit model For example if you consider the 2D fit model of the effective mass Mgs and energy difference AE of a B meson See examples testAFitProjectionPlot cc and examples mes_de model txt then it is sufficient to specify the following FitConfiguration specify the variables to use in the fit variables bMes bDeltaE specify the names of the signal and background components components signal continuum Bbg0 fitOptions etrmh set the limits and initial values of the variables used bMes 5 2700 0 L 5 25 5 29 B 30 bDeltaE 0 0000 0 L 0 3 0 3 B 30 set the component types Signal default continuum default
13. syntax in the configuration file Each yield has its own blinding string blinding state and a scale factor used to compute the blinding offset For ex ample if we consider the signal fit component its yield has the name signalYield The blinding parameters associated with signalYield are signalYield BlindingType signalYield BlindingString and signalYield BlindingScale The possible values of the BlindingType are blind and un blind all lowercase The blinding string is any user defined string and the scale is used to tune the size of the random offset The RooUnblindOffset class is used to implement blind yields in AFit 4 9 Replacing Variables In A PDF Often we want to construct a complicated PDF where several shapes have a common parameter for example a kinematic endpoint in Mgs as described by an ARGUS PDF shape The data are all constrained by the same endpoint and so in principle should have the same RooRealVar parameter assigned as this common parameter for all components that are described by an ARGUS PDF in the fit Normally this will not be the case each ARGUS PDF will have its own endpoint It is possible to override this behaviour by substituting parameters when the PDF is being built The order in which this is done is important the PDF you take the new parameter from must have been built before the PDF you are currently trying to modify so this has to appear earlier in the list of components or variables Tf this i
14. to the pdf Figure 24 shows the result of running this macro RooRealVar x x 0 0 5 5 AFitPdfFactory fact AFitAbsPdfBuilder bld AFitAbsPdfBuilder fact makePdf pdf gaussian x RooAbsPdf pdf RooAbsPdf bld gt getPdf O AFitProjectionPlot plotter RooDataSet data pdf gt generate RooArgSet x 1000 RooPlot frame plotter makePlot x data pdf frame gt Draw Section 6 3 describes an example that uses a conditional variable in the PDF It is necessary to provide prototype data sets for all conditional variables when projecting such a PDF 5 2 1 Example Enhancing the signal for a 2D fit model If one has a more complicated model for example a 2D fit to a sample of data for example the effective mass Mgs and energy difference AE of a B meson it is possible to enhance the signal by cutting on variables not projected in the fit to data An example of this would be the following 38 Entries 4 4 2 0 2 4 Figure 24 An example of using the AFitProjectionPlot class to plot a Gaussian PDF and generated data set RooPlot frame plotter makePlot mes gt 5 27 DeltaE data pdf where the RooDataSet passed to the makePlot function will automatically have the cut applied to it in order to produce the plot with a correct normalization One can see the effect of making such a cut by comparing the distribution of
15. AFit User Guide Adrian Bevan Fergus Wilson December 24 2010 Abstract AFit provides a high level user interface to RooFit and TMVA Complicated maximum likelihood fits can be set up using a text file without the need to write a lot of code and modified quickly to refine an analysis There are a number of utilities to facilitate further analysis of the likelihood fit such as toy MC and plotting interfaces The TMVAInterface provided allows a user to configure which classifiers to run with which variables It is then possible to append RooDataSets with the output classifier MVAs for inclusion in a fit to the data A number of examples are provded in the last section of this user guide Contents 1 Introduction to AFit 3 1 1 Platform Requirements niis wfc a amp Bip A eae AO 4 2 PDFs 4 2 1 Parameter naming convention sooo e 4 2 2 MATUS A wise ise Rast eae th tes atta Sees ek MOA RAS a ATA A ATA 5 2 3 Breit Wigner ci ce Bote a ee ee ea a Pee A eG 6 2 3 1 Non relativistic Breit Wigner or Cauchy function 6 2 3 2 Relativistic Breit Wigner with a Blatt Weisskopf Form Factor 6 24 B ki Function est ats es ah Rae ek ee ee AA A 8 2 5 Chebychev Polynomial o 9 230 Crystal Ball io ts a TAE NOS 10 2 Decay Models sac a ea A bas a a BS 11 2 7 1 The Decay Modelo csi eee a ad a a ee ae 11 2 7 2 The BDecay Model 2 2 emir aca ee PR ee ae 11 2 7 3 The BCPGenDecay Model e
16. B 30 The type of each fit component is specified see Section 3 for a list of possible fit component options signal default continuum default Bbg0 default The final part of the FitConfiguration block specifies the fit yields and allowed ranges for each of the fit components signalYield 500 00 10 000 L 100 10000 continuumYield 2000 00 10 000 L 100 10000 Bbg0Yield 50 000 10 000 L 100 10000 Each fit component specified has to have the functional form of the PDFs defined The available options are listed in Table 1 For example the signal PDF for this example has a Gaussian for the mpgs distribution and a Landau function for the AE 51 signal signal_bMes type gaussian signal_bDeltaE_type landau where each PDF type is chosen by specifying a value to the label given by lt component name gt _ lt variable name gt type The remainder of the configuration file specifies the functional form of the other fit components components and the PDF parameters for each PDF of each variable for the fit The AFitMaster class us used in order to construct this fit model at the start of examples rareBdecay cc RooAbsPdf AFitMaster master AFit example rareBdecay txt pdf master getPdf The remainder of the example macro uses this PDF to generate a sample of simulated data using the AFitToy class See section 5 5 and to plot the pd
17. F name myGaussian then the name of the mean of the Gaussian will be myGaussian_mean See Table 1 for a list of variable names for each PDF The following code shows how to make a Gaussian PDF and then change the value of the parameters RooRealVar x x 4 0 4 0 The Gaussian PDF with default settings AFitGaussian gausshape x mypdfname re define default settings gaussshape setParameter mean 175 0 gaussshape setParameter width 3 0 RooAbsPdf gausPDF gausshape getPdf When building an AFitProdPdf the naming convention for the PDF follows the extended rule of i PDF component name followed by ii discriminating variable name followed by iii parameter variable name In this case there will be an underscore between i and ii For example the signal component for a one dimensional fit in some variable x will have the PDF component name signal If this is described by a Gaussian distribution then the mean of that distribution will have the parameter name signal xmean These conventions are imposed in order to help users configure their fits efficently 2 2 Argus This is the distribution used for a parametric description of a combinatorial background shape as first used by the ARGUS Collaboration 6 1 P m mo 7 gt my1 m moY exp 1 m mg O m lt mo 1 where 0 m lt mo 1 and 0 m gt m
18. Fs made using this function include time dependent CP asymmetry measurement PDFs where the PDF is split according to physics flavour tag categories Section 6 4 describes an example of how to make a model that is split by several categories 4 3 The makeConditionalPdf function The makeConditional Pdf member function of AFitMaster is used to read in the configuration file specified in the constructor and build a conditional PDF according to the configuration 4 4 The fitParameters function The fitParameters member function of AFitMaster reads in reference data specified in the configuration file for a given fit component and fits PDFs to the reference data for that fit component The data files to use when fitting parameters are specified in the configuration file block PDF Param Files The following example shows a signal data file being specified as an ascii file called signal txt with the discriminating variables as uncorrelated The keyword data in the comma separated list is the name of the data to be read so this is a dummy variable for an ascii file PDF Param Files Signal_referencedata signal txt ascii data uncorrelated If discriminating variables in the fit are specified as uncorrelated in the configuration file then the PDF parameters for each discriminating variable will be fitted separately for a fit component Otherwise all parameters for all PDFs in a component will be fitted simultaneously 32 When
