# Rs 1 0 2_Hypotestwithshapes¶

The macro creates a simple signal model and two background models, which are added to a RooWorkspace. The macro creates a toy dataset, and then uses a RooStats ProfileLikleihoodCalculator to do a hypothesis test of the background-only and signal+background hypotheses. In this example, shape uncertainties are not taken into account, but normalization uncertainties are.

Author: Kyle Cranmer
This notebook tutorial was automatically generated with ROOTBOOK-izer from the macro found in the ROOT repository on Friday, September 18, 2020 at 09:43 AM.

In :
%%cpp -d
#include "RooDataSet.h"
#include "RooRealVar.h"
#include "RooGaussian.h"
#include "RooProdPdf.h"
#include "RooProduct.h"
#include "TCanvas.h"
#include "RooChebychev.h"
#include "RooAbsPdf.h"
#include "RooFit.h"
#include "RooFitResult.h"
#include "RooPlot.h"
#include "RooAbsArg.h"
#include "RooWorkspace.h"
#include "RooStats/ProfileLikelihoodCalculator.h"
#include "RooStats/HypoTestResult.h"
#include <string>


In :
%%cpp -d
// This is a workaround to make sure the namespace is used inside functions
using namespace RooFit;
using namespace RooStats;


See below for implementation

In :
void AddModel(RooWorkspace *);
void DoHypothesisTest(RooWorkspace *);
void MakePlots(RooWorkspace *);


In :
%%cpp -d
{

// Make models for signal (Higgs) and background (Z+jets and QCD)
// In real life, this part requires an intelligent modeling
// of signal and background -- this is only an example.

// set range of observable
Double_t lowRange = 60, highRange = 200;

// make a RooRealVar for the observable
RooRealVar invMass("invMass", "M_{inv}", lowRange, highRange, "GeV");

// --------------------------------------
// make a simple signal model.
RooRealVar mH("mH", "Higgs Mass", 130, 90, 160);
RooRealVar sigma1("sigma1", "Width of Gaussian", 12., 2, 100);
RooGaussian sigModel("sigModel", "Signal Model", invMass, mH, sigma1);
// we will test this specific mass point for the signal
mH.setConstant();
// and we assume we know the mass resolution
sigma1.setConstant();

// --------------------------------------
// make zjj model.  Just like signal model
RooRealVar mZ("mZ", "Z Mass", 91.2, 0, 100);
RooRealVar sigma1_z("sigma1_z", "Width of Gaussian", 10., 6, 100);
RooGaussian zjjModel("zjjModel", "Z+jets Model", invMass, mZ, sigma1_z);
// we know Z mass
mZ.setConstant();
// assume we know resolution too
sigma1_z.setConstant();

// --------------------------------------
// make QCD model
RooRealVar a0("a0", "a0", 0.26, -1, 1);
RooRealVar a1("a1", "a1", -0.17596, -1, 1);
RooRealVar a2("a2", "a2", 0.018437, -1, 1);
RooRealVar a3("a3", "a3", 0.02, -1, 1);
RooChebychev qcdModel("qcdModel", "A  Polynomial for QCD", invMass, RooArgList(a0, a1, a2));

// let's assume this shape is known, but the normalization is not
a0.setConstant();
a1.setConstant();
a2.setConstant();

// --------------------------------------
// combined model

// Setting the fraction of Zjj to be 40% for initial guess.
RooRealVar fzjj("fzjj", "fraction of zjj background events", .4, 0., 1);

// Set the expected fraction of signal to 20%.
RooRealVar fsigExpected("fsigExpected", "expected fraction of signal events", .2, 0., 1);
fsigExpected.setConstant(); // use mu as main parameter, so fix this.

