RooFit/RooStats TutorialCAT 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

C++ command line interface & macros

Data management & histogramming

Graphics interface

I/O support

MINUIT

ToyMC dataGeneration

Data/ModelFitting

Data Modeling

Model Visualization

Shared library: libRooFit.so

RooFit purpose - Data Modeling for Physics Analysis

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

Distribution of observables x

Determination of p,q

Fit model to data

Define data model

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, 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

A n

a l y

s i s

c

y c

l e

Data modeling – OO representation

• Mathematical objects are represented as C++ objects

variable RooRealVar

function RooAbsReal

PDF RooAbsPdf

space point RooArgSet

list of space points RooAbsData

integral RooRealIntegral

RooFit classMathematical concept

),;( qpxF

px,

x

dxxfx

xmax

min

)(

)(xf

kx

Model building – (Re)using standard components

• RooFit provides a collection of compiled standard PDF classes

RooArgusBG

RooPolynomial

RooBMixDecay

RooHistPdf

RooGaussian

BasicGaussian, Exponential, Polynomial,…

Physics inspiredARGUS,Crystal Ball, Breit-Wigner, Voigtian,B/D-Decay,….

Non-parametricHistogram, KEYS

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

RooBMixDecay

RooPolynomial

RooHistPdf

RooArgusBG

Model building – (Re)using standard components

• Most physics models can be composed from ‘basic’ shapes

RooAddPdf+

RooGaussian

RooBMixDecay

RooPolynomial

RooHistPdf

RooArgusBG

RooGaussian

Model building – (Re)using standard components

• Most physics models can be composed from ‘basic’ shapes

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)

g(x;m,s)m(y;a0,a1)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– Based on ROOT TFormula class

– 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

Goodness-of-fit measure

-log(Likelihood)

Extended –log(L)

Chi2

User Defined

(add custom/penalty terms to any of these)

InterfaceOne-line: RooAbsPdf::fitTo(…)

Interactive: RooMinuit class

OutputModifies parameter objects of PDF

Save snapshot of initial/final parameters, correlation matrix, fit status etc…

Sample interactive MINUIT sessionRooNLLVar nll(“nll”,”nll”,pdf,data) ;

RooMinuit m(nll) ;

m.hesse() ;

x.setConstant() ;

y.setVal(5) ;

m.migrad() ;

m.minos()

RooFitResult* r = m.save() ;

Data typeBinned

Unbinned

Weighted unbinned Access any of MINUITsminimization methods

Change and fix param. values,using native RooFit interface during fit session

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 mesRooPlot* frame = mes.frame() ;

Project the data on the mes viewdata->plotOn(frame) ;

Project the PDF on the mes viewpdf->plotOn(frame) ;

Project the bkg. PDF componentpdf->plotOn(frame,Components(“bkg”))

Draw the view on a canvasframe->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 parametersand>80 floating parameters have been used (published physics result)

– Very easy to use – Most operations are one-liners

RooAbsPdf

RooDataSet

RooAbsData

gauss.fitTo(data)

data = gauss.generate(x,1000)

Fitting Generating

Advanced features – Task automation

• Support for routine task automation, e.g. goodness-of-fit study

Input model Generate toy MC Fit model

Repeat N times

Accumulatefit statistics

Distribution of- parameter values- parameter errors- parameter pulls

// Instantiate MC study managerRooMCStudy mgr(inputModel) ;

// Generate and fit 100 samples of 1000 eventsmgr.generateAndFit(100,1000) ;

// Plot distribution of sigma parametermgr.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!

RooFit users tutorial

The basics

Probability density functions & likelihoods

The basics of OO data modeling

The essential ingredients: PDFs, datasets, functions

Outline of the hands-on part

1. Guide you through the fundamentals of RooFit

2. Look at some sample composite data models1. Still quite simple, all 1-dimensional

3. Try to do at least one ‘advanced topic’, preferably more1. 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

4. Copy roofit_tutorial.tar.gz from ~mbaak/public/1. Untar roofit_tutorial.tar in your favorite directory on lxplus

2. Contents of the tutorial setup

tutorial/setup.shtutorial/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++ nameName 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

RooRealVar x(“x”,“x observable”,-10,10) ;

RooRealVar mean(“mean”,“mean”,0.0,-10,10) ;RooRealVar width(“width”,“width”,3.0,0.1,10.) ;

RooGaussian gauss(“gauss”,”Gaussian”, x, mean, width) ;

Initial value

Allowed range

Allowed range

Try these commands in an interactive root session.