19. MuonLifetime txt 5This member function takes a pointer to a TObjArray as an argument for the list of input files to merge This TObjArray can be made from a comma separated list of files in a TString using TString Tokenize 48 If you run the MuonLifetime cc example on the sample of data provided you will obtain the result 7 2 09 0 02 us which accounts for the interaction of y in matter The distribution of the data and the fitted PDF are shown in Figure 30 10 Events 0 39 10 10 T A eH i 4 6 8 10 12 14 16 18 20 Figure 30 Fitting the y lifetime using the MuonLifetime cc example The fit configuration of this example can be found in examples MuonLifetime txt The first line after FitConfiguration specifies the variable name t for the time difference between the two signals from the photomultiplier tube FitConfiguration variables t Each fit component these are single background listed in the configuration file option components needs to have its form defined see signal etc listed below The names are defined by the comma separated values in the components list components signal background J The fit range and number of bins used when plotting the discriminating variables is also defined t 0 5 0 01 L 0 5 20 0 B 50 j The type of each fit component is specified see Section 3 for a list of possible fit component options 49
20. alModelData_plot_bMes RooCurve curve frame getCurve TotalModelProjected RooHist PullPlot plotter makePullPlot datahist curve PullPlot Draw Ax which will result in a pull plot like Figure 27 being draw pulls 3 i E 2 mi a ee 1 5 25 5 255 5 26 5 265 5 27 5 275 5 28 5 285 5 29 Figure 27 An example of using the AFitProjectionPlot class to make a pull plot 5 2 4 Likelihood Projection Plots One way to enhance the signal content of a data sample when making a projection is to cut on a likelihood ratio of the projected variables this ratio does not use information from the plotted variable The likelihood ratio computed in AFitProjectionPlot is the log of the ratio of signal to total likelihoods The following code snippet illustrates how to make a liklihood ratio projection plot AFitProjectionPlot plotter RooPlot llrframe plotter makeLRProjectionFrame pdf data varToPlot sigCompName RooPlot projframe plotter makeLRProjection pdf data varToPlot sigCompName cutVal where pdf is the total PDF data is the data set to use when making the projection varToPlot is a TString whose value is the variable to plot and sigCompName is the component signal Al to enhance The argument cutVal is the minimum value of the likelihood ratio for projected events Figure 28 shows the likelihood ratio distribution and
21. ariable a conditional variable a PDF type and a PDF name as arguments Once you have built your resolution model it is possible to use it as the PDF uses a conditional variable you need to construct a prototype For example if you have a TTree called tree you can construct a RooDataSet and o At prototype with the following RooDataSet data rds the data tree RooArgSet deltat deltatErr RooDataSet proto data gt reduce RooArgSet deltatErr and subsequently fit the data and make a plot of the data pdf gt fitTo data trh AFitProjectionPlot plotter RooPlot frame plotter makePlot deltat amp data pdf proto where you note that the prototype is required for plotting An example macro demonstrating the use of the resolution function PDF with a test data sample can be found in the examples directory as resolutionFunction cc and resolutionFunctionData root 6 4 RooSimultaneous splitting a PDF by categories The AFitMaster makeSimPdf member function allows a user to construct a RooAbsPdf that is subsequently split by one or more RooCategories according to a specified rule The example described below can be found in the files simpdf cc and simpdf txt Before defining how to split a PDF up according to categories the fit model needs to be defined in the normal way This example uses two two discriminating variables mpg and AE and two fit components signal and background The signal co
22. ariables the AFitToy class can be used to generate toy Monte Carlo samples of events as follows AFitMaster master AFit example MuonLifetime txt RooAbsPdf pdf master getPdf Get the time variable from the AFitMaster RooArgSet compSet pdf gt getComponents RooArgSet parSet pdf gt getParameters compSet RooRealVar t RooRealVar parSet gt find t make the interface used to run toys AFitToy toy for int i 0 i lt 10 i generate the toy data toy setSeed i RooDataSet data toy generateToySample pdf RooArgSet t 1000 0 fit the toy data RooFitResult r pdf fitTo data etrm where the muon lifetime fit example has been used for this toy see Section 6 1 The argu ments to generateToySample are a pointer to the RooAbsPdf to fit the set of variables to generate the number of events to generate and the prototype to use for any conditional vari ables that you want to generate The total number of events generated in each toy sample will have a Poisson mean of 1000 This default behaviour can be switched off by calling the toy setPoisson kFALSE before generating If one does this then each toy will have exactly 100 events generated There are more sophisticated toy generation functions to call The highest level one is illustrated in the following muon lifetime fit example 46 AFitToy toy toy setO0utputDir toy
23. composite This component model is the sum of several indivudual components added with a given weight ing The total pdf can be written in terms of each of the components C as PDF a 1 fi fn Co a firCi z faCnlz 38 where the f are the fractions of the components with from 1 through to n The fraction of the the zeroth component is given by one minus the sum of the fi 4 Building a Fit model All applications in AFit start from a fit model that is constructed by the AFitMaster class This class is responsible for reading a configuration file that specifies the lists of discriminating 31 variables and fit components as well as the individual PDF types for each discriminating variable used in every fit component Once this has been done the fit model will be constructed The rest of this section describes additional functionality provided by this class Several examples are described in Section 6 4 1 The makePdf function The makePdf member function of AFitMaster is used to read in the configuration file specified in the constructor and build a PDF according to the configuration 4 2 The makeSimPdf function The makeSimPdf member function of AFitMaster is used to read in the configuration file spec ified in the constructor and build a PDF according to the configuration The pdf constructed using makePdf is split according to rules specified for one or more RooCategory variables Ex amples of PD
