FourBinInstructional.C

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//  This example is a generalization of the on/off problem.  

/*
FourBin Instructional Tutorial:
 authors: 
 Kyle Cranmer <cranmer@cern.ch>
 Tanja Rommerskirchen <tanja.rommerskirchen@cern.ch>

 date: June 1, 2010

 This example is a generalization of the on/off problem.  
It's a common setup for SUSY searches.  Imagine that one has two
variables "x" and "y" (eg. missing ET and SumET), see figure.  
The signal region has high values of both of these variables (top right).
One can see low values of "x" or "y" acting as side-bands.  If we
just used "y" as a sideband, we would have the on/off problem.  
 - In the signal region we observe non events and expect s+b events.  
 - In the region with low values of "y" (bottom right) 
   we observe noff events and expect tau*b events.  
Note the significance of tau.  In the background only case:
   tau ~ <expectation off> / <expectation on>
If tau is known, this model is sufficient, but often tau is not known exactly.
So one can use low values of "x" as an additional constraint for tau.  
Note that this technique critically depends on the notion that the 
joint distribution for "x" and "y" can be factorized.  
Generally, these regions have many events, so it the ratio can be
measured very precisely there.  So we extend the model to describe the 
left two boxes... denoted with "bar".
  - In the upper left we observe nonbar events and expect bbar events
  - In the bottom left we observe noffbar events and expect tau bbar events
Note again we have:
   tau ~ <expecation off bar> / <expectation on bar>
One can further expand the model to account for the systematic associated 
to assuming the distribution of "x" and "y" factorizes (eg. that 
tau is the same for off/on and offbar/onbar). This can be done in several
ways, but here we introduce an additional parameter rho, which so that
one set of models will use tau and the other tau*rho. The choice is arbitary,
but it has consequences on the numerical stability of the algorithms.  
The "bar" measurements typically have more events (& smaller relative errors).
If we choose <expectation noffbar> = tau * rho * <expectation noonbar>, the
product tau*rho will be known very precisely (~1/sqrt(bbar)) and the contour 
in those parameters will be narrow and have a non-trivial tau~1/rho shape.
However, if we choose to put rho on the non/noff measurements (where the 
product will have an error ~1/sqrt(b)), the contours will be more ameanable 
to numerical techniques.  Thus, here we choose to define
   tau := <expecation off bar> / (<expectation on bar>)
   rho := <expecation off> / (<expectation on> * tau)

^ y
|
|---------------------------+
|               |           |
|     nonbar    |    non    |
|      bbar     |    s+b    |
|               |           |
|---------------+-----------|
|               |           |
|    noffbar    |    noff   |
|    tau bbar   | tau b rho |
|               |           |
+-----------------------------> x


Left in this way, the problem is under-constrained.  However, one may
have some auxiliary measurement (usually based on Monte Carlo) to
constrain rho.  Let us call this auxiliary measurement that gives
the nominal value of rho "rhonom".  Thus, there is a 'constraint' term in 
the full model: P(rhonom | rho).  In this case, we consider a Gaussian
constraint with standard deviation sigma.

In the example, the initial values of the parameters are
  - s    = 40
  - b    = 100
  - tau  = 5
  - bbar = 1000
  - rho  = 1
  (sigma for rho = 20%)
and in the toy dataset:
   - non = 139 
   - noff = 528
   - nonbar = 993
   - noffbar = 4906
   - rhonom = 1.27824

Note, the covariance matrix of the parameters has large off-diagonal terms.  
Clearly s,b are anti-correlated.  Similary, since noffbar >> nonbar, one would
expect bbar,tau to be anti-correlated.  

This can be seen below.
            GLOBAL      b    bbar   rho      s     tau
        b  0.96820   1.000  0.191 -0.942 -0.762 -0.209
     bbar  0.91191   0.191  1.000  0.000 -0.146 -0.912
      rho  0.96348  -0.942  0.000  1.000  0.718 -0.000
        s  0.76250  -0.762 -0.146  0.718  1.000  0.160
      tau  0.92084  -0.209 -0.912 -0.000  0.160  1.000

Similarly, since tau*rho appears as a product, we expect rho,tau 
to be anti-correlated. When the error on rho is significantly 
larger than 1/sqrt(bbar), tau is essentially known and the 
correlation is minimal (tau mainly cares about bbar, and rho about b,s).  
In the alternate parametrizaton (bbar* tau * rho) the correlation coefficient 
for rho,tau is large (and negative).


The code below uses best-practices for RooFit & RooStats as of 
June 2010.  It proceeds as follows:
 - create a workspace to hold the model
 - use workspace factory to quickly create the terms of the model
 - use workspace factory to define total model (a prod pdf)
 - create a RooStats ModelConfig to specify observables, parameters of interest
 - add to the ModelConfig a prior on the parameters for Bayesian techniques
   - note, the pdf it is factorized for parameters of interest & nuisance params
 - visualize the model
 - write the workspace to a file
 - use several of RooStats IntervalCalculators & compare results
*/

#include "TStopwatch.h"
#include "TCanvas.h"
#include "TROOT.h"
#include "RooPlot.h"
#include "RooAbsPdf.h"
#include "RooWorkspace.h"
#include "RooDataSet.h"
#include "RooGlobalFunc.h"
#include "RooFitResult.h"
#include "RooRandom.h"
#include "RooStats/ProfileLikelihoodCalculator.h"
#include "RooStats/LikelihoodInterval.h"
#include "RooStats/LikelihoodIntervalPlot.h"
#include "RooStats/BayesianCalculator.h"
#include "RooStats/MCMCCalculator.h"
#include "RooStats/MCMCInterval.h"
#include "RooStats/MCMCIntervalPlot.h"
#include "RooStats/ProposalHelper.h"
#include "RooStats/SimpleInterval.h"
#include "RooStats/FeldmanCousins.h"
#include "RooStats/PointSetInterval.h"

using namespace RooFit;
using namespace RooStats;

void FourBinInstructional(bool doBayesian=false, bool doFeldmanCousins=false, bool doMCMC=false){
  
  // let's time this challenging example
  TStopwatch t;
  t.Start();

  // set RooFit random seed for reproducible results
  RooRandom::randomGenerator()->SetSeed(4357);

  // make model
  RooWorkspace* wspace = new RooWorkspace("wspace");
  wspace->factory("Poisson::on(non[0,1000], sum::splusb(s[40,0,100],b[100,0,300]))");
  wspace->factory("Poisson::off(noff[0,5000], prod::taub(b,tau[5,3,7],rho[1,0,2]))");
  wspace->factory("Poisson::onbar(nonbar[0,10000], bbar[1000,500,2000])");
  wspace->factory("Poisson::offbar(noffbar[0,1000000], prod::lambdaoffbar(bbar, tau))");
  wspace->factory("Gaussian::mcCons(rhonom[1.,0,2], rho, sigma[.2])");
  wspace->factory("PROD::model(