RooFit/RooStats Tutorial CAT Meeting, June 2009 Presented by: Max Baak Thanks to: Wouter Verkerke, Kyle Cranmer for examples!

Structure of RooFit/RooStats tutorial A tutorial in two sessions. Part one (Monday, 10h30): Introduction to RooFit Entry-level exercises Aimed for beginners Part two (Friday, 10h00): Introduction to RooStats (statistics extension to RooFit) (Selection of) Advanced and new features of RooFit Also useful for experienced users

RooFit: Your toolkit for data modeling What is RooFit? A powerful toolkit for modeling and fitting the expected distribution(s) of events in a physics analysis Very easy to setup large-scale fit in structured, transparent fashion. Primarily targeted to high-energy physicists using ROOT But, even used in financial world. Originally developed for the BaBar collaboration by Wouter Verkerke and David Kirkby, back in year 2000. Wouter is main developer Included with ROOT since v5.xx Core code is very mature, stable Continuous development, addition of more-powerful features. Standard in CMS!

Documentation Main sources of documentation: http://root.cern.ch/drupal/content/users-guide See for RooFit documentation (150+ pages) $ROOTSYS/tutorials/roofit/ See for example macros http://root.cern.ch/root/Reference.html See for (latest) class descriptions. RooFit classes start with “Roo”. RooFit code itself is structured and well documented! http://root.cern.ch/root/roottalk/roottalk09/ Browse though RootTalk Bug Wouter Verkerke directly 

Implementation – Add-on package to ROOT Shared library: libRooFit.so Data Modeling ToyMC data Generation Model Visualization Data/Model Fitting MINUIT C++ command line interface & macros Data management & histogramming I/O support Graphics interface

RooFit purpose - Data Modeling for Physics Analysis    Distribution of observables x Define data model Probability Density Function F(x; p, q) Physical parameters of interest p Other parameters q to describe detector effect (resolution,efficiency,…) Normalized over allowed range of the observables x w.r.t the parameters p and q     Fit model to data   Determination of p,q

Data modeling - Desired functionality Building/Adjusting Models Easy to write basic PDFs ( normalization) Easy to compose complex models (modular design) Reuse of existing functions Flexibility – No arbitrary implementation-related restrictions A n a l y s i s c y c l e Using Models Fitting : Binned/Unbinned (extended) MLL fits, Chi2 fits Toy MC generation: Generate MC datasets from any model Visualization: Slice/project model & data in any possible way Speed – Should be as fast or faster than hand-coded model

Data modeling – OO representation Mathematical objects are represented as C++ objects Mathematical concept RooFit class variable RooRealVar function RooAbsReal PDF RooAbsPdf space point RooArgSet integral RooRealIntegral list of space points RooAbsData

Model building – (Re)using standard components RooFit provides a collection of compiled standard PDF classes RooBMixDecay Physics inspired ARGUS,Crystal Ball, Breit-Wigner, Voigtian, B/D-Decay,…. RooPolynomial RooHistPdf Non-parametric Histogram, KEYS RooArgusBG RooGaussian Basic Gaussian, Exponential, Polynomial,… PDF Normalization By default RooFit uses numeric integration to achieve normalization Classes can optionally provide (partial) analytical integrals Final normalization can be hybrid numeric/analytic form

Model building – (Re)using standard components Most physics models can be composed from ‘basic’ shapes RooBMixDecay RooPolynomial RooHistPdf RooArgusBG RooGaussian + RooAddPdf

Model building – (Re)using standard components Most physics models can be composed from ‘basic’ shapes RooBMixDecay RooPolynomial RooHistPdf RooArgusBG RooGaussian * RooProdPdf

Model building – (Re)using standard components Building blocks are flexible Function variables can be functions themselves Just plug in anything you like Universally supported by core code (PDF classes don’t need to implement special handling) m(y;a0,a1) g(x;m,s) g(x,y;a0,a1,s) RooPolyVar m(“m”,y,RooArgList(a0,a1)) ; RooGaussian g(“g”,”gauss”,x,m,s) ;