24. ction checks to see if a file exists before trying to add it to a chain and any zombie files are skipped automatically 5 6 Toy Monte Carlo Validation Embedded Toys The previous section summarised tools used to run Toy Monte Carlo validation studies where the likelihood is used to generate an ensemble of data samples to fit back Any deviation from the input results would be a result of the intrinsic bias of the fit In defining the likelihood we make assumptions about many things including the correlations between discriminating variables in the fit These assumptions can be tested by embedding simulated data samples obtained from the Full Monte Carlo simulation for an experiment The AFitToy utility has several member functions to facilitate performing such a toy These are generateEmbeddedToySample 47 and generateEmbeddedToys Given a large data sample of events from a Full Monte Carlo simulation one can create a number of subsamples with a predetermined number of events using the following toy generateEmbeddedToys sourcedata 1000 1 10 where this example used 1000 events per sample and generates 10 samples with numbers 1 through 10 The output data files are written to the directory specified by setOutputDir and have file names that are lt iToy gt _embtoydata root Once several files have been generated for example signal and background they can be merged together using the mergeFiles member function S 6 Example
25. ctly 2 9 Exponential The exponential function defined as P x 7 e 15 where y is the slope of the exponential This PDF has a type exponential when using the AFitPdfFactory If the category fitLifetime has the value true then the slope of the exponential is replaced by 1 y where y is fitted as a lifetime and the PDF becomes Pana 16 12 e o ance PDF o N al appear roof Disa ojectio o Pro pS ce a lot A MOE 1 j D a AE AMET jj 02 03 04 05 06 07 0 8 0 9 1 v energy Figure 7 An example of the neutrino disappearance PDF shape The irregular oscillation amplitude visible for low y energy is a plotting artifact A RooPlot of e nential PDF gt o 2 o o E Projection of Expo o A 0 02 5 4 3 2 1 0 1 2 3 4 5 Figure 8 An example of the exponential PDF shape The fitLifetime category has the name lt pdfname gt _ lt variablename gt fitLifetime This option is useful when trying to determine the lifetime of exponentially decaying sample of data Figure 8 shows an example of the exponential function PDF 13 The following code snippet will make an exponential PDF with a decay constant of 1 0 RooRealVar x x 0 0 20 0 The Exponential PDF AFitExponential expshape x mypdfname RooAbsPdf expPDF expshape getPdf
26. data before and after applying it This particular example is available in examples testAFitProjectionPlot cc and Figure 25 shows the distribution of generated data before and after applying the cut of Mgs gt 5 27 to it 5 2 2 Example Plotting an asymmetry There is an interface method for plotting asymmetries on data using the AFitProjectionPlot class For example one can make an asymmetry plot as a function of At deltat for a time dependent CP analysis where there is a flavour tag variable tag a data sample data a prototype with conditional variables proto that includes the flavour tag using the function call RooPlot frame plotter makeAsymmetryPlot deltat tag data pdf proto where the end result should look similar to Figure 26 39 Entries 100 0 2 0 0 2 A 0 Discriminating Variable Discriminating Variable Figure 25 An example of using the AFitProjectionPlot to make signal enhanced projections left before and right after cutting on Mgs 0 5 Asymmetry o 0 5 10 At Figure 26 An example of using the AFitProjectionPlot to make a At asymmetry plot 5 2 3 Pull Plots Given a RooPlot that has both a data set and a curve plotted on it you can produce a pull plot that is a plot of the difference between the curve and the data points normalized to the 40 error on the data by doing the following RooHist datahist frame getHist Tot
27. dimensional component PDFs that can be constructed Section 3 how to use the AFitMaster to construct a PDF model Section 4 the various utilities available Section 5 and working examples Section 6 If you are already very familiar with the use of RooFit and maximum likelihood fitting you might consider starting to work through the examples section before reading all of the earlier sections of this user guide The main underlying statistical techniques used by RooFit and AFit are described in more detail in the following references 4 5 as well as other books on statistics and data analysis 1 1 Platform Requirements The AFit package has been tested using e gcc version 3 4 6 20060404 e The current beta version of AFit is being tested using ROOT version 5 26 00 e Scientific Linux 2 PDFs This section summarises the library of PDFs that are implemented in AFit There are code snippets illustrating how to use the different PDFs described below However for more compli cated likelihood fit models it is more practical to use the AFitMaster class as an interface to the PDF classes described here See Section 4 2 1 Parameter naming convention The parameter names for all simple PDF components described in this section follow the same rule The parameter name is constructed from two parts separated by a underscore i the PDF name folllowed by ii the parameter variable name For example if we consider a Gaussian PDF with a PD
28. e 2 3 1 The default fit component The default fit component is a product of one dimensional PDFs which can be expressed as PDF z PDF x1 x PDF2 a2 x x PDF ay 35 The configuration file excerpt for a default component looks like signal signal_x_type gaussian Signal_y_type gaussian Signal_q_type gaussian where the term in square brackets is the component name This is followed by a list of variables with the naming convention of the component name followed by the discriminating variable name followed by _type The value assigned to this variable has to be one of the PDF types listed in Table 1 3 2 Scalar to Vector Vector decays vvpolarisation The decay of a scalar particle to two vector particles results in the final state being a superpo sition of three amplitudes One of these corresponds to the longitudinal polarisation and the other two are transverse polarisations In the vvpolarisation component model the longitudinal and transverse polarisations are combined by the fraction of longitudinally polarised events fz In order to use the physical value of fz in a fit one has to specify the reconstruction efficiency for the longitudinal and transverse events So the PDF is given by P x PL 1 17 Pr a 36 where PL x is the PDF for the longitudinal polarisation Pr x is the PDF for the transverse polarisation and fe f is the observed fractional difference between the two polarisati
29. e using the PDF factory method makeConditionalPdf PDF Pdf Factory Label Variable Names Argus argus endpt xi power Breit Wigner breitwigner m0 Gamma Relativistic Breit Wigner relbreitwigner mass width radius spin mass_a mass_b Bukin bukin Xp sigp xi rhol rho2 Chebychev Polynomial chebyN pi Crystal Ball cbshape mean width alpha n Decay decay tau BCPDecay cpdecay tau dm S C avgMistag delMistag mu BDecay bdecay tau dm f0 fl f2 f3 dgamma Disappearance disappearance theta amp L deltamsqr Exponential exponential constant Gaussian gaussian mean width Asymmetric Gaussian agaussian mean widthL widthR Generic PDF generic his Gounaris Sakurai gounarissakurai mass width radius spin mass_a mass_b Helicity helicity Histogram 1dhist KEYS 1dkeys ate Landau landau mean sigma Novosibirsk novosibirsk peak width tail Polynomial polyN pi PSF psf PSFcoef N _PSFlimit_N Resolution resolution sigMeani sigSigi coreFrac tailFrac scaleCoreMean scaleCoreWidth scaleTailMean scaleTailWidth useTruthModel Sigmoid sigmoid a b Step Veto step limits veto Voigtian voigtian mean width sigma Algorithm Composite Add PDF add x y Composite Multiply PDF Multi dimensional PDFs multiply x y 29 3 Fit Components A fit component is a multi dimensional PDF defined for the set of variables x The types of fit component can be built are listed below and summarised in Tabl