// Introduce mu: the signal strength in units of the expectation.
// eg. mu = 1 is the SM, mu = 0 is no signal, mu=2 is 2x the SM
RooRealVar mu("mu", "signal strength in units of SM expectation", 1, 0., 2);

// Introduce ratio of signal efficiency to nominal signal efficiency.
// This is useful if you want to do limits on cross section.
RooRealVar ratioSigEff("ratioSigEff", "ratio of signal efficiency to nominal signal efficiency", 1., 0., 2);
ratioSigEff.setConstant(kTRUE);

// finally the signal fraction is the product of the terms above.
RooProduct fsig("fsig", "fraction of signal events", RooArgSet(mu, ratioSigEff, fsigExpected));

// full model
RooAddPdf model("model", "sig+zjj+qcd background shapes", RooArgList(sigModel, zjjModel, qcdModel),
RooArgList(fsig, fzjj));

// interesting for debugging and visualizing the model
//  model.printCompactTree("","fullModel.txt");
//  model.graphVizTree("fullModel.dot");

wks->import(model);
}


In :
%%cpp -d
{

Int_t nEvents = 150;
RooAbsPdf *model = wks->pdf("model");
RooRealVar *invMass = wks->var("invMass");

RooDataSet *data = model->generate(*invMass, nEvents);

wks->import(*data, Rename("data"));
}


In :
%%cpp -d
void DoHypothesisTest(RooWorkspace *wks)
{

// Use a RooStats ProfileLikleihoodCalculator to do the hypothesis test.
ModelConfig model;
model.SetWorkspace(*wks);
model.SetPdf("model");

// plc.SetData("data");

ProfileLikelihoodCalculator plc;
plc.SetData(*(wks->data("data")));

// here we explicitly set the value of the parameters for the null.
// We want no signal contribution, eg. mu = 0
RooRealVar *mu = wks->var("mu");
//   RooArgSet* nullParams = new RooArgSet("nullParams");
RooArgSet poi(*mu);
RooArgSet *nullParams = (RooArgSet *)poi.snapshot();
nullParams->setRealValue("mu", 0);

// plc.SetNullParameters(*nullParams);
plc.SetModel(model);
// NOTE: using snapshot will import nullparams
// in the WS and merge with existing "mu"
// model.SetSnapshot(*nullParams);

plc.SetNullParameters(*nullParams);

// We get a HypoTestResult out of the calculator, and we can query it.
HypoTestResult *htr = plc.GetHypoTest();
cout << "-------------------------------------------------" << endl;
cout << "The p-value for the null is " << htr->NullPValue() << endl;
cout << "Corresponding to a significance of " << htr->Significance() << endl;
cout << "-------------------------------------------------\n\n" << endl;
}


In :
%%cpp -d
void MakePlots(RooWorkspace *wks)
{

// Make plots of the data and the best fit model in two cases:
// first the signal+background case
// second the background-only case.

// get some things out of workspace
RooAbsPdf *model = wks->pdf("model");
RooAbsPdf *sigModel = wks->pdf("sigModel");
RooAbsPdf *zjjModel = wks->pdf("zjjModel");
RooAbsPdf *qcdModel = wks->pdf("qcdModel");

RooRealVar *mu = wks->var("mu");
RooRealVar *invMass = wks->var("invMass");
RooAbsData *data = wks->data("data");

// --------------------------------------
// Make plots for the Alternate hypothesis, eg. let mu float

mu->setConstant(kFALSE);

model->fitTo(*data, Save(kTRUE), Minos(kFALSE), Hesse(kFALSE), PrintLevel(-1));

// plot sig candidates, full model, and individual components
new TCanvas();
RooPlot *frame = invMass->frame();
data->plotOn(frame);
model->plotOn(frame);
model->plotOn(frame, Components(*sigModel), LineStyle(kDashed), LineColor(kRed));
model->plotOn(frame, Components(*zjjModel), LineStyle(kDashed), LineColor(kBlack));
model->plotOn(frame, Components(*qcdModel), LineStyle(kDashed), LineColor(kGreen));

frame->SetTitle("An example fit to the signal + background model");
frame->Draw();
//  cdata->SaveAs("alternateFit.gif");