Model building – Expression based components RooFormulaVar – Interpreted real-valued function Based on ROOT TFormula class Ideal for modifying parameterization of existing compiled PDFs RooGenericPdf – Interpreted PDF User expression doesn’t need to be normalized Maximum flexibility RooBMixDecay(t,tau,w,…) RooFormulaVar w(“w”,”1-2*D”,D) ; RooGenericPdf f("f","1+sin(0.5*x)+abs(exp(0.1*x)*cos(-1*x))",x)

Using models – Fitting options Fitting interface is flexible and powerful, many options supported Data type Binned Unbinned Weighted unbinned Sample interactive MINUIT session RooNLLVar nll(“nll”,”nll”,pdf,data) ; RooMinuit m(nll) ; m.hesse() ; x.setConstant() ; y.setVal(5) ; m.migrad() ; m.minos() RooFitResult* r = m.save() ; Access any of MINUITs minimization methods Goodness-of-fit measure -log(Likelihood) Extended –log(L) Chi2 User Defined (add custom/penalty terms to any of these) Change and fix param. values, using native RooFit interface during fit session Output Modifies parameter objects of PDF Save snapshot of initial/final parameters, correlation matrix, fit status etc… Interface One-line: RooAbsPdf::fitTo(…) Interactive: RooMinuit class

Using models – Fitting speed & optimizations RooFit delivers per-fit tailored optimization without user overhead! Benefit of function optimization traditionally a trade-off between Execution speed (especially in fitting) Flexibility/maintainability of analysis user code Optimizations usually hard-code assumptions… Evaluation of –log(L) in fits lends it well to optimizations Constant fit parameters often lead to higher-level constant PDF components PDF normalization integrals have identical value for all data points Repetitive nature of calculation ideally suited for parallelization. RooFit automates analysis and implementation of optimization Modular OO structure of PDF expressions facilitate automated introspection Find and pre-calculate highest level constant terms in composite PDFs Apply caching and lazy evaluation for PDF normalization integrals Optional automatic parallelization of fit on multi-CPU hosts Optimization concepts are applied consistently and completely to all PDFs Speedup of factor 3-10 typical in realistic complex fits

Using models – Plotting RooPlot – View of 1 datasets/PDFs projected on the same dimension  Create the view on mes RooPlot* frame = mes.frame() ;  Project the data on the mes view data->plotOn(frame) ;  Project the PDF on the mes view pdf->plotOn(frame) ;  Project the bkg. PDF component pdf->plotOn(frame,Components(“bkg”))  Draw the view on a canvas frame->Draw() ; Axis labels auto-generated

Using models - Overview All RooFit models provide universal and complete fitting and Toy Monte Carlo generating functionality Model complexity only limited by available memory and CPU power models with >16000 components, >1000 fixed parameters and>80 floating parameters have been used (published physics result) Very easy to use – Most operations are one-liners Fitting Generating data = gauss.generate(x,1000) RooAbsPdf gauss.fitTo(data) RooDataSet RooAbsData

Advanced features – Task automation Support for routine task automation, e.g. goodness-of-fit study Accumulate fit statistics Input model Generate toy MC Fit model Distribution of - parameter values - parameter errors - parameter pulls Repeat N times // Instantiate MC study manager RooMCStudy mgr(inputModel) ; // Generate and fit 100 samples of 1000 events mgr.generateAndFit(100,1000) ; // Plot distribution of sigma parameter mgr.plotParam(sigma)->Draw()

RooStats What is RooStats? Set of statistical tools on top of RooFit (& ROOT). Joint, open project between LHC experiments and ROOT. Code is developing quickly. Goals Enable the combining of results of multiple measurements/experiments, including syst. uncertainties. Standard in CMS! Various tools to determine sensitivity and limits. Techniques ranging from Bayesian to fully Frequentist.