30. energy Correlation matrix follows 1 0 0157139 0 0157139 1 where the order of the columns and rows is the same as the order of variables in the comma separated list 5 1 2 Spearmans Rank Correlation Coefficients The matrix of Spearmans rank correlation coefficients rs can be calculated in a similar way using AFitStatTools st st correlation tree mass energy kTRUE or AFitStatTools st st spearmans_rank_correlation tree mass energy where the correlation r is given by 6 4 n n 1 30 rg 1 where n is the number of data points the d are difference between the integer ranks of the i event for the two variables x and y and this quantity is summed over all events 37 5 1 3 Miscellaneous The AFitStatTools class has member functions that perform a number of different calculations including e Binomial error a a2 X sum e x probability 5 2 Projection Plots The class AFitProjectionPlot can be used in order to make projections of a PDF on a RooPlot If a RooDataSet is provided the data will be plotted on the RooPlot as well as the projection over the PDF This class can also be used in order to compute a pull plot from the difference between data points and the PDF curve The following is an example of using this plot class to make a plot of a Gaussian PDF for the variable x where a generated data set of 1000 events is plotted on the RooPlot in addition
31. ertainty on the discriminating variable and the u and o are the means and widths of the it Gaussian where i core tail outlier It is possible to multiply the mean and with of the core and tail Gaussians by a x however the default behaviour is not to do this In order to scale the mean and width parameters by the error on x the following parameters should be set to yes scaleCoreMean scaleTailMean scaleCoreWidth scaleTailWidth An example of the resolution function PDF for the proper time difference distribution At in BaBar is shown in Figure 19 for B ntr Monte Carlo simulated data a e P Entries o Terre Figure 19 An example of the resolution function PDF shape The figure on the right shows the same plot on a log scale A resolution function PDF can be constructed using the following steps 24 the discriminating variable is deltat and the conditional variable is is the error on deltat deltatErr RooRealVar deltat reso_dt dt 10 0 10 0 RooRealVar deltatErr deltaterr sdt 1 2 0 0 2 50 AFitResolution resoBld kdeltat amp deltatErr Reso RooAbsPdf pdf resoBld getPdf and there is a detailed example of how to use the AFitResolution class in Section 6 3 2 22 Sigmoid This PDF has a type sigmoid when using the AFitPdfFactory The function implemented is 1 P
32. f and simulated data using the AFitProjectionPlot class See section 5 2 6 3 Fitting a At resolution function The Resolution PDF described in Section 2 is used to model the resolution on the proper time difference At between the decays of two neutral B mesons in an event at the B Factories The resolution function parameters are scaled multiplied by the error on o At on an event by event basis In order to use this PDF you need to define the discriminating variable and if appropriate the corresponding conditional variable v At Having done this you can instantiate the AFitResolution class RooRealVar deltat reso_dt dt 10 0 10 0 RooRealVar deltatErr deltaterr sdt 1 2 0 0 2 50 AFitResolution resoBld amp deltat amp deltatErr SigReso As this example requires that the mean and core parameters are scaled by a At we must set the following resoBld resoBld resoBld resoBld Having configured the PDF setParameter scaleCoreMean yes setParameter scaleTailMean yes setParameter scaleCoreWidth yes setParameter scaleTailWidth yes it is possible to now make it using 52 RooAbsPdf pdf resoBld getPdf All of these steps can be replaced by a configuration file with the appropriate settings and the use of the AFitMaster member function makeConditionalPdf The makeConditionalPdf function takes a discriminating v
33. iables to train are specified in a comma separated list with pairs of data variable name Type TMVA recognises types of I and F to distinguish between different integer variables int long int etc and floating point variables float double Once finished training MVAs TMVA writes output to a sub directory called weights The known MVA types are Cuts Likelihood HMatrix Fisher CFMIpANN TMIpANN BDT RuleFit SVM MLP See the TMVA User Guide for more information on these 11 Other options that are recognised by AFit are described below e factoryOptions This variable can be set to any of the options passed to the TMVA factory on instantiation e trainingOptions This variable can be set to define the options used when calling factory PrepareTraining e preselectionCut This variable can be set to define the preselection cut used when calling factory PrepareTrainingAndTestTree e analysisprefix This is the name of the TMVA Factory instance used also the weight file prefix name Once the training configuration file has been prepared this can be used by typing the following AFitTMVAInterface a myConfigurationFile txt a trainMethods TMVA will run the specified classifiers with the specified variables found in the source files Once you ve inspected the output of TMVA and decided which classifier s you want to consider using in your fit you can compute classifiers for RooDataSets usi
34. ies The format of the split rules is the category name followed by a colon and then a comma sepa rated list of PDF parameters that should be split by this category Note that it is not possible to split a parameter by more than one category In this example the tagcat splitting rule has been specified using splitRule and the decay splitting rule was specified using decay_splitrule Once the data card has been properly defined the simultaneous pdf is built using AFitMaster master macros simpdf txt RooAbsPdf simpdf master getSimPdf The second example of using a RooSimultaneous to split a PDF by category is based on the previous discussion see simpdf2 cc and simpdf2 txt This is a signal plus background model as above where the PDF is split by decay mode There are four different signal decays under consideration J YK J pK9 Ke n n J pK Y 2S K The parameters that are split by the decay category are signal yield background yield and the AF width Figure 31 shows the mpgs and AB distributions of four of the five decay channels based using a simulated data sample In order to generate the simulated data a prototype data set containing the decay category is constructed The total number of entries in the data set of each decay type corresponds to the sum of signal and background Having prepared the prototype data set the AFitToy utility see Section 5 5 is used to generate the data sample to f
35. ion 0 5 RooAddPdf pdf pdf RooArgList gaussPDF argusPDF sigfrac Int_t nToGen 1000 RooDataSet data RooDataSet pdf generate RooArgSet x nToGen Double t nSig sigfrac getVal nToGen Double_t nBG nToGen nSig compute the likelihood ratio AFitLRPlot lrplot lrplot makePlot gaussPDF argusPDF new RooArgSet x data nSig nBG 5 4 TMVA Interface The class AFitTMVA Interface is a utility to facilitate the computation of an MVA within an analysis framework See Ref 11 for more information on the TMVA package The TMVA training options are steered from a text file or the same PDF data card that you use for your fit The text file block is denoted by TMVAInterface The following is an example configuration used for computing a Fisher and training a Boosted Decision Tree with TMVA TMVAInterface sigFile tmva_fsig root bgFile tmva_fbg root dataName data outputFileName tmva_out root trainingMethod Fisher BDT variables a F b F 44 The signal and background files are specified by assigning values to sigFile and bgFile It is assumed that the input data come from a RooDataSet with a name given by the value of dataName The output file has the default value of tmva_out root More than one training method can be booked for the variables going into the MVA This is done by assigning the appropriate method to a comma separated list of methods The var
