Detailed Description

'Number Counting Example' RooStats tutorial macro #100

This tutorial shows an example of a combination of two searches using number counting with background uncertainty.

The macro uses a RooStats "factory" to construct a PDF that represents the two number counting analyses with background uncertainties. The uncertainties are taken into account by considering a sideband measurement of a size that corresponds to the background uncertainty. The problem has been studied in these references:

After using the factory to make the model, we use a RooStats ProfileLikelihoodCalculator for a Hypothesis test and a confidence interval. The calculator takes into account systematics by eliminating nuisance parameters with the profile likelihood. This is equivalent to the method of MINOS.

pict1_rs_numberCountingCombination.C.png

Processing /builddir/build/BUILD/root-6.10.00/tutorials/roostats/rs_numberCountingCombination.C...
[1mRooFit v3.60 -- Developed by Wouter Verkerke and David Kirkby[0m 
                Copyright (C) 2000-2013 NIKHEF, University of California & Stanford University
                All rights reserved, please read http://roofit.sourceforge.net/license.txt
[#0] WARNING:ObjectHandling -- NumberCountingPdfFactory: changed value of tau_0 to 100.01 to be consistent with background and its uncertainty.  Also stored these values of tau into workspace with name . tau_0ExpectedNumberCountingData if you test with a different dataset, you should adjust tau appropriately.
[#0] WARNING:ObjectHandling -- NumberCountingPdfFactory: changed value of tau_1 to 100.01 to be consistent with background and its uncertainty.  Also stored these values of tau into workspace with name . tau_1ExpectedNumberCountingData if you test with a different dataset, you should adjust tau appropriately.
[#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 will be evaluated in cache-and-track mode: (sigRegion_0,sideband_0,sigRegion_1,sideband_1)
[#1] INFO:Minization -- 
  RooFitResult: minimized FCN value: 17.6316, estimated distance to minimum: 4.65724e-14
                covariance matrix quality: Full, accurate covariance matrix
                Status : MINIMIZE=0 
    Floating Parameter    FinalValue +/-  Error   
  --------------------  --------------------------
                   b_0    1.0000e+02 +/-  1.14e+00
                   b_1    1.0000e+02 +/-  1.45e+00
          masterSignal    1.0000e+00 +/-  2.46e-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: 19.9696, estimated distance to minimum: 1.4638e-07
                covariance matrix quality: Full, accurate covariance matrix
                Status : MINIMIZE=0 
    Floating Parameter    FinalValue +/-  Error   
  --------------------  --------------------------
                   b_0    1.0020e+02 +/-  9.96e-01
                   b_1    1.0010e+02 +/-  9.95e-01
-------------------------------------------------
The p-value for the null is 0.015294
Corresponding to a significance of 2.16239
-------------------------------------------------
[#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 will be evaluated in cache-and-track mode: (sigRegion_0,sideband_0,sigRegion_1,sideband_1)
[#1] INFO:Minization -- 
  RooFitResult: minimized FCN value: 17.6316, estimated distance to minimum: 4.62904e-07
                covariance matrix quality: Full, accurate covariance matrix
                Status : MINIMIZE=0 
    Floating Parameter    FinalValue +/-  Error   
  --------------------  --------------------------
                   b_0    1.0000e+02 +/-  9.99e-01
                   b_1    1.0000e+02 +/-  9.96e-01
          masterSignal    9.9967e-01 +/-  4.78e-01
[#1] INFO:Minization -- RooProfileLL::evaluate(nll_TopLevelPdf_ExpectedNumberCountingData_Profile[masterSignal]) Creating instance of MINUIT
[#1] INFO:Minization -- RooProfileLL::evaluate(nll_TopLevelPdf_ExpectedNumberCountingData_Profile[masterSignal]) determining minimum likelihood for current configurations w.r.t all observable
[#1] INFO:Minization -- RooProfileLL::evaluate(nll_TopLevelPdf_ExpectedNumberCountingData_Profile[masterSignal]) minimum found at (masterSignal=1.00002)
.
[#1] INFO:Minization -- RooProfileLL::evaluate(nll_TopLevelPdf_ExpectedNumberCountingData_Profile[masterSignal]) Creating instance of MINUIT
[#1] INFO:Minization -- RooProfileLL::evaluate(nll_TopLevelPdf_ExpectedNumberCountingData_Profile[masterSignal]) determining minimum likelihood for current configurations w.r.t all observable
[#0] ERROR:InputArguments -- RooArgSet::checkForDup: ERROR argument with name masterSignal is already in this set
[#1] INFO:Minization -- RooProfileLL::evaluate(nll_TopLevelPdf_ExpectedNumberCountingData_Profile[masterSignal]) minimum found at (masterSignal=1.00007)
..........................................................................................................................................................................................................lower limit on master signal = 0.089069
upper limit on master signal = 2.00127
-------------------------------------------------
Consider this parameter point:
RooRealVar::masterSignal = 0 +/- 0.477956  L(0 - 3) 
It is NOT in the interval.
-------------------------------------------------
-------------------------------------------------
Consider this parameter point:
RooRealVar::masterSignal = 2 +/- 0.477956  L(0 - 3) 
It IS in the interval.
-------------------------------------------------

