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

2. 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

3. 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!

4. Documentation Main sources of documentation:
See for RooFit documentation (150+ pages)
$ROOTSYS/tutorials/roofit/
See for example macros
See for (latest) class descriptions. RooFit classes start with “Roo”.
RooFit code itself is structured and well documented!
Browse though RootTalk
Bug Wouter Verkerke directly 

5. 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

6. 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

7. 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

8. 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

9. 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

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

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

12. 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) ;

13. 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)

14. 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

15. 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

16. 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

17. 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

18. 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()

19. 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.

20. RooStats documentation
Mailing list:

21. 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)

22. 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!

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

24. 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
 Source this setup script first!
 This presentation
 Macros to be used in this tutorial
Open in your favorite browser

25. 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”) ;

26. 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

27. 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 …

28. 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

29. 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

30. 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

31. 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"

32. 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

33. 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

34. 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

35. 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

36. 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() ;

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

38. 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

39. 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.

40. 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:

41. 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

42. Building realistic models

43. 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 ...

44. 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

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

46. 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

47. 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

48. 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

49. 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

50. 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

51. 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

52. 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

53. 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.

54. 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?

55. 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.

56. 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.)

57. 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
 Source this setup script first!
 This presentation
 Macros to be used in second part of the tutorial
Open in your favorite browser

58. 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"),...) ;

59. 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

60. 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?

61. 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) *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

62. 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.

63. 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

64. 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.

65. 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

66. 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.

67. 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.

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

69. RooStats: Weighted events and samples
Typical use-cases of sample or event-weights:
Combination of MC samples with different luminosities
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

70. 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?

71. 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)) ;

72. 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.

73. 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()

74. 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

75. 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_ 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!