36. it and plot The projections are made using the AFitProjection utility see Section 5 2 6 5 Time dependent CP asymmetry fit In order to fit for time dependent CP asymmetries in Y 45 Bop decays one can use the AFitBCPGenDecay class Before instantiating this class a resolution function needs to be set up see Sec 6 3 The example cpfit cc illustrates how to set up a time dependent fit for a signal decay like B gt J VK Figure 32 shows the resulting distributions of At for B and B tagged events as well as the time dependent CP asymmetry 54 1 Mes e 10 L 5 28 L 5 28 5 29 23 5 26 mes GH 10 fi L 0 1 AE fev 0 1 0 1 AE bev 0 1 100 Entries w S T L AE fev fi AE amp Gevy Figure 31 The top four mgg and bottom four AF distributions for the four decay channels used in the example The decay channels are left to right top to bottom J YK JJ YK K9 rn J pK p 28 K2 One final finesse that is required for a realistic time dependent CP asymmetry fit is to split the final PDF by flavour tagging category and to ensure that all appropriate parameters are split accordingly w Aw u and any resolution function parameters that are different for different flavour tagging categories see Sec 6 4 The example cpfit_tagging cc and associated configuration file illustrate how to exte
37. later fitting and validations Figure 14 shows an example of this type of PDF given an input histogram The data in the figure are the result of randomly filling the histogram according to a Gaussian distribution with a mean of 0 5 and width of 0 1 A RooPlot of x Events 0 04 Figure 14 An example of the KEYS PDF shape solid compared with the original data points In order to build a KEYS PDF for a discriminating variable x you need to enter the following AFitKeys pdfbld x pdf pdfbld setParameter file myfile root pdfbld setParameter tree treename pdfbld setParameter rho 1 pdfbld setParameter mirror NoMirror RooAbsPdf pdf pdfbld getPdf where datafile sets the ROOT file containing the TTree treename used to construct the PDF p 19 is a smoothing parameter and mirror is a RooCategory that defines how edge effects are dealt with by RooKeysPdf The allowed options for mirror are NoMirror MirrorLeft MirrorRight MirrorBoth MirrorAsymLeft MirrorAsymLeftRight MirrorAsymRight MirrorLeftAsymRight MirrorAsymBoth The default is p 1 and mirror set to NoMirror 2 17 Landau This PDF has a type landau when using the AFitPdfFactory and Figure 15 shows an example of the Landau PDF shape The functional form of the Landau distribution is given in Ref 10 and this is often
38. mponent has a Gaussian PDF for each of the discriminating variables whereas the background mpgs distribution is described by an Argus PDF and the background AF distribution is described by a polynomial similar to the example in Section 6 2 In order to specify how the PDF is modified according to different categories these categories and splitting rules have to be specified in the FitConfiguration data card block For this example the pdf will be split according to signal decay type the decay category and by the flavour tag the tagcat category The following snipped of the example data card defines how the PDF is modified 53 catVars tagcat decay splitRule tagcat signal bMes mean tagcat_categories Lepton Kaon1 Kaon2 KaonPion Pion Other NoTag decay_categories Jpsik0S Jpsik0OS_pi0pi0O JPsik 0 Psi2SK0S Chic1K0S decay_splitrule decay signal_bDeltaE_width The catVars variable is followed by a comma separated list of category variables that the PDF will be split by The individual categories are themselves defined by the category name categories variables The individual category labels are numerically numbered 0 1 2 for each of the cat egory variables The splitting rule that defines what PDF parameters are split by what category needs to be defined This rule can either be defined using the splitRule variable of the data card or by the individual category name _splitrule variables for each of the categor
39. nd the simple macro described above in order to split by tagging category The parameters that are 59 deltat data pl lot2_dokaty data_plott Figure 32 The At distribution for top B and middle B tagged events as well as bottom the time dependent CP asymmetry split by the tagging category are a tagging efficiency associated with the signal yield which is called signal_tageff as well as the mistag parameters w and Aw 7 Acknowledgements The following people have contributed to the development of this package Fergus Wilson Rutherford Appleton Laboratory References 1 The RooFit web page is http roofit sourceforge net 2 The MINUIT user guide can be obtained from http wwasdoc web cern ch wwwasdoc minuit minmain 3 The ROOT web page is root cern ch 4 Statistics A Guide to the Use of Statistical Methods in the Physical Sciences R Barlow John Wiley amp Sons Ltd 1989 5 Statistical Data Analysis G Cowan OUP 1998 6 H Albrecht et al ARGUS Collaboration Phys Lett B241 1990 278 56 7 M J Oreglia Ph D Thesis SLAC 236 Appendix D 1980 J E Gaiser Ph D Thesis SLAC 255 Appendix F 1982 T Skwarnicki Ph D Thesis DESY F31 86 02 Appendix E 1986 G J Gounaris and J J Sakurai Phys Rev Lett 21 244 1968 K S Cranmer Comp Phys Comm 136 198 2001 L Landau J Phys USSR 8 1944 201 see also W Allison and J Cobb Ann
40. ne shown in the Figure 22 A RooPlot of Discriminating Variable 5 9 bo k po o ction of Fit Master Model y Proje o o o ens TIT TITT TTT TIT ryt tty ttre 0 04 ee O E Jt tt rt Jott t pri 0 01 02 03 04 05 06 07 08 09 1 Discriminating Variable Figure 18 An example of the Parametric Step Function shape Instantiate the discriminating variable RooRealVar x x 0 0 1 0 Set the relative weights of each bin RooRealVar n0 n0 0 1 RooRealVar n1i n1 0 2 RooRealVar n2 n2 0 3 RooRealVar n3 n3 0 4 RooRealVar n4 n4 0 1 RooArgList coefList coefList add n0 coefList add n1 coefList add n2 coefList add n3 coefList add n4 Set the bin boundaries there are N 1 boundaries Note that the first and last boundary match the limits of the discriminating variable x TArrayD limits limits Set 6 limits 0 0 0 limits 1 0 5 limits 2 0 6 limits 3 0 7 limits 4 0 9 limits 5 1 0 RooParametricStepFunction pdf pdf ParametricStepFunction PDF x coefList limits 5 2 21 Resolution The resolution function implemented in AFit is a triple Gaussian function of the form Plz pi Tepe canal t a n Lore core fiailGtail x o x Htail Ctail 1 Joore ftail Goutlier x Houtlier Coutlier 28 where x is the discriminating variable o x is the unc
41. ng the AFitTMVAInterface runReader member function This takes a RooDataSet as an argument and will add columns to the data set for each classifier specified in the configuration file The new RooDataSet is passed back to the user for storage In the case that there are several files that you wish to process you can use the overloaded function of the same name that takes two string arguments the first string is the initial file and the second string is the target file 5 5 Toy Monte Carlo Validation of the likelihood The AFitToy class has been implemented in order to simplify the process of running toy Monte Carlo validations of the likelihood function The process of running a toy Monte Carlo study involves 45 1 Setting the random number seed for use in generation this ensures that one is able to reproduce exactly the same sample of data using the same sequence of random numbers each time One will have to use different seeds for different toy Monte Carlo samples 2 Determining the number of events to generate If one analyses an ensemble of toys it is necessary to generate a mean number of events according to a Poissonian distribution 3 Generating the simulated data sample there are several ways of doing this depending on the type of toy to be performed and the preferred context 4 Fitting the simulated data sample 5 Persisting the results of the fit for further analysis Given a pdf and a RooArgSet of discriminating v