RooStats documentation http://twiki.cern.ch/twiki/bin/view/RooStats/ Mailing list: roostats-development@cern.ch

Combination of measurements: An Example Example shows opening (fake) Atlas and CMS measurements, and performing a combined fit to a common parameter with a profile likelihood. (thanks to Kyle Cranmer)

Appetizer for first part of tutorial Featuring: The basic RooFit toolkit Convolutions of functions Calculate the P-value of your model. Modelling the top mass spectrum A combined fit to signal and control samples Unbinned efficiency curve fit And much more!

The basics RooFit users tutorial Probability density functions & likelihoods The basics of OO data modeling The essential ingredients: PDFs, datasets, functions

Outline of the hands-on part Guide you through the fundamentals of RooFit Look at some sample composite data models Still quite simple, all 1-dimensional Try to do at least one ‘advanced topic’, preferably more Tutorial 8: Calculating the P-value of your analysis. P-Value = How often does an equivalent data sample with no signal mimic the signal you observe Tutorial 9: Fit to a top mass distribution Tutorial 10: Simultaneous fit to signal and control samples Copy roofit_tutorial.tar.gz from ~mbaak/public/ Untar roofit_tutorial.tar in your favorite directory on lxplus Contents of the tutorial setup tutorial/setup.sh tutorial/docs/roofit_tutorial.ppt tutorial/macros http://root.cern.ch/root/html/ClassIndex.html  Source this setup script first!  This presentation  Macros to be used in this tutorial Open in your favorite browser

Loading RooFit into ROOT >source setup.sh (in the tutorial/ directory) Make sure libRooFit.so is in $ROOTSYS/lib Start ROOT In the ROOT command line load the RooFit library Normally, this happens automatically. gSystem->Load(“libRooFit”) ;

Creating a variable – class RooRealVar Creating a variable object Every RooFit objects must have a unique name! RooRealVar mass(“mass”,“m(e+e-)”,0,1000) ; C++ name Name Title Allowed range

Creating a probability density function First create the variables you need Then create a function object Give variables as arguments to link variables to a function Try these commands in an interactive root session. Allowed range RooRealVar x(“x”,“x observable”,-10,10) ; RooRealVar mean(“mean”,“mean”,0.0,-10,10) ; RooRealVar width(“width”,“width”,3.0,0.1,10.) ; Allowed range Initial value RooGaussian gauss(“gauss”,”Gaussian”, x, mean, width) ; Continue typing commands till slide 34 …

Making a plot of a function First create an empty plot A frame is a plot associated with a RooFit variable Draw the empty plot on a ROOT canvas RooPlot* frame = x.frame() ; frame->Draw() Plot range taken from limits of x

Making a plot of a function (continued) Draw the (probability density) function in the frame Update the frame in the ROOT canvas gauss.plotOn(frame) ; frame->Draw() Axis label from gauss title Unit normalization

Interacting with objects Changing and inspecting variables Draw another copy of gauss width.getVal() ; (const Double_t) 3.00 width = 1.0 ; (const Double_t) 1.00 gauss.plotOn(frame) ; frame->Draw() macro/tut0.C

Inspecting composite objects Inspecting the structure of gauss Inspecting the contents of frame gauss.printCompactTree() ; 0x10b95fc0 RooGaussian::gauss (gauss) [Auto] 0x10b90c78 RooRealVar::x (x) 0x10b916f8 RooRealVar::mean (mean) 0x10b85f08 RooRealVar::width (width) frame->Print(“v”) RooPlot::frame(10ba6830): "A RooPlot of "x"" Plotting RooRealVar::x: "x" Plot contains 2 object(s) (Options="L") RooCurve::curve_gaussProjected: "Projection of gauss"

Data Unbinned data is represented by a RooDataSet object Class RooDataSet is RooFit interface to ROOT class TTree RooDataSet RooRealVar y RooDataSet associates a RooRealVar with column of a TTree Association by matching TTree Branch name with RooRealVar name RooRealVar x TTree row x y 1 0.57 4.86 2 5.72 6.83 3 2.13 0.21 4 10.5 -35. 5 -4.3 -8.8

