RooFit Tutorial – Topical Lectures June 2007 Tutorial by Wouter Verkerke Presented by Max Baak

RooFit: Your toolkit for data modeling What is it? A powerful toolkit for modeling the expected distribution(s) of events in a physics analysis Primarily targeted to high-energy physicists using ROOT Originally developed for the BaBar collaboration by Wouter Verkerke and David Kirkby. Included with ROOT v5.xx Documentation: http://root.cern.ch/root/Reference.html – for latest class descriptions. RooFit classes start with “Roo”. http://roofit.sourceforge.net – for documentation and tutorials Bug Wouter 

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

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

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 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 RooFit delivers per-fit tailored optimization without user overhead!

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()

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’ 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 2. Tutorial 9: Fit the top mass Copy tutorial.tar.gz from /project/atlas/users/mbaak/TPL/ Untar tutorial.tar in your home directory Contents of the tutorial setup tutorial/docs/roofit_tutorial.ppt /macros http://root.cern.ch/root/html514/ClassIndex.html  This presentation  Macros to be used in this tutorial Open in your favorite browser

Loading RooFit into ROOT >source setup.csh (in the tutorial/ directory) Make sure libRooFit.so is in $ROOTSYS/lib Start ROOT In the ROOT command line load the RooFit library Or start root in the tutorial/ directory. 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 function Allowed range RooRealVar mass(“mass”,“mass”,-10,10) ; RooRealVar mean(“mean”,“mean”,0.0,-10,10) ; RooRealVar width(“width”,“sigma”,3.0,0.1,10.) ; Allowed range Initial value RooGaussian gauss(“gauss”,”Gaussian”, mass, mean, width) ;

Making a plot of a function First create an empty plot A frame is a plot associated with a RooFit variable (mass in this case) Draw the empty plot on a ROOT canvas RooPlot* frame = mass.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 sigma.getVal() ; (const Double_t) 3.00 sigma = 1.0 ; (const Double_t) 1.00 gauss->plotOn(frame) ; frame->Draw()

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::sigma (sigma) 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 TFile f(“tut0.root”) ; f.ls() ; root [1] .ls TFile** tut1.root TFile* tut1.root KEY: TTree xtree;1 xtree macros/tut0.root RooDataSet data(“data”,”data”,x,xtree) ; RooFit Variable in dataset Imported TTree

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 pdf->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 sigma.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 HESSE)

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 In macro/tut1.C uncomment two lines below // Make plot and see what happens

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 doc/html/ClassIndex.html (open with internet explorer)

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 it ...

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 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 macros/tut7.C Experiment with lowering number of signal events

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 last 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

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

Advanced examples See the macros/examples/ directory.