42. nstantiate and multiply together The functional form of this PDF is given by Pla P x Po x Pale 33 The multiplication is implemented by using a RooGenericPDF and the limit of the number of PDFs that can be multiplied together is governed by the computing resources Figure 23 shows a helicity PDF multiplied by a polynomial 27 0 035 0 03 LTPP TTT TTT Normalised PDF 0 025 0 02 0 015 0 01 _ _ m 0 005 o 1 1 1 1 08 0 6 0 4 0 2 0 0 2 x Figure 23 An example of the Multiply PDF shape solid for a helicity distribution dotted multiplied by a polynomial dashed 2 27 Multi dimensional PDFs Multi dimensional PDFs can be constructed from the product of their individual components using an AFitProdPdf P x PDF x1 x PDF x2 xX X PDFy ay 34 Multidimensional PDFs can be created using the AFitPdfFactory member function makePdf TString name RooArgList amp discVarList where the first argument is the name of the PDF and the second argument is a RooArgList of discriminating variables 2 28 PDF summary Table 1 summarises the different types of PDF available in AFit This table shows the name of the function the abbreviation of the function name that is used in the AFitPdfFactory and the list of parameter names required by the pdf 28 Table 1 The different types of PDF available in AFit tdenotes a conditional PDF that can be mad
43. nted in order to study angular correlations in components of type vvpolarisation For the helicity angle 0 the longitudinally polarised part of the vvpolarisation component should have a cos 6 distribution of the form xsqr and the transversely polarised part of the vvpolarisation component should have a cos 6 distribution of the form 1 xsqr In practice any PDF of this form should be modulated by an acceptance function A RooPlot of Projection of o o o o N o N oa oe o o a 0 01 0 005 O TTT PTT Try Tr ery Tere Se Figure 12 An example of the helicity PDF shape for dashed longitudinal and solid transverse polarisation forms The following code snippet will make a helicity PDF shape with an xsqr distribution AFitHelicity hbld x pdf RooAbsPdf h hbld getPdf 17 2 15 Histogram non parametric PDF It is possible to construct a non parametric PDF based on an input histogram using the RooHist Pdf class The AF it wrapper of this class is AFitHist The value of the PDF for a given value of x corresponds to the normalised bin content of the histogram used to define the PDF unless interpolation is used between adjacent bins In order to build a Histogram PDF for a discriminating variable x you need to enter the following AFitHist pdfbld x pdf pdfbld datafile setVal myfile root pdfbld histname setVal his
44. o 0 This PDF has a type argus when using the AFitPdfFactory see below The following is an example of the code necessary to make an ARGUS PDF in AFit the discriminating variable x is the beam constrained B meson mass RooRealVar x x 5 25 5 29 The AUGUS PDF AFitArgus argus x arguspdf RooAbsPdf argusPDF argus getPdf By default the ARGUS PDF will have an endpoint my 5 29 and slope 50 as shown in Figure 1 A RooPlot of en o gt ejectien of Argus Euncti 2 e o h o Pr 0 002 Figure 1 An example of the ARGUS PDF shape 2 3 Breit Wigner 2 3 1 Non relativistic Breit Wigner or Cauchy function The non relativistic Breit Wigner function also called a Cauchy function is given by 1 2 E m 41 29 A and is shown in Figure 2 Here mo is the position of the peak and T is the width of the peak P x mo T The following code snippet is sufficient to build a non relativistic Breit Wigner PDF bwPDF By default the mass and width of the PDF are 0 770 and 0 150 respectively The variable names for the mass and width of the PDF are the name of the pdf in this case bwpdf followed by mO and width x is the resonance mass e g m_rho RooRealVar x x 0 5 1 0 AFitBreitWigner bw x bwpdf RooAbsPdf bwPDF bw getPdf 2 3 2 Relativistic Breit Wigner
45. o data points and the shaded regions correspond to the simulated data generated from the PDF If the PDF describes the data properly the data and toy data distributions will agree and the signal should peak near R 1 and the background should peak near R 0 The Figure clearly shows a background component that peaks near R 1 which indicates that there is no way to distinguish between the signal and part of the background using that particular fit configuration Number of Entries t 0 0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 1 Likelihood Ratio Figure 29 The likelihood ratio R for a sample of Monte Carlo compared to a PDF The signal component of the PDF is shown in green and the background component is shown in red The code required to make the plot of Figure 29 available in erample likelihoddRatio cc and is shown here 43 the discriminating variable x is the beam constrained B meson mass RooRealVar x x mass GeV 5 25 5 29 The background PDF AFitArgus argus x arguspdf RooAbsPdf argusPDF argus getPdf The signal PDF AFitGaussian gauss x gausspdf gauss setParameter mean 5 28 gauss setParameter width 0 01 RooAbsPdf gaussPDF gauss getPdf set the fraction of signal at 50 so there is 90 background and generate 1000 toy events from the composite PDF RooRealVar sigfrac sigfrac fract
46. o use this class The functional form of this PDF is e7At ro P At o At E 1F Aw 1 2w x Spsin Am At Cy cos Am At TBO R At o At where P describes BO B tagged events Tpo is the B lifetime w is the mistag probability Aw is the mistag probability difference between B and B tagged events Ama is the B B mixing frequency and S and C are the CP asymmetry parameters The resolution function R is convolved with the physical time dependence as indicated It is possible to blind S and C by setting the blindingState to blind ensuring that the blind StringS and blindStringC variables have been set appropriately NOTE When using this model it is imperative that the RooAbsPdf of the reso lution function you intend to use is instantiated before the BCPGenDecay model s RooAbsPdf 2 8 Disappearance PDF The neutrino disappearance probability is given by P E 0 A Am L 1 sin 20 sin AAm L E 14 where is the mixing angle A 1 267 given current data L is the baseline of the neutrino experiment km and E is the neutrino energy GeV The class AFitDisapperance is the implementation of this PDF The corresponding distribution produced by this class is shown in Figure 7 for L 265 A 1 267 and Am 0 0024 The observed PDF for low neutrino energy lt 0 1 GeV is an artifact of the granularity of the RooCurve used to create the plot The numerical PDF is reproduced corre
47. ons The effective parameter fe f is related to the physical parameter fzr via eff JE bo 1 frjer er ft 20 where e and er are the efficiencies for the longitudinal and transverse polarisations The configuration file excerpt for a vvpolarisation component looks like 30 Table 2 The different types of fit component available Component Name Description default A product of one dimensional PDFs in x vvpolarisation PDFs for longitudinal and transfers components coupled by fz composite a weighted sum of components vvsignal vvsignal_polarisationfrac 1 0 0 1 L 0 0 1 0 vvsignal_effLong 0 30 C vvsignal_effTran 0 40 C vvsignal_long vvsignal_long x_type gaussian vvsignal_long_y_type gaussian vvsignal_long q type landau vvsignal_tran vvsignal_ tran x type gaussian vvsignal_tran_y_type gaussian vvsignal_tran_q type landau where the term in square brackets is the component name For the vvsignal component the only parameters that need to be configured are the value of fr which is given by vvsignal polarisationfrac and the efficiencies for the longitudinal and transverse polarisations given by vvsignal_effLong and vvsignal_effTran respectively in this example The longitudinal and transverse polarisa tions are identified with the other two component names Each of these polarisations is modelled by a default PDF type as described above 3 3 Composite component model
48. ples 48 6 1 Fitting the muon lifetime e 48 6 2 Simple rare B decay search at BaBar or Belle 50 6 3 Fitting a At resolution function a 52 6 4 RooSimultaneous splitting a PDF by categories oaoa aaa 53 6 5 Time dependent CP asymmetry fit aoaaa a 54 7 Acknowledgements 56 1 Introduction to AFit This package has been developed in order to add a layer of abstraction on top of RooFit 1 and simplify complex maximum likelihood fit based data analysis The underlying function minimiser is MINUIT 2 The aim of AFit is to provide a general fit framework that can be used in order to analyse data but without having to write code in order to set the fit up In order to do set up a general fit you can configure AFit using a datacard In addition to this high level abstraction it is also possible to use the individual wrapper classes to the RooFit Probability Density Functions PDFs when addressing simple problems The default version of RooFit to use with AFit is the version bundled with ROOT 3 AFit provides a higher level interface to RooFit that facilitates using a text configuration file to construct a complicated multidimensional likelihood fit model There are also a number of tools and interfaces provided to simplify validation of the likelihood fit as well as performing analysis of the data The rest of this document summarises the available PDFs Section 2 different types of multi