#include "RooStats/ProfileLikelihoodCalculator.h"
#include "RooStats/NumberCountingPdfFactory.h"
#include "RooStats/ConfInterval.h"
#include "RooStats/HypoTestResult.h"
#include "RooStats/LikelihoodIntervalPlot.h"
#include "RooRealVar.h"
// use this order for safety on library loading
using namespace RooFit;
using namespace RooStats;
// declare three variations on the same tutorial
void rs_numberCountingCombination_expected();
void rs_numberCountingCombination_observed();
void rs_numberCountingCombination_observedWithTau();
// -------------------------------
// main driver to choose one
void rs_numberCountingCombination(int flag=1)
{
   if(flag==1)
      rs_numberCountingCombination_expected();
   if(flag==2)
      rs_numberCountingCombination_observed();
   if(flag==3)
      rs_numberCountingCombination_observedWithTau();
}
// -------------------------------
void rs_numberCountingCombination_expected()
{
   /////////////////////////////////////////
   // An example of a number counting combination with two channels.
   // We consider both hypothesis testing and the equivalent confidence interval.
   /////////////////////////////////////////
   /////////////////////////////////////////
   // The Model building stage
   /////////////////////////////////////////
   // Step 1, define arrays with signal & bkg expectations and background uncertainties
   Double_t s[2] = {20.,10.};           // expected signal
   Double_t b[2] = {100.,100.};         // expected background
   Double_t db[2] = {.0100,.0100};      // fractional background uncertainty
   // Step 2, use a RooStats factory to build a PDF for a
   // number counting combination and add it to the workspace.
   // We need to give the signal expectation to relate the masterSignal
   // to the signal contribution in the individual channels.
   // The model neglects correlations in background uncertainty,
   // but they could be added without much change to the example.
   NumberCountingPdfFactory f;
   RooWorkspace* wspace = new RooWorkspace();
   f.AddModel(s,2,wspace,"TopLevelPdf", "masterSignal");
   // Step 3, use a RooStats factory to add datasets to the workspace.
   // Step 3a.
   // Add the expected data to the workspace
   f.AddExpData(s, b, db, 2, wspace, "ExpectedNumberCountingData");
   // see below for a printout of the workspace
   //  wspace->Print();  //uncomment to see structure of workspace
   /////////////////////////////////////////
   // The Hypothesis testing stage:
   /////////////////////////////////////////
   // Step 4, Define the null hypothesis for the calculator
   // Here you need to know the name of the variables corresponding to hypothesis.
   RooRealVar* mu = wspace->var("masterSignal");
   RooArgSet* poi = new RooArgSet(*mu);
   RooArgSet* nullParams = new RooArgSet("nullParams");
   nullParams->addClone(*mu);
   // here we explicitly set the value of the parameters for the null
   nullParams->setRealValue("masterSignal",0);
   // Step 5, Create a calculator for doing the hypothesis test.
   // because this is a
   ProfileLikelihoodCalculator plc( *wspace->data("ExpectedNumberCountingData"),
                                    *wspace->pdf("TopLevelPdf"), *poi, 0.05, nullParams);
   // Step 6, Use the Calculator to get a HypoTestResult
   HypoTestResult* htr = plc.GetHypoTest();
   assert(htr != 0);
   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;
   /* expected case should return:
      -------------------------------------------------
      The p-value for the null is 0.015294
      Corresponding to a significance of 2.16239
      -------------------------------------------------
   */
   //////////////////////////////////////////
   // Confidence Interval Stage