Creating a dataset from a TTree First open file with TTree Create RooDataSet from tree macros/tut1.root TFile f(“tut1.root”) ; f.ls() ; root [1] .ls TFile** tut1.root TFile* tut1.root KEY: TTree xtree;1 xtree xtree->Print() ; RooDataSet data(“data”,”data”,xtree,x) ; Imported TTree RooFit Variable in dataset

Drawing a dataset on a frame Create new plot frame, draw RooDataSet on frame, draw frame RooPlot* frame2 = x.frame() ; data.plotOn(frame2) ; frame2->Draw() ; Note Poisson Error bars

Overlaying a PDF curve on a dataset Add PDF curve to frame gauss.plotOn(frame2) ; frame2->Draw() ; Unit normalized PDF automatically scaled to dataset But shape is not right! Lets fit the curve to the data

Fitting a PDF to an unbinned dataset Fit gauss to data Behind the scenes RooFit constructs the Likelihood from the PDF and the dataset RooFit passes the Likelihood function to MINUIT to minimize RooFit extracts the result from MINUIT and stores in the RooRealVar objects that represent the fit parameters Draw the result gauss.fitTo(data) ; gauss.plotOn(frame2) ; frame2->Draw() ;

Looking at the fit results Look again at the PDF variables Results from MINUIT back-propagated to variables width.Print() ; RooRealVar::sigma: 1.9376 +/- 0.043331 (-0.042646, 0.044033) L(-10 – 10) mean.Print() ; RooRealVar::mean: -0.0843265 +/- 0.061273 (-0.061210, 0.061361) L(-10 - 10) Adjusted value Symmetric error (from HESSE) Asymmetric error (from MINOS, not shown by default)

Putting it all together A self contained example to construct a model, fit it, and plot it on top of the data void fit(TTree* dataTree) { // Define model RooRealVar x(“x”,”x”,-10,10) ; RooRealVar sigma(“sigma”,”sigma”,2,0.1,10) ; RooRealVar mean(“mean”,”mean”,-10,10) ; RooGaussian gauss(“gauss”,”gauss”,x,mean,sigma) ; // Import data RooDataSet data(“data”,”data”,dataTree,x) ; // Fit data gauss.fitTo(data) ; // Make plot RooPlot* frame = x.frame() ; data.plotOn(frame) ; gauss.plotOn(frame) ; frame->Draw() ; } macro/tut1.C See next slide for instructions

Putting it all together A self contained example to construct a model, fit it, and plot it on top of the dataset. macro/tut1.C root [0] TFile f("tut1.root") root [1] .L tut1.C root [2] fit(xtree) In macro/tut1.C uncomment two lines below // Make plot and see what happens (From hereon you can modify the macros directly yourself.) gauss.fitTo(data,Minos()); gauss.fitTo(data,Hesse()); // default // (See RooMinuit.cxx for // all possible fit options) Edit the macro to switch between Hesse and Minos minimization.

Building composite PDFS RooFit has a collection of many basic PDFs. RooArgusBG - Argus background shape RooBifurGauss - Bifurcated Gaussian RooBreitWigner - Breit-Wigner shape RooCBShape - Crystal Ball function RooChebychev - Chebychev polynomial RooDecay - Simple decay function RooExponential - Exponential function RooGaussian - Gaussian function RooKeysPdf - Non-parametric data description RooPolynomial - Generic polynomial PDF RooVoigtian - Breit-Wigner (X) Gaussian HTML class documentation in: http://root.cern.ch/root/html/ROOFIT_ROOFIT_Index.html

Building realistic models You can combine any number of the preceding PDFs to build more realistic models RooRealVar x(“x”,”x”,-10,10) // Construct background model RooRealVar alpha(“alpha”,”alpha”,-0.3,-3,0) ; RooExponential bkg(“bkg”,”bkg”,x, alpha) ; // Construct signal model RooRealVar mean(“mean”,”mean”,3,-10,10) ; RooRealVar sigma(“sigma”,”sigma”,1,0.1,10) ; RooGaussian sig(“sig”,”sig”,x,mean,sigma) ; // Construct signal+background model RooRealVar sigFrac(“sigFrac”,”signal fraction”,0.1,0,1) ; RooAddPdf model(“model”,”model”,RooArgList(sig,bkg),sigFrac) ; // Plot model RooPlot* frame = x.frame() ; model.plotOn(frame) ; model.plotOn(frame,Components(bkg),LineStyle(kDashed)) ; frame->Draw() ; macro/tut2.C

