Presented by: Max Baak
Thanks to: Wouter Verkerke,
Kyle Cranmer for examples!

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!

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
ToyMC data
Generation
Model
Visualization
Data/Model
Fitting
C++ command line interface & macros
Data management & histogramming
MINUIT
I/O support
Graphics interface

RooFit purpose - Data Modeling for Physics Analysis
Distribution of observables
 
Define data model
Probability Density Function
  
• 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
 and


Fit model to data
Determination of
 

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
Using Models
 Fitting : Binned/Unbinned (extended) MLL fits, Chi 2 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 variable function
PDF
F space point x ,
(
 x p
(
 x ; f
 p , q
)

 x x max integral x
 f ( x ) dx
 x k
)
RooFit class
RooRealVar
RooAbsReal
RooAbsPdf
RooArgSet
RooRealIntegral
RooAbsData

Model building – (Re)using standard components
• RooFit provides a collection of compiled standard PDF classes
RooBMixDecay
RooPolynomial
RooHistPdf
RooArgusBG
RooGaussian
Physics inspired
ARGUS,Crystal Ball,
Breit-Wigner, Voigtian,
B/D-Decay,….
Non-parametric
Histogram, KEYS
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

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;a
0
,a
1
) g(x; m ,s)
RooPolyVar m (“m”,y,RooArgList(a0,a1)) ;
RooGaussian g(“g”,”gauss”,x, m ,s) ; g(x, y ; a
0
,a
1
,s)

Model building – Expression based components
• RooFormulaVar – Interpreted real-valued function
– Based on ROOT TFormula class
– Ideal for modifying parameterization of existing compiled PDFs
RooBMixDecay(t,tau, w ,…)
RooFormulaVar w (“w”,”1-2*D”,D) ;
• RooGenericPdf – Interpreted PDF
– Based on ROOT TFormula class
– User expression doesn’t need to be normalized
– Maximum flexibility
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
Goodness-of-fit measure
 -log(Likelihood)
 Extended –log(L)
 Chi 2
 User Defined
 (add custom/penalty terms to any of these)
Sample interactive MINUIT session
RooNLLVar nll(“nll”,”nll”,pdf,data) ;
RooMinuit m(nll) ;
Access any of MINUITs minimization methods m.hesse() ; x.setConstant() ; y.setVal(5) ; m.migrad() ; m.minos()
Change and fix param. values, using native RooFit interface during fit session
RooFitResult* r = m.save() ;
Interface
 One-line: RooAbsPdf::fitTo(…)
 Interactive: RooMinuit class
Output
 Modifies parameter objects of PDF
 Save snapshot of initial/final parameters, correlation matrix, fit status etc…

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
Input model
Generate toy MC Fit model
Accumulate fit statistics
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()

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.

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

• 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)

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!

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

1. Guide you through the fundamentals of RooFit
2. Look at some sample composite data models
1. Still quite simple, all 1-dimensional
3. Try to do at least one ‘advanced topic’, preferably more
1. 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 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



Source this setup script first!
This presentation
Macros to be used in this tutorial http://root.cern.ch/root/html/ClassIndex.html
Open in your favorite browser

• >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 gSystem->Load(“libRooFit”) ;
– Normally, this happens automatically.

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

• First create the variables you need Try these commands in an
Allowed range interactive root session.
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
• Then create a function object
RooGaussian gauss(“gauss”,”Gaussian”, x , mean , width ) ;
– Give variables as arguments to link variables to a function
Continue typing commands till slide 34 …

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

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

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

• Inspecting the structure of gauss gauss.printCompactTree() ;
0x10b95fc0 RooGaussian::gauss (gauss) [Auto]
0x10b90c78 RooRealVar::x (x)
0x10b916f8 RooRealVar::mean (mean)
0x10b85f08 RooRealVar::width (width)
• Inspecting the contents of frame 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"
(Options="L") RooCurve::curve_gaussProjected: "Projection of gauss"

• Unbinned data is represented by a RooDataSet object
• Class RooDataSet is RooFit interface to ROOT class TTree
RooDataSet
RooRealVar y
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
RooDataSet a RooRealVar column of a associates with
TTree
Association by matching TTree
Branch name with RooRealVar name

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

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

• 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

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

• Look again at the PDF 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)
– Results from MINUIT back-propagated to variables

• A self contained example to construct a model, fit it, and plot it on top of the data void fit(TTree* dataTree) {
// Define model macro/tut1.C
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() ; See next slide for instructions

• 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
(From hereon you can modify the macros below // Make plot and see what happens 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.

