RooStats_RooStatsTutorialsJune2013

Table of Contents

RooFit/RooStats tutorials for INFN School of Statistics 2013.......................................................................1

Introduction.............................................................................................................................................1

Getting started with the software......................................................................................................1

Terminology and Conventions.........................................................................................................2

RooFit Exercises: Create and fitting a model..................................................................................................3

Exercise 1: Gaussian model and fit it to random generated data............................................................3

Exercise 2: Fit of a Signal over an Exponential Background.................................................................5

Exercise 3: Fit to a Combined model......................................................................................................8

RooStats Exercises...........................................................................................................................................13

Exercise 1: Create aModelConfig from the Gaussian plus Exponential Model used in the RooFit

Exercise 2..............................................................................................................................................13

Exercise 2: Profile Likelihood Calculator.............................................................................................14

Exercise 3a: Bayesian MCMC Calculator............................................................................................16

Exercise 3b: Bayesian Numerical Calculator.......................................................................................19

Exercise 4: Create a Poisson Counting Model......................................................................................22

Exercise 5: Compute a 95% Upper Limit with the Counting Model....................................................24

Exercise 6: Compute the significance (Hypothesis Test).....................................................................25

Exercise 6b: Compute significance (p-value) as function of the signal mass.......................................29

Exercise 7: Compute Limits using Hypothesis Test Inversion (CLs Limits).......................................31

Exercise 7b: Using theStandardHypoTestInvDemo.C tutorial............................................................35

RooFit/RooStats tutorials for INFN School of

Statistics 2013

This is a set of RooFit/RooStats tutorials given at INFN school of statistics (3-7 June 2013) , see the schoolagenda.

The required materials used during the RooFit/RooStats tutorials is described. The materials consist ofexercises that can be solved from what has been presented in the lecture. Two levels of help are provided: ahint and a full solution.

Hint Hide

Here the hint is shown.

Solution Hide

Here the solution is shown.

The hint allows to create the ROOT macro required to complete the exercise step by step, while the solutionprovides the full running code. The hint consists of code snippets, which can be combined to write a fullROOT macro to solve the exercise. If you are not able to write the macro yourself, because you find toodifficult or because the time is not sufficient, you can go directly to the solution, which will contain therunning code and (hopefully) you will be able to run the macro and get the result of the tutorial exercise.You can find the single code attached (links at the end of the page) or you can use the provided tar file. Somepoints linked to the exercises are optional and marked with a icon: they are useful to scrutinise in moredetail some aspects. Try to solve or look at them if you have the time.

Additional material can be found on:

the ROOT Web page;

the RooFit Web page and User Guide,Quick Start Guide;

the RooStats Twiki WebHome where you find links also to past tutorials;

the lecture slides (see Timetable for June 7);

RooFit and RooStats tutorials available in $ROOTSYS/tutorials/roofit and

$ROOTSYS/tutorials/roostats.

Introduction

Getting started with the software

A few advice words before starting. The tutorials are based on the ROOT version 5.34, possibly the latest one,

5.34.07 which is also installed in the Virtual Machine available for the school.

See the computer setup instructions if you need to install the virtual machine.

If you are going to use your installed version of ROOT, make sure you have configured with RooFit (./configure --enable-roofit). You can check if roofit is installed in your version of

ROOT by typing the command root-config --has-roofit. The answer must be yes. If not,you must re-configure ROOT and rebuild it or install a new binary version. See again the computersetup instructions for a detailed description on how to install and configure ROOT.

A basic knowledge of ROOT is assumed, in particular how to run a ROOT macro in interpreted mode(CINT) and in compiled mode (ACLIC)

root> .x myMacro.C // interpreted

root> .x myMacro.C+ // compiled

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It is recommended to add the following lines at the beginning of your ROOT macro, that will load

automatically the Roofit and Roostats libraries (note that the RooStats calculator are defined in the

RooStats namespace):

using namespace RooFit;

using namespace RooStats;

Avoid running a macro multiple times in the same ROOT CINT sessions (it might unfortunately crash

sometimes if you do it, because some objects might not be properly deleted in the macro)

Note that Roostats methods start with capital letter (as in ROOT) while Roofit ones start with lower

letter

The Roostats calculator are quite verbose, you can suppress the output by doing:

RooMsgService::instance().setGlobalKillBelow(RooFit::WARNING) ;

Some links to documentation:

ROOT reference guide: http://root.cern.ch/drupal/content/reference-guide

Roofit user's manual: http://root.cern.ch/drupal/content/users-guide

ROOT, Roofit, Roostats tutorials: http://root.cern.ch/root/html/tutorials/ or within the local ROOT

installation under $ROOTSYS/tutorials

Terminology and Conventions

Here we give pragmatic definitions for a few basic concepts that we will use.