Building realistic models

Sampling ‘toy’ Monte Carlo events from model Just like you can fit models, you can also sample ‘toy’ Monte Carlo events from models RooDataSet* mcdata = model.generate(x,1000) ; RooPlot* frame2 = x.frame() ; mcdata->plotOn(frame2) ; model->plotOn(frame2) ; frame2->Draw() ; Try this yourself ...

RooAddPdf can add any number of models RooRealVar x("x","x",0,10) ; // Construct background model RooRealVar alpha("alpha","alpha",-0.7,-3,0) ; RooExponential bkg1("bkg1","bkg1",x,alpha) ; // Construct additional background model RooRealVar bkgmean("bkgmean","bkgmean",7,-10,10) ; RooRealVar bkgsigma("bkgsigma","bkgsigma",2,0.1,10) ; RooGaussian bkg2("bkg2","bkg2",x,bkgmean,bkgsigma) ; // Construct signal model RooRealVar mean("mean","mean",3,-10,10) ; RooRealVar width("width","width",0.5,0.1,10) ; RooBreitWigner sig("sig","sig",x,mean,width) ; // Construct signal+2xbackground model RooRealVar bkg1Frac("bkg1Frac","signal fraction",0.2,0,1) ; RooRealVar sigFrac("sigFrac","signal fraction",0.5,0,1) ; RooAddPdf model("model","model",RooArgList(sig,bkg1,bkg2), RooArgList(sigFrac,bkg1Frac)) ; RooPlot* frame = x.frame() ; model.plotOn(frame) ; model.plotOn(frame,Components(RooArgSet(bkg1,bkg2)),LineStyle(kDashed)) ; frame->Draw() ; macros/tut3.C

RooAddPdf can add any number of models Try adding another signal term

Extended Likelihood fits Regular likelihood fits only fit for shape Number of coefficients in RooAddPdf is always one less than number of components Can also do extended likelihood fit Fit for both shape and observed number of events Accomplished by adding ‘extended likelihood term’ to regular LL Extended term automatically constructed in RooAddPdf if given equal number of coefficients & PDFS

Extended Likelihood fits and RooAddPdf How to construct an extended PDF with RooAddPdf Fitting with extended model // Construct extended signal+2xbackground model RooRealVar nbkg1(“nbkg1",“number of bkg1 events",300,0,1000) ; RooRealVar nbkg2(“nbkg2",“number of bkg2 events",200,0,1000) ; RooRealVar nsig( “nsig",“number of signal events",500,0,1000) ; RooAddPdf emodel(“emodel",“emodel",RooArgList(sig, bkg1, bkg2), RooArgList(nsig,nbkg1,nbkg2)) ; Previous model sigFrac bkg1Frac Add extended term sigFrac bkg1Frac ntotal New representation nsig nbkg1 nbkg2 macros/tut4.C emodel.fitTo(data,”e”) ; Look at sum, expected errors, and correlations between fitted event numbers Include extended term in fit

Switching gears Hands-on exercise so far designed to introduce you to basic model building syntax Real power of RooFit is in using those models to explore your analysis in an efficient way No time in this short session to cover this properly, so next slide just gives you a flavor of what is possible Multidimensional models, selecting by likelihood ratio Demo on ‘task automation’ as mentioned in last slide of introductory slide

Multi-dimensional PDFs RooFit handles multi-dimensional PDFs as easily as 1D PDFs Just use class RooProdPdf to multiply 1D PDFS Case example: selecting B+  D0 K+ Three discriminating variables: mES, DeltaE, m(D0) Look at example model, fit, plots in * * Signal Model * * Background Model Run example model, fit, plots in: macros/tut5.C