// --------------------------------------
// Do Fit to the Null hypothesis.  Eg. fix mu=0

mu->setVal(0);          // set signal fraction to 0
mu->setConstant(kTRUE); // set constant

model->fitTo(*data, Save(kTRUE), Minos(kFALSE), Hesse(kFALSE), PrintLevel(-1));

// plot signal candidates with background model and components
new TCanvas();
RooPlot *xframe2 = invMass->frame();
data->plotOn(xframe2, DataError(RooAbsData::SumW2));
model->plotOn(xframe2);
model->plotOn(xframe2, Components(*zjjModel), LineStyle(kDashed), LineColor(kBlack));
model->plotOn(xframe2, Components(*qcdModel), LineStyle(kDashed), LineColor(kGreen));

xframe2->SetTitle("An example fit to the background-only model");
xframe2->Draw();
//  cbkgonly->SaveAs("nullFit.gif");
}


The main macro.

Create a workspace to manage the project.

In :
RooWorkspace *wspace = new RooWorkspace("myWS");

RooFit v3.60 -- Developed by Wouter Verkerke and David Kirkby
Copyright (C) 2000-2013 NIKHEF, University of California & Stanford University



Add the signal and background models to the workspace

In :
AddModel(wspace);

[#1] INFO:ObjectHandling -- RooWorkspace::import(myWS) importing RooAddPdf::model
[#1] INFO:ObjectHandling -- RooWorkspace::import(myWS) importing RooGaussian::sigModel
[#1] INFO:ObjectHandling -- RooWorkspace::import(myWS) importing RooRealVar::invMass
[#1] INFO:ObjectHandling -- RooWorkspace::import(myWS) importing RooRealVar::mH
[#1] INFO:ObjectHandling -- RooWorkspace::import(myWS) importing RooRealVar::sigma1
[#1] INFO:ObjectHandling -- RooWorkspace::import(myWS) importing RooProduct::fsig
[#1] INFO:ObjectHandling -- RooWorkspace::import(myWS) importing RooRealVar::mu
[#1] INFO:ObjectHandling -- RooWorkspace::import(myWS) importing RooRealVar::ratioSigEff
[#1] INFO:ObjectHandling -- RooWorkspace::import(myWS) importing RooRealVar::fsigExpected
[#1] INFO:ObjectHandling -- RooWorkspace::import(myWS) importing RooGaussian::zjjModel
[#1] INFO:ObjectHandling -- RooWorkspace::import(myWS) importing RooRealVar::mZ
[#1] INFO:ObjectHandling -- RooWorkspace::import(myWS) importing RooRealVar::sigma1_z
[#1] INFO:ObjectHandling -- RooWorkspace::import(myWS) importing RooRealVar::fzjj
[#1] INFO:ObjectHandling -- RooWorkspace::import(myWS) importing RooChebychev::qcdModel
[#1] INFO:ObjectHandling -- RooWorkspace::import(myWS) importing RooRealVar::a0
[#1] INFO:ObjectHandling -- RooWorkspace::import(myWS) importing RooRealVar::a1
[#1] INFO:ObjectHandling -- RooWorkspace::import(myWS) importing RooRealVar::a2


Add some toy data to the workspace

In :
AddData(wspace);

[#1] INFO:ObjectHandling -- RooWorkspace::import(myWS) importing dataset modelData
[#1] INFO:ObjectHandling -- RooWorkSpace::import(myWS) changing name of dataset from  modelData to data


Inspect the workspace if you wish wspace->Print();

Do the hypothesis test

In :
DoHypothesisTest(wspace);