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() ;

Plot range taken from limits of x

frame->Draw()

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) ;

Axis label from gauss title

Unit normalization

frame->Draw()

Interacting with objects

• Changing and inspecting variables

• Draw another copy of gauss

width.getVal() ;(const Double_t) 3.00

width = 1.0 ;

width.getVal() ;(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" (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

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

TTree

RooDataSet

RooRealVar xRooRealVar y

RooDataSet associatesa RooRealVar withcolumn of a TTree

Association by matching TTree Branch name with RooRealVarname

Creating a dataset from a TTree

• First open file with TTree

• Create RooDataSet from tree

TFile f(“tut1.root”) ;f.ls() ;root [1] .lsTFile** tut1.root TFile* tut1.root KEY: TTree xtree;1 xtreextree->Print() ;

RooDataSet data(“data”,”data”,xtree,x) ;

RooFit Variable in datasetImported TTree

macros/tut1.root

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 normalizedPDF automaticallyscaled to dataset

But shape is not right!Lets fit the curveto the data

Fitting a PDF to an unbinned dataset

• Fit gauss to 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.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 Symmetricerror

(from HESSE)

Asymmetricerror

(from MINOS, not shownby 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 slidefor instructions

Putting it all together

• A self contained example to construct a model, fit it, and plot it on top of the dataset.

root [0] TFile f("tut1.root") root [1] .L tut1.C root [2] fit(xtree)

macro/tut1.C

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

Edit the macro to switch between Hesse and Minos minimization.

gauss.fitTo(data,Minos());gauss.fitTo(data,Hesse()); // default

// (See RooMinuit.cxx for// all possible fit options)

(From hereon you can modify the macros directly yourself.)

Building composite PDFS

• RooFit has a collection of many basic PDFs.

RooArgusBG - Argus background shapeRooBifurGauss - Bifurcated GaussianRooBreitWigner - Breit-Wigner shapeRooCBShape - Crystal Ball functionRooChebychev - Chebychev polynomialRooDecay - Simple decay functionRooExponential - Exponential functionRooGaussian - Gaussian functionRooKeysPdf - Non-parametric data descriptionRooPolynomial - Generic polynomial PDFRooVoigtian - 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 modelRooRealVar alpha(“alpha”,”alpha”,-0.3,-3,0) ;RooExponential bkg(“bkg”,”bkg”,x, alpha) ;

// Construct signal modelRooRealVar mean(“mean”,”mean”,3,-10,10) ;RooRealVar sigma(“sigma”,”sigma”,1,0.1,10) ;RooGaussian sig(“sig”,”sig”,x,mean,sigma) ;

// Construct signal+background modelRooRealVar sigFrac(“sigFrac”,”signal fraction”,0.1,0,1) ;RooAddPdf model(“model”,”model”,RooArgList(sig,bkg),sigFrac) ;

// Plot modelRooPlot* 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

)log()),(log()(log expexp NNNpxgpL obsD

i

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 modelsigFrac

bkg1Frac

Add extended termsigFrac

bkg1Fracntotal

New representationnsig

nbkg1nbkg2

emodel.fitTo(data,”e”) ;

Include extended term in fit

macros/tut4.C

Look at sum, expected errors, and correlations between fitted event numbers

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

1. Multidimensional models, selecting by likelihood ratio

2. 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.CRun example model, fit, plots in:

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

macros/tut6.C

Plain projection of mESof previous excercise

Nsig = 91 ± 10

Result from 3D fit

Close to sqrt(N)

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, (Nsig), pull(Nsig)

• RooFit can do all this for you with 2 lines of code!– Try out the example in

macros/tut7.CExperiment 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 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

• 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

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.shtutorial/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/rf707_kernelestimation.C

macros2/rf603_multicpu.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 observablex 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

Exercise: Add a third measurement to the combination.

macros2/combination.C

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

• 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)

macros2/rs102_hypotestwithshapes.C

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

• 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

Apply the ProfileLikelihoodCalculator to compare with the HybridCalculator signal significance

macros2/rf604_constraints.C

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!