Selecting by Likelihood ratio Plain projection of multi-dimensional PDF and dataset often don’t do justice to analyzing power of PDF You don’t see selecting power of PDF in dimensions that are projected out Possible solution: don’t plot all events, but show only events passing cut of signal,bkg likelihood ratios constructed from PDF dimensions that are not shown in the plot Plain projection of mES of previous excercise Result from 3D fit Nsig = 91 ± 10 Close to sqrt(N) macros/tut6.C

Next topic: How stable is your fit When looking at low statistics fit, you’ll want to check explicitly Is your fit stable and unbiased Check by running through large set of toy MC samples Fit each sample, accumulate fit statistics and make pull distribution Technical procedure Generate toy Monte Carlo sample with desired number of events Fit for signal in that sample Record number of fitted signal events Repeat steps 1-3 often Plot distributions of Nsig, s(Nsig), pull(Nsig) RooFit can do all this for you with 2 lines of code! Try out the example in Experiment with lowering number of signal events macros/tut7.C

How often does background mimic your signal? Useful quantity in determining importance of your signal: the P-value P-Value: How often does a data sample of comparable statistics with no signal mimic the signal yield you observe Tells you how probable it is that your peak is the result of a statistical fluctuation of the background Procedure very similar to previous exercise First generate fake ‘data’, fit data to determine ‘data signal yield’ Generate toy Monte Carlo sample with 0 signal events Fit for signal in that sample Record number of fitted signal events Repeat steps 1-3 often See what fraction of fits result in a signal yield exceeding your ‘observed data yield’ Try out the example in macros/tut8.C

Top mass fit macros/tut9.C Set up you own top mass fit! Fit the top quark mass distribution in For the top signal (around 160 GeV/c2), use a Gaussian. For the background, try out Chebychev polynomial (RooChebychev) Polynomial (RooPolynomial) macros/tut9.C Minumum number of background terms needed? Which background description works better? Why? Look at correlation matrix.

Simultaneous fit to signal and control sample(s) Often useful to split data sample into various categories in a fit Signal region / control sample(s), number of good jets, b-tag / b-veto, fiducial volumes, etc. Categories may be overlapping Assigning of categories done using ‘RooCategory’ objects Roofit: Easy to make simultaneous fit to various categories Use full statistical power of entire sample. Correlation of fit parameters automatically propagated! Very powerful technique. Try out example in Simultanous fit to signal region and bkg control sample, using a RooCategory macros/tut10.C Add a third category & sample that contains a control Gaussian shape with the same width (but different mean) as needed in the signal region. How does the simultaneous fit improve?

Convolution of pdfs RooFit can do both analytical and numerical convolutions. Various analytical convolutions provided. Eg. Exponential and Gaussian – see class: RooDecay Numerical convolutions done with Fast Fourier transforms Need the FFTW library. Often as fast as analytical convolutions! Try out example: macros/tut11.C Replace the Landau with a Breit-Wigner function. Add a second, wider exponential. Do the new fit to a toy sample.

Unbinned efficiency curve fit Statistical error often not properly accounted for when performing a binned efficiency curve fit. Binomial errors do not go to zero close when eff=0 or eff=1. Proper implementation: unbinned efficiency curve fit, possible in RooFit For an unbinned efficiency fit, see: macros/tut12.C Use a RooMCStudy to proof that the pull distributions of the fit parameters are as expected. (See also tutorial 8.)

Outline of hands-on part 2 A few advanced RooFit examples. Several RooStats examples. Copy roofit_tutorial.tar.gz from ~mbaak/public/ Untar roofit_tutorial.tar in your favorite directory on lxplus Contents of the tutorial setup: tutorial/setup.sh tutorial/docs/roofit_tutorial.ppt tutorial/macros2 http://root.cern.ch/root/html/ClassIndex.html  Source this setup script first!  This presentation  Macros to be used in second part of the tutorial Open in your favorite browser