49. ppet will make a Chebychev Polynomial of order 3 Int_t iOrder 3 AFitCheby chebybld x pdf iOrder RooAbsPdf cheby chebybld getPdf 3For example see http mathworld wolfram com ChebyshevPolynomialoftheFirstKind html 2 6 Crystal Ball Crystal Ball CB distribution The CB shape is a Gaussian with an exponential tail as defined in 7 The functional form implemented in RooFit is given by x e7 m mo 207 m gt mo as 12 Anja exp 0 2 N mo m o nfa a 1 P m mo 0 0 n 1 mim lt mo ao 13 where we use the abbreviation alpha a n n Mean my and resn o This PDF has a type cbshape when using the AFitPdfFactory see below The following is an example of the code necessary to make a Crystal Ball PDF in AFit and Figure 5 shows the corresponding PDF distribution RooRealVar x x 5 0 5 0 The Crystal Ball PDF AFitCBShape cbshape x mypdfname RooAbsPdf cbPDF cbshape getPdf A RooPlot of e PDF 2 e a p o N al Prejection of CBSha o N 5 4 3 2 1 0 1 2 3 4 5 Figure 5 An example of the Crystal Ball PDF shape 10 2 7 Decay Models 2 7 1 The Decay Model The AFitDecay class is the wrapper for the RooDecay class This is used to model lifetime decay of non mixing particles as a function of At The functional form of this PDF i
50. s All example macros and configurations can be found in the AFit example directory These examples are ready to run from the same directory that contains the AFit package This assumes that once you have compiled AFit that you then load the shared library into root prior to running these examples 6 1 Fitting the muon lifetime The muon lifetime can be fitted from data taken with a simple experiment where slow moving muons from cosmic rays are trapped in a scintillator and subsequently decay 12 This process results in to light pulses detected by a photomultiplier tube and the different in time t between the two pulses is recorded The start time of the clock is the time when the muon enters the detector and the stop time is that when the muon decays into an electron and two neutrinos In addition to this signal process there is a background which is assumed to be uniform in t So the PDF for this problem is given by P Noignale 7 Npbackground 41 This is a one dimensional problem with being the sole discriminating variable There are two components signal and background The configuration file for this example is MuonLifetime txt So the signal component is described by an exponential function using a lifetime rather than a constant as the parameter to be determined from data and the background component is described by a polynomial of order one with a coefficient of zero The configuration file for this example can be found in
51. s e lAt r P At o At R At o At where 7 is the lifetime of the particle The resolution function R is convolved with the physical time dependence as indicated Figure 6 illustrates the shape of this PDF A RooPlot of A t a o o o o o Events 40 2 2500 2000 1500 1000 500 pte Poe Pa Pa PP loss 0 8 6 4 2 0 2 4 6 8 10 At 1O mb Figure 6 An example of the Decay PDF shape NOTE When using this model it is imperative that the RooAbsPdf of the resolu tion function you intend to use is instantiated before the Decay models RooAbsPdf 2 7 2 The BDecay Model The AFitBDecay class is the wrapper for the RooBDecay class This is used to model lifetime decay of non mixing particles as a function of At The functional form of this PDF is the most general description of B decay time distribution with effects of CP violation mixing and life time differences Dilution is not explicitly included in this pdf See the RooFit documentation for more information NOTE When using this model it is imperative that the RooAbsPdf of the reso lution function you intend to use is instantiated before the BCPGenDecay model s RooAbsPdf 11 2 7 3 The BCPGenDecay Model The AFitBCPGenDecay class is the wrapper for the time dependent CP asymmetry RooBCPGenDecay class This is used to measure CP asymmetries in Y 45 Bop decays at BaBar Section 6 5 illustrates how t
52. s satisfied then you can assign a parameter substitution by adding the following line to the appropriate section describing your PDF in the data card mypdfname_variablesToReplace oldVarA newVarA oldVarB newVarB 35 where the comma separated list provides pairs of old variables that will be replaced by new ones 4 10 Computing systematic uncertainties The systematic uncertainties on a fit result from a constant parameter p in the likelihood model is given by the shift on the nominal fitted value when p is varied by o p The computeSystematicError function within AFitMaster can be used to compute such an un certatinty When this function is called three fits to the data are performed i the nominal fit ii a fit with p set to p o p and iii a fit with p set to p o p The shifts in all parameters allowed to vary in the fit are reported in a specified text file For example using the rare B decay model one can compute the systematic uncertainty from a generated toy Monte Carlo simulated data sample using the following AFitMaster master AFit example rareBdecay txt RooAbsPdf pdf master getPdf RooArgSet amp varsToGen master getDiscVarSet AFitToy toy RooDataSet data toy generateToySample pdf varsToGen 1000 0 master computeSystematicError data continuum bMesxi testSyst txt The output file from this test is the following Bbg0Yield
53. t pdfbld order setVal 2 RooAbsPdf pdf pdfbld getPdf where the histogram needed to define the PDF is called hist and can be found in the ROOT file myfile root An assertion is made if the specified histogram does not exist Figure 13 shows an example of this type of PDF given an input histogram The data in the histogram are the result of randomly filling the histogram according to a Gaussian distribution with a mean of 0 5 and width of 0 1 A RooPlot of x N o o o Events 0 05 o o o o 40 20 Mia tiiiirtiiiitiviitiiiitiy 0 01 02 03 04 05 06 07 08 0 9 1 Figure 13 An example of the Histogram PDF shape solid compared with the original data points 18 2 16 KEYS non parametric PDF The KEYS algorithm Kernal Estimation of Your Shapes 9 can be used to obtain a smoothed non parametric PDF representation of a sample of data or Monte Carlo simulated data The use of the KEYS PDF is similar to that of the histogram PDF described above The main difference is that fits using a KEYS pdf will be a lot slower than those using a histogram PDF The reason for this is that a Gaussian kernel is computed for each event in the data set used to construct a KEYS PDF and the PDF is evaluated at each point by computing a sum over all events In practice it is usually beneficial to compute a KEYS PDF once and then store the output shape as a histogram for use in all
54. te energy and the beam energy in the eFe CM frame Here the Bro momentum pg and four momentum of the initial state Ei pi are defined in the laboratory frame and Ef is the Brec energy in the ete CM frame The distribution of mpgs AE peaks at the B mass near zero for signal events and does not peak for background We can simultaneously fit the mgs and AE to isolate our signal In order to do this we first need to decide how many fit components there are secondly we need to determine what 50 the functional forms of the PDFs we will use to describe these components The example examples rareBdecay cc assumes that there is a signal component as well as a background from B decays and a background from e e qq events where q u d s c The fit configuration of this example can be found in examples rareBdecay txt The first line after FitConfiguration specifies the variable names bMes mpg and bDeltaE AL FitConfiguration variables bMes bDeltaE Each fit component these are single continuum and Bbg0 listed in the configuration file option components needs to have its form defined see signal etc listed below The names are defined by the comma separated values in the components list components signal continuum Bbg0 j The fit range and number of bins used when plotting the discriminating variables is also defined bMes 5 2700 O L 5 25 5 29 B 30 bDeltaE 0 0000 0 L 0 3 0 3