   // Step 8, Here we re-use the ProfileLikelihoodCalculator to return a confidence interval.
   // We need to specify what are our parameters of interest
   RooArgSet* paramsOfInterest = nullParams; // they are the same as before in this case
   plc.SetParameters(*paramsOfInterest);
   LikelihoodInterval* lrint = (LikelihoodInterval*) plc.GetInterval();  // that was easy.
   lrint->SetConfidenceLevel(0.95);
   // Step 9, make a plot of the likelihood ratio and the interval obtained
   //paramsOfInterest->setRealValue("masterSignal",1.);
   // find limits
   double lower = lrint->LowerLimit(*mu);
   double upper = lrint->UpperLimit(*mu);
   LikelihoodIntervalPlot lrPlot(lrint);
   lrPlot.SetMaximum(3.);
   lrPlot.Draw();
   // Step 10a. Get upper and lower limits
   cout << "lower limit on master signal = " <<  lower << endl;
   cout << "upper limit on master signal = " <<  upper << endl;
   // Step 10b, Ask if masterSignal=0 is in the interval.
   // Note, this is equivalent to the question of a 2-sigma hypothesis test:
   // "is the parameter point masterSignal=0 inside the 95% confidence interval?"
   // Since the significance of the Hypothesis test was > 2-sigma it should not be:
   // eg. we exclude masterSignal=0 at 95% confidence.
   paramsOfInterest->setRealValue("masterSignal",0.);
   cout << "-------------------------------------------------" << endl;
   std::cout << "Consider this parameter point:" << std::endl;
   paramsOfInterest->first()->Print();
   if( lrint->IsInInterval(*paramsOfInterest) )
      std::cout << "It IS in the interval."  << std::endl;
   else
      std::cout << "It is NOT in the interval."  << std::endl;
   cout << "-------------------------------------------------\n\n" << endl;
   // Step 10c, We also ask about the parameter point masterSignal=2, which is inside the interval.
   paramsOfInterest->setRealValue("masterSignal",2.);
   cout << "-------------------------------------------------" << endl;
   std::cout << "Consider this parameter point:" << std::endl;
   paramsOfInterest->first()->Print();
   if( lrint->IsInInterval(*paramsOfInterest) )
      std::cout << "It IS in the interval."  << std::endl;
   else
      std::cout << "It is NOT in the interval."  << std::endl;
   cout << "-------------------------------------------------\n\n" << endl;
   delete lrint;
   delete htr;
   delete wspace;
   delete poi;
   delete nullParams;
   /*
   // Here's an example of what is in the workspace
   //  wspace->Print();
   RooWorkspace(NumberCountingWS) Number Counting WS contents
   variables
   ---------
   (x_0,masterSignal,expected_s_0,b_0,y_0,tau_0,x_1,expected_s_1,b_1,y_1,tau_1)
   p.d.f.s
   -------
   RooProdPdf::joint[ pdfs=(sigRegion_0,sideband_0,sigRegion_1,sideband_1) ] = 2.20148e-08
   RooPoisson::sigRegion_0[ x=x_0 mean=splusb_0 ] = 0.036393
   RooPoisson::sideband_0[ x=y_0 mean=bTau_0 ] = 0.00398939
   RooPoisson::sigRegion_1[ x=x_1 mean=splusb_1 ] = 0.0380088
   RooPoisson::sideband_1[ x=y_1 mean=bTau_1 ] = 0.00398939
   functions
   --------
   RooAddition::splusb_0[ set1=(s_0,b_0) set2=() ] = 120
   RooProduct::s_0[ compRSet=(masterSignal,expected_s_0) compCSet=() ] = 20
   RooProduct::bTau_0[ compRSet=(b_0,tau_0) compCSet=() ] = 10000
   RooAddition::splusb_1[ set1=(s_1,b_1) set2=() ] = 110
   RooProduct::s_1[ compRSet=(masterSignal,expected_s_1) compCSet=() ] = 10
   RooProduct::bTau_1[ compRSet=(b_1,tau_1) compCSet=() ] = 10000
   datasets
   --------
   RooDataSet::ExpectedNumberCountingData(x_0,y_0,x_1,y_1)
   embedded pre-calculated expensive components
   -------------------------------------------
   */
}
void rs_numberCountingCombination_observed()
{
   /////////////////////////////////////////
   // The same example with observed data in a main