[#1] INFO:Minization -- createNLL: caching constraint set under name CONSTR_OF_PDF_model_FOR_OBS_invMass with 0 entries
[#0] PROGRESS:Minization -- ProfileLikelihoodCalcultor::DoGLobalFit - find MLE
[#0] PROGRESS:Minization -- ProfileLikelihoodCalcultor::DoMinimizeNLL - using Minuit / Migrad with strategy 1
[#1] INFO:Minization -- RooMinimizer::optimizeConst: activating const optimization
[#1] INFO:Minization --  The following expressions have been identified as constant and will be precalculated and cached: (sigModel,zjjModel,qcdModel)
[#1] INFO:Minization --
RooFitResult: minimized FCN value: 717.039, estimated distance to minimum: 8.90226e-10
covariance matrix quality: Full, accurate covariance matrix
Status : MINIMIZE=0

Floating Parameter    FinalValue +/-  Error
--------------------  --------------------------
fzjj    3.1152e-01 +/-  5.03e-02
mu    1.0968e+00 +/-  3.03e-01

[#0] PROGRESS:Minization -- ProfileLikelihoodCalcultor::GetHypoTest - do conditional fit
[#0] PROGRESS:Minization -- ProfileLikelihoodCalcultor::DoMinimizeNLL - using Minuit / Migrad with strategy 1
[#1] INFO:Minization -- RooMinimizer::optimizeConst: activating const optimization
[#1] INFO:Minization --
RooFitResult: minimized FCN value: 723.97, estimated distance to minimum: 2.09862e-09
covariance matrix quality: Full, accurate covariance matrix
Status : MINIMIZE=0

Floating Parameter    FinalValue +/-  Error
--------------------  --------------------------
fzjj    2.6213e-01 +/-  5.18e-02

-------------------------------------------------
The p-value for the null is 9.83108e-05
Corresponding to a significance of 3.72332
-------------------------------------------------



Make some plots

In :
MakePlots(wspace);

[#1] INFO:Minization -- createNLL picked up cached consraints from workspace with 0 entries
[#1] INFO:Minization -- RooMinimizer::optimizeConst: activating const optimization
[#1] INFO:Minization --  The following expressions have been identified as constant and will be precalculated and cached: (sigModel,zjjModel,qcdModel)
[#1] INFO:Minization -- RooMinimizer::optimizeConst: deactivating const optimization
[#1] INFO:Plotting -- RooAbsPdf::plotOn(model) directly selected PDF components: (sigModel)
[#1] INFO:Plotting -- RooAbsPdf::plotOn(model) indirectly selected PDF components: ()
[#1] INFO:Plotting -- RooAbsPdf::plotOn(model) directly selected PDF components: (zjjModel)
[#1] INFO:Plotting -- RooAbsPdf::plotOn(model) indirectly selected PDF components: ()
[#1] INFO:Plotting -- RooAbsPdf::plotOn(model) directly selected PDF components: (qcdModel)
[#1] INFO:Plotting -- RooAbsPdf::plotOn(model) indirectly selected PDF components: ()
[#1] INFO:Minization -- createNLL picked up cached consraints from workspace with 0 entries
[#1] INFO:Minization -- RooMinimizer::optimizeConst: activating const optimization
[#1] INFO:Minization --  The following expressions have been identified as constant and will be precalculated and cached: (sigModel,zjjModel,qcdModel)
[#1] INFO:Minization -- RooMinimizer::optimizeConst: deactivating const optimization
[#1] INFO:Plotting -- RooAbsPdf::plotOn(model) directly selected PDF components: (zjjModel)
[#1] INFO:Plotting -- RooAbsPdf::plotOn(model) indirectly selected PDF components: ()
[#1] INFO:Plotting -- RooAbsPdf::plotOn(model) directly selected PDF components: (qcdModel)
[#1] INFO:Plotting -- RooAbsPdf::plotOn(model) indirectly selected PDF components: ()


Cleanup

In :
delete wspace;


Draw all canvases

In :
%jsroot on
gROOT->GetListOfCanvases()->Draw()