55. the corresponding before after signal enhancement projections of the data Both the total and signal PDFs are plotted on the RooPlot of the projection made using the makeLRProjection function 150F Entries B Discriminating Variable N E So o o o Events 0 0132021 o o o 40 20 i 4 hanson goal i 1 log Lsig Ltotal Events 0 00133333 NN 5 5 28 5 285 29 Discriminating Variable Figure 28 The distribution of top the projection of mpgs middle likelihood ratio bottom the projection after cutting on the likelihood ratio to enhance signal 42 5 3 Likelihood Ratio Plots A test that can be made between the a complicated PDF and a reference data sample is to compute the ratio of the likelihood of an event to be signal Lsig normalised to the total likelihood of an event to either signal or background Lsig Log The class AFitLRPlot computes this ratio for a sample of simulated data generated from the PDF toy data and compares this to a reference sample of data The likelihood for this example is one dimensional the discriminating variable is the B meson mass The signal PDF is a Gaussian distribution centred at 5 28 GeV c with a width of 10 MeV c and the background distribution is described by an Argus function see below Figure 29 shows the likelihood ratio L sig Rao Lig Log 40 for Monte Carl
56. tructor of the class For example in order to make a generic PDF for the function f x 1 2 1 one can do the following 15 RooRealVar x x 5 0 5 0 AFitGeneric genshape x mypdfname 0 0 0 1 RooAbsPdf genericPdf genshape getPdf The corresponding shape of this generic PDF example is shown in Figure 11 0 03 Figure 11 An example of the generic PDF shape for the function f x x 2 1 2 13 Gounaris Sakurai lineshape PDF This is similar in shape to the relativistic Breit Wigner described in Sec 2 3 2 Gounaris Sakurai GS distribution is a model of the P wave rr scattering amplitude 8 The parameters mean and width of the resonance are mg and To The GS PDF used is for A and is defined as 1 d To moy m r g meT2 m 18 GS m mo To J R where m2 F Toza KORC AnG m 8 k2 rm0 dh ds amg 19 _ 2k m vs 2k m h s VS A A 20 dh ds szmg h m 6r mo 2m8 mms 21 16 mo t 2k mo mo m2mo a an mo 2rk mo rk3 mo 22 es m ga ne van 0 e 24 T m is the mass dependent width of Eq 23 s m and k s is defined in Eq 5 2 14 Helicity PDF This PDF has type helicity when using the AFitPdfFactory The functional form of the helicity is depends on the value of the type string specified prior to building the PDF and this PDF is impleme
57. used to describe the fluctuations in energy loss of a charged particle passing through a thin layer of material A RooPlot of 19 024 022 3 30 02 G 2 018 3 016 o 3 014 5 3 012 0 01 0 008 0 006 0 004 0 002 5 4 3 2 1 0 1 2 3 4 5 Figure 15 An example of the Landau PDF shape The following code snippet will construct a Landau PDF with a mean of 0 0 and a width of 1 0 AFitLandau landaubld x pdf RooAbsPdf landau landaubld getPdf 2 18 Novosibirsk This PDF has a type novosibirsk when using the AFitPdfFactory and Figure 16 shows an example of the Novosibirsk PDF shape The functional form of the Novosibirsk function is 2 2 2 Pj pr 25 20 sinh AV In 4 AVvin4d xo is the peak position is the width of the peak and A is a parameter describing the tail of the distribution dy 1 A x 20 0 x 26 A RooPlot of Shape BDF o o a gt e o e o o N N a KE p aaa arpa o mb oa o Projectiomof Novosikirsk o y o o al 4 3 2 1 0 1 2 3 4 5 1 a Figure 16 An example of the Novosibirsk PDF shape The following code snippet will construct a Novosibirsk PDF with a mean of 0 0 a width of 1 0 and a tail parameter of 1 0 AFitNovosibirskShape novbld x pdf RooAbsPdf nov novbld getPdf 2 19 Polynomial The polynomial is defined as N E Ple pi
58. where p p and x x for x lt x and p rhog and x x2 when x gt x2 This function is shown in Figure 4 for 0 2 tp 0 5 op 0 1 p 0 1 and p2 0 2 Ar 2ln2 F 1 10 212 ap opV2In3 Ee F 10 The parameters p and 0 are the peak position and width FWHM 2 35 and is an asymmetry parameter The numerical integral of the Bukin function does not always converge so you should check to make sure that the PDF is correctly evaluated by plotting the fitted result if you use this PFD The following code snippet will make a bukin pdf with the same PDF parameters as used in Figure 4 P T Xp Op E p Ap exp 1 0 5 0 0 5 1 Figure 4 An example of the Bukin PDF shape RooRealVar x x 0 0 1 0 AFitBukin bukin x bpdf RooAbsPdf bukinPDF bukin getPdf 2 5 Chebychev Polynomial Chebychev Polynomial distribution The Cheby shape is given by Plz pi 1 ON 11 i l where the parameters p are coefficints of the functions T and the T are define elsewhere The RooFit user guide advises that one should use a Chebychev Polynomial over a polynomial wherever possible as the coefficients of a Chebychev Polynomial have smaller correlations than the coefficients of a polynomial The corollory of this is that fits behave better when using a Chebychev Polynomial than when using the equivalent polynomial The following code sni
59. with a Blatt Weisskopf Form Factor The relativistic Breit Wigner form implemented in AFit is described in the following m2 m mg m T2 m P m Mo Do J R 6 Figure 2 An example of the Breit Wigner PDF shape where the mass dependent width is given by 2J 1 Vim To aoo FRE pe 1 2 1 2 k m z a met pe 5 and the functions F are the spin dependent Blatt Weisskopf form factors PMs 1 6 Fa 7 F x 8 In Eq 4 mo is the mass of the resonance To is its width J is its spin and R is the interaction radius The event generator on BaBar EvtGen uses a range parameter of 3 0GeV7 0 6 fm The parameters Ma and my are the masses of the daughters of the decaying resonance The corresponding distribution for a p meson described by this PDF is shown in Fig 3 where mo 0 77GeV c To 0 15GeV J 1 R 3 0GeV and mg my 0 135GeV c The allowed values of J are 0 1 and 2 The following code snippet illustrates how to instantiate an AFitRelBreit Wigner 0 04 Entries o N 0 01 00 4 0 6 0 8 1 12 Figure 3 An example of the relativistic Breit Wigner PDF shape for a p meson RooRealVar x x 0 0 1 0 AFitRelBreitWigner bw x bwpdf RooAbsPdf bwPDF bw getPdf 2 4 Bukin Function The Bukin function is given by EF I E 21 VIP ME 9 VERTE n VeHI E tem
60. x a b 1 ea xt b gt 29 where the coefficients a and b are an exponent scale factor and x offset respectively Figure 20 shows an example of the Sigmoid PDF shape A RooPlot of Oo 50 02 9 018 E 9 016 o 9 014 2 012 Oo 0 70 01 a 0 008 0 006 0 004 0 002 5 4 3 2 1 0 1 2 3 4 5 Figure 20 An example of the sigmoid PDF shape The following code snippet will make a sigmoid function PDF with a 1 0 and b 0 0 AFitSigmoid sigbld x pdf RooAbsPdf sigmoid sigbld getPdf 25 2 23 Step Function It can be useful to impose a sharp cut off to a PDF In such cases it is useful use a step function PDF where Pie 1 30 between za and x and P x 0 elsewhere Similarly it can also be useful to veto a region of x using the complement of the step function Figure 21 shows an example of the step and veto functions where x 2 and xp 3 A RooPlot of Discriminating Variable 2 e a Projection of e 2 2 e p S a Sn SS T T T T T T 3 5 4 4 5 5 3 5 4 4 5 5 Discriminating Variable Discriminating Variable Figure 21 An example of left a step and right a veto function The following code snippet will make a step function PDF with a step between x 0 0 and x 1 0 RooRealVar x x 1 0 2 0 AFitStepFunction stepbld x pdf RooAbsPdf step stepbld getPdf

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