   // measurement and an background-only auxiliary
   // measurement with a factor tau more background
   // than in the main measurement.
   /////////////////////////////////////////
   // The Model building stage
   /////////////////////////////////////////
   // Step 1, define arrays with signal & bkg expectations and background uncertainties
   // We still need the expectation to relate signal in different channels with the master signal
   Double_t s[2] = {20.,10.};           // expected signal
   // Step 2, use a RooStats factory to build a PDF for a
   // number counting combination and add it to the workspace.
   // We need to give the signal expectation to relate the masterSignal
   // to the signal contribution in the individual channels.
   // The model neglects correlations in background uncertainty,
   // but they could be added without much change to the example.
   NumberCountingPdfFactory f;
   RooWorkspace* wspace = new RooWorkspace();
   f.AddModel(s,2,wspace,"TopLevelPdf", "masterSignal");
   // Step 3, use a RooStats factory to add datasets to the workspace.
   // Add the observed data to the workspace
   Double_t mainMeas[2] = {123.,117.};      // observed main measurement
   Double_t bkgMeas[2] = {111.23,98.76};    // observed background
   Double_t dbMeas[2] = {.011,.0095};       // observed fractional background uncertainty
   f.AddData(mainMeas, bkgMeas, dbMeas, 2, wspace,"ObservedNumberCountingData");
   // see below for a printout of the workspace
   //  wspace->Print();  //uncomment to see structure of workspace
   /////////////////////////////////////////
   // The Hypothesis testing stage:
   /////////////////////////////////////////
   // Step 4, Define the null hypothesis for the calculator
   // Here you need to know the name of the variables corresponding to hypothesis.
   RooRealVar* mu = wspace->var("masterSignal");
   RooArgSet* poi = new RooArgSet(*mu);
   RooArgSet* nullParams = new RooArgSet("nullParams");
   nullParams->addClone(*mu);
   // here we explicitly set the value of the parameters for the null
   nullParams->setRealValue("masterSignal",0);
   // Step 5, Create a calculator for doing the hypothesis test.
   // because this is a
   ProfileLikelihoodCalculator plc( *wspace->data("ObservedNumberCountingData"),
                                    *wspace->pdf("TopLevelPdf"), *poi, 0.05, nullParams);
   wspace->var("tau_0")->Print();
   wspace->var("tau_1")->Print();
   // Step 7, Use the Calculator to get a HypoTestResult
   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;
   /* observed case should return:
      -------------------------------------------------
      The p-value for the null is 0.0351669
      Corresponding to a significance of 1.80975
      -------------------------------------------------
   */
   //////////////////////////////////////////
   // Confidence Interval Stage
   // Step 8, Here we re-use the ProfileLikelihoodCalculator to return a confidence interval.
   // We need to specify what are our parameters of interest
   RooArgSet* paramsOfInterest = nullParams; // they are the same as before in this case
   plc.SetParameters(*paramsOfInterest);
   LikelihoodInterval* lrint = (LikelihoodInterval*) plc.GetInterval();  // that was easy.
   lrint->SetConfidenceLevel(0.95);
   // Step 9c. Get upper and lower limits
   cout << "lower limit on master signal = " <<   lrint->LowerLimit(*mu ) << endl;
   cout << "upper limit on master signal = " <<   lrint->UpperLimit(*mu ) << endl;
   delete lrint;
   delete htr;
   delete wspace;
   delete nullParams;
   delete poi;
}
void rs_numberCountingCombination_observedWithTau()
{
   /////////////////////////////////////////