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

• You can combine any number of the preceding PDFs to build more realistic models
RooRealVar x(“x”,”x”,-10,10) macro/tut2.C
// 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() ;

• 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 ...

RooRealVar x("x","x",0,10) ;
// Construct background model
RooRealVar alpha("alpha","alpha",-0.7,-3,0) ;
RooExponential bkg1("bkg1","bkg1",x,alpha) ; macros/tut3.C
// 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() ;

Try adding another signal term

• 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
 log

L (
 p )

  
D log( g (
 x i
,
 p ))

N exp

N obs log( N exp
)
• Extended term automatically constructed in RooAddPdf if given equal number of coefficients & PDFS

• How to construct an extended PDF with RooAddPdf
// 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
• Fitting with extended model emodel.fitTo(data, ”e” ) ; macros/tut4.C
Include extended term in fit
Look at sum, expected errors, and correlations between fitted event numbers

• 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
1. Multidimensional models, selecting by likelihood ratio
2. Demo on ‘task automation’ as mentioned in last slide of introductory slide

• 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)
Signal Model * *
Background Model
* * macros/tut5.C

• 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
Plain projection of mES of previous excercise
Result from 3D fit
Nsig = 91 ± 10
Close to sqrt(N)
– 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 macros/tut6.C

• 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 N sig
, s (N sig
), pull(N sig
)
• 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

• 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

• Set up you own top mass fit!
• Fit the top quark mass distribution in macros/tut9.C
• For the top signal (around 160 GeV/c 2 ), use a Gaussian.
• For the background, try out
– Chebychev polynomial (RooChebychev)
– Polynomial (RooPolynomial)
Minumum number of background terms needed?
Which background description works better?
Why? Look at correlation matrix.

• Often useful to split data sample into various categories in a fit
– Signal region / control sample(s), number of good jets, b-tag / bveto, 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 macros/tut10.C
– Simultanous fit to signal region and bkg control sample, using a
RooCategory
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?

• 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.

• 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.)

1. A few advanced RooFit examples.
2. 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
 Source this setup script first!
 This presentation
 Macros to be used in second part of the tutorial http://root.cern.ch/root/html/ClassIndex.html
Open in your favorite browser

• 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"),...) ;

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

• Often useful to split data sample into various categories in a fit
– Signal region / control sample(s), number of good jets, b-tag / bveto, 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 macros/tut10.C
– Simultanous fit to signal region and bkg control sample, using a
RooCategory
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?

• 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: macros2/rf104_classfactory.C
• 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

• 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.

• Taking derivatives and integrals of pdfs/functions.
macros2/rf111_derivatives.C
• Morphing between pdfs
– RooLinearMorph macros2/rf705_linearmorph.C
• Parallel fitting and plotting macros2/rf603_multicpu.C
– 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/rf707_kernelestimation.C

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

• 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 macros2/rf303_conditional.C
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/rf306_condpereventerrors.C
macros2/rf307_fullpereventerrors.C

• 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: macros2/rf502_wspacewrite.C
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: macros2/rf502_wspaceread.C
.. to read the workspace, and pick up where you left off!
Study the macro to see how easy this is done.
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.

• Ask your neighbor for the workspace file (‘measurement’) he/she has just created.
• Run: macros2/rf502_wspacewrite2.C
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: macros2/combination.C
(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.
Exercise: Add a third measurement to the combination.

“Workspaces are the future of digital publishing.”

• 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: macros2/topmassfit.C
• See next slide …

• 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?

// 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)) ;

• 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.
macros2/rs301_splot.C
• 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.
Replace the isolation observable & pdf by antoher observable you are interested in, for example the trigger efficiency category & pdf from tut12.

• Profile-likelihood test calculator macros2/rs102_hypotestwiths hapes.C
– 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()

• HybridCalculator macros2/rs201_hybridcalc ulator.C
– 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: macros2/rf604_constraints.C
to add a Gaussian bkg constraint directly to the likelihood sum.
Apply the ProfileLikelihoodCalculator to compare with the HybridCalculator signal significance

• 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!