Root news Root v5.24 will come out next Wednesday. This contains RooFit v3.00 New RooStats functionality & examples. Example cool, new RooFit functionality: choose between different fit minimizers Such as: Minuit2 GSLMultiMin pdf->fitTo(data,Minimizer("GSLMultiMin","conjugatefr"),...) ;

This RooFit/RooStats tutorial session Featuring: Making your own pdf Adaptive kernel pdfs Morphing between datasets Working with workspaces Combination of measurements Profile likelihood scans Fitting of negative weights sPlots Hypothesis testing

Leftover: Simultaneous fit to several samples Often useful to split data sample into various categories in a fit Signal region / control sample(s), number of good jets, b-tag / b-veto, fiducial volumes, etc. Categories may be overlapping Assigning of categories done using ‘RooCategory’ objects Roofit: Easy to make simultaneous fit to various categories Use full statistical power of entire sample. Correlation of fit parameters automatically propagated! Very powerful technique. Try out example in Simultanous fit to signal region and bkg control sample, using a RooCategory macros/tut10.C Add a third category & sample that contains a control Gaussian shape with the same width (but different mean) as needed in the signal region. How does the simultaneous fit improve?

Making your own PDF/Function RooFit contains ‘factories’ that make it very easy for you to create a new pdf or function. Run the following macro and take a look at the contensts: Use the functionality RooClassFactory::makePdfInstance to make your own Breit-Wigner function. 1. / ((x-m)*(x-m) + 0.25*w*w) The proper normalization is automatically done by RooFit … Note the produced, corresponding .cxx and .h file! Use your Breit-Wigner function to generate and fit a Z spectrum. Mz = 90.2 GeV, GammaZ = 2.5 GeV macros2/rf104_classfactory.C

A Few Cool Examples You Should Really See Unfortunately we do not have time to go through all features of RooFit … Next follows a selection of powerful examples. Please go through the macros to see what they do. Ask any related questions you may have.

More RooFit Examples Taking derivatives and integrals of pdfs/functions. Morphing between pdfs RooLinearMorph Parallel fitting and plotting For comparison, do same macro with only 1 cpu-core. Adaptive kernel estimation. The following pdfs allow you to model models any dataset. Just plug your dataset into the pdf. RooKeysPdf (1-dimensional), RooNDKeysPdf (n-dimensional) Great for: modeling control samples or difficult correlations! Great for generating realistic Toy MC samples from data/full-MC! macros2/rf111_derivatives.C macros2/rf705_linearmorph.C macros2/rf603_multicpu.C macros2/rf707_kernelestimation.C

Morphing with Keys pdfs The macro loads two Higgs datasets, one for m(H)=130 GeV, and one for m(H) = 170 GeV. macros2/morph_keys.C Using the previous example in rf705_linearmorph.C, plot the approximated Higgs mass distributions for m(H) = 140,150,160 GeV.

Conditional pdfs A conditional pdf describes x, given the observable y. Pdf ( x | y ), eg: a mass resolution function, given the mass error. For an example conditional pdfs, see: Here the mean of a Gaussian for observable x depends on observable y. When plotting the distribution of x, one needs to project over the distribution of y. Note for the plotting: model.plotOn(xframe,ProjWData()) Other detailed examples. These show decay distributions with a Gaussian resolution function with per-event fit errors. macros2/rf303_conditional.C macros2/rf306_condpereventerrors.C macros2/rf307_fullpereventerrors.C

RooStats: Workspaces RooFit allows you to store an entire analysis into a ‘workspace’ object, that can be stored in a root file. This includes: pdfs, observables, functions, datasets. Try out: This stores the file: rf502_workspace.root Study the macro how to add an object to a workspace. You can then read back the workspace in a new session. Try out: .. to read the workspace, and pick up where you left off! Study the macro to see how easy this is done. macros2/rf502_wspacewrite.C macros2/rf502_wspaceread.C For the next exercise, rewrite out the workspace, where you change all initial values of the fit parameters, except for the ‘mean’ parameter. Eg sigma, bkgfrac, etc. Reduce the number of signal events.