   // The same example with observed data in a main
   // measurement and an background-only auxiliary
   // measurement with a factor tau more background
   // than in the main measurement.
   /////////////////////////////////////////
   // The Model building stage
   /////////////////////////////////////////
   // Step 1, define arrays with signal & bkg expectations and background uncertainties
   // We still need the expectation to relate signal in different channels with the master signal
   Double_t s[2] = {20.,10.};           // expected signal
   // Step 2, use a RooStats factory to build a PDF for a
   // number counting combination and add it to the workspace.
   // We need to give the signal expectation to relate the masterSignal
   // to the signal contribution in the individual channels.
   // The model neglects correlations in background uncertainty,
   // but they could be added without much change to the example.
   NumberCountingPdfFactory f;
   RooWorkspace* wspace = new RooWorkspace();
   f.AddModel(s,2,wspace,"TopLevelPdf", "masterSignal");
   // Step 3, use a RooStats factory to add datasets to the workspace.
   // Add the observed data to the workspace in the on-off problem.
   Double_t mainMeas[2] = {123.,117.};      // observed main measurement
   Double_t sideband[2] = {11123.,9876.};    // observed sideband
   Double_t tau[2] = {100.,100.}; // ratio of bkg in sideband to bkg in main measurement, from experimental design.
   f.AddDataWithSideband(mainMeas, sideband, tau, 2, wspace,"ObservedNumberCountingDataWithSideband");
   // see below for a printout of the workspace
   //  wspace->Print();  //uncomment to see structure of workspace
   /////////////////////////////////////////
   // The Hypothesis testing stage:
   /////////////////////////////////////////
   // Step 4, Define the null hypothesis for the calculator
   // Here you need to know the name of the variables corresponding to hypothesis.
   RooRealVar* mu = wspace->var("masterSignal");
   RooArgSet* poi = new RooArgSet(*mu);
   RooArgSet* nullParams = new RooArgSet("nullParams");
   nullParams->addClone(*mu);
   // here we explicitly set the value of the parameters for the null
   nullParams->setRealValue("masterSignal",0);
   // Step 5, Create a calculator for doing the hypothesis test.
   // because this is a
   ProfileLikelihoodCalculator plc( *wspace->data("ObservedNumberCountingDataWithSideband"),
                                    *wspace->pdf("TopLevelPdf"), *poi, 0.05, nullParams);
   // Step 7, Use the Calculator to get a HypoTestResult
   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;
   /* observed case should return:
      -------------------------------------------------
      The p-value for the null is 0.0352035
      Corresponding to a significance of 1.80928
      -------------------------------------------------
   */
   //////////////////////////////////////////
   // Confidence Interval Stage
   // Step 8, Here we re-use the ProfileLikelihoodCalculator to return a confidence interval.
   // We need to specify what are our parameters of interest
   RooArgSet* paramsOfInterest = nullParams; // they are the same as before in this case
   plc.SetParameters(*paramsOfInterest);
   LikelihoodInterval* lrint = (LikelihoodInterval*) plc.GetInterval();  // that was easy.
   lrint->SetConfidenceLevel(0.95);
   // Step 9c. Get upper and lower limits
   cout << "lower limit on master signal = " <<   lrint->LowerLimit(*mu ) << endl;
   cout << "upper limit on master signal = " <<   lrint->UpperLimit(*mu ) << endl;
   delete lrint;
   delete htr;
   delete wspace;
   delete nullParams;
   delete poi;
}

Author: Kyle Cranmer

Definition in file rs_numberCountingCombination.C.