observable - something you measure in an experiment, for example a particle's momentum. Often, a

function of measured quantities, for example an invariant mass of several particles

global observable or auxiliary observable - an observable from another measurement, for example,

the integrated luminosity

model - a set of probability density functions (PDFs), which describe distributions of the observables

or functions of observables

model parameter - any variable in your PDF expression, which is not an observable

parameter of interest (POI) - a model parameter that you study, for example a cross section of your

signal process

nuisance parameter - every other model parameter, which is not your parameter of interest

data or dataset - a set of values of observables, either measured in an experiment, or simulated

likelihood - a model computed for a particular dataset

hypothesis - a particular model, with specified observables, POI, nuisance parameters, and prior

PDFs (in case of Bayesian inference)

prior PDF - a probability density for an observable or a model parameter, which is known a priori,

i.e. before a measurement is considered. This is a Bayesian concept exclusively. Prior has no meaning

or place in a frequentist type of inference

Bayesian - a type of statistical inference that usually produces probability of the hypothesis given the

data. Requires a prior.

frequentist - a type of statistical inference that usually produces probability of the data given the

hypothesis.

RooFit Exercises: Create and fitting a model

The aim of the RooFit exercises is to learn how to build a RooFit model, how to fit the model on a set of datathat we will generate and how to plot the result. We will also learn how to save the full RooFit workspace

containing the model and data in a file which we can read back later, for example when we will do the

RooStats exercises. We will learn also how to generate the data from the model. The same generated data wewill then use to fit the model

Using the snippets of code, one can build a full ROOT macro which can be run interactively in ROOT or

compiled using ACLIC.

Exercise 1: Gaussian model and fit it to random generated

data

We will start with a very simple exercise. If you already know a bit of RooFit you might skip this exercise andgo directly to the next one. The aim of this exercise is to show and make you practise the basic functionalityof RooFit.

We will create a Gaussian model from which we will generate a pseudo-data set and then we will fit this dataset. After fitting, we will plot the result. You can find the instruction for each steps in the lecture slides.

Start creating the Gaussian model using the workspace factory, with the syntax introduced in the

lecture slides.

Use the generate(...) method of the RooAbsPdf class to generate a data set with 1000 events.

Plot the data set using RooPlot in a ROOT Canvas.

Fit the generate data using the fitTo(...) method of the RooAbsPdf class

Plot the resulting fit function on top of the data

Hint Hide

Creating the model

Use the syntax of the RooWorkspace factory to create first the Gaussian p.d.f. function with the

variables (observables and parameters).

Every variable in RooFit, needs to be created with a defined range [xmin,xmax]. Use very large value

if you don't know the range. If you provided only a value, the variable is considered constant. If you

provide only the minimum and the maximum, the initial value will be taken in the middle of the

range. It is important, to avoid undesired side effect, to have the initial value within the given range.

Note one can create using the workspace factory syntax any existing RooFit pdf class by using the Pdf

class name stripped by the Roo suffix. The exact signature required (variables to be passed) is defined

in the Pdf class constructor (You can browse the reference documentation here.

You need to define the [value, min, max] of a variable only the first time you create in the factory.

Afterwards you can reference the variable by its name.

// create Gaussian Pdf

RooWorkspace w("w");

w.factory("Gaussian:pdf(x[-10,10],mu[1,-1000,1000],s[2,0,1000])");

Retrieving objects from the workspace.

After you have created the variable and p.d.f in the workspace, you can access their pointers, by using

RooWorkspace::var for variables or RooWorkspace::pdf for p.d.f. Example:

// retrieving model components

RooAbsPdf * pdf = w.pdf("pdf"); // access object from workspace

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RooRealVar * x = w.var("x"); // access object from workspace

Generating a dataset:

We need to pass the observable we want to generate the data on (x) and the number of events

Note that we can generate also binned data sets. In that case we have two options:

use a RooDataHist and generateBinned: RooDataHist * binData =

pdf->generateBinned(*x, 1000);.

use the option AllBinned() which will generate a weighted data set: RooDataSet *

data = pdf->generate(*x, NumEvents(1000), AllBinned());.

// generate an unbinned dataset of 1000 events

RooDataSet * data = pdf->generate( *x, 1000);

Plot the data:

// create a RooPlot object and plot the data

RooPlot * plot = x->frame();

data->plotOn(plot);

Fit the data: you need to call the RooAbsPdf::fitTo.

// fit the data and save its result. Use the optional =Minuit2= minimiser

RooFitResult * res = pdf->fitTo(*data, Minimizer("Minuit2","Migrad"), Save(true) );

Plot the resulting fit function in the same plot

pdf->plotOn(plot);

plot->Draw(); // to show the RooPlot in the current ROOT Canvas

Solution Hide

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Gaussian Fit Result:

Fit of the Gaussian model using RooFit