RooStats: Combination of measurements Ask your neighbor for the workspace file (‘measurement’) he/she has just created. Run: This creates a second workspace, rf502_workspace2.root, which contains a second measurement. Now pretend these are two Higgs measurements! ;-) To calculate the average Higgs mass, run the script: (see next slide for result) Study this script: the combined fit is a full, proper profile likelihood fit! (Both measurements are completely refit!) What’s the 95% confidence region of ‘mean’? Rule for combining measurements: parameters with identical names are assumed to be the same parameter. macros2/rf502_wspacewrite2.C macros2/combination.C Exercise: Add a third measurement to the combination.

RooStats: Profile likelihood scan “Workspaces are the future of digital publishing.”

RooStats: Weighted events and samples Typical use-cases of sample or event-weights: Combination of MC samples with different luminosities MC@NLO events: positive and negative event weights When using event weights in unbinned maximum likelihood fit: Minimum found is correct Associated errors are incorrect, unless calculated properly Eg when using negative event weights, statistical error are typically underestimated. RooFit can do the proper error calculation! Try: See next slide … macros2/topmassfit.C

RooStats: Weighted events and samples Continue with macro: macros2/topmassfit.C Turn off the usage of event weights in the fit and in the plot. (See next slide for instructions.) How do the statistical errors change? Can you explain the change in behaviour?

RooStats: How to use (event-) weights in RooFit // set the weight observable dataset->setWeightVar(weightvar) ; // default option: errors from original HESSE error matrix // errors are “as expected on data”, but do not reflect correct // MC statistics model.fitTo(*data,SumW2Error(kFALSE)) ; // sum-of-weights corrected HESSE error matrix // errors correspond to true MC statistics model.fitTo(*data,SumW2Error(kTRUE)) ; // plot weighted events data->plotOn(frame,DataError(RooAbsData::SumW2)) ;

RooStats: sPlots sPlots is a technique to unfold two distributions, eg. signal and background events, when making a plot. It’s not a supersymmetric plot ;-) In this macro, the distribution of interest is the electron isolation, for Z->ee vs QCD. To make sPlots for the isolation, a ‘control’ discriminator is needed to unfold the signal and bkg distributions. In this example, provided by a mass fit. Based on the control variable, an s-eventweight is assigned for each event, which is used to draw the plots. macros2/rs301_splot.C Replace the isolation observable & pdf by antoher observable you are interested in, for example the trigger efficiency category & pdf from tut12.

RooStats: Profile Likelihood hypothesis test macros2/rs102_hypotestwithshapes.C Profile-likelihood test calculator RooStats::ProfileLikelihoodCalculator The ProfileLikelihoodCalculator makes a profile likelihood scan in the fraction of signal events (‘mu’). See function: DoHypothesisTest() Using a Gaussian interpretation (Wilk’s Theorem), the LL-ratio at zero signal gets converted into a P-value (=significance) Try to make a Profile likelihood scan of ‘mu’ to test the Gaussian interpretation (see also: macros2/combination.C), and calculate the significance yourself. Do this in the function: MakePlots()

RooStats: HybridCalculator RooStats::HypoTestCalculator A hybrid Frequentist and Bayesian tool. The tool integrate over nuisance (bkg) parameters using a Freq. technique. The macro has a (Gaussian) Bayesian prior for the number of bkg events, but is Frequentist (ie. toy MC) to get -2lnQ distributions from S&B and B-only samples. See: to add a Gaussian bkg constraint directly to the likelihood sum. macros2/rs201_hybridcalculator.C macros2/rf604_constraints.C Apply the ProfileLikelihoodCalculator to compare with the HybridCalculator signal significance

Further reading There are more (advanced) RooFit features and examples worth demonstrating than one can fit in two brief tutorial sessions. I have tried to show a (popular) snapshot of all possibilities. You are encouraged to take a look at: The RooFit documentation (docs/RooFit_Users_Manual_2.91-33.pdf) The examples in the directory: examples/roofit/ … to experience the full power of RooFit and RooStats ! I hope you’ve enjoyed the tutorials and will continue to keep on using RooFit and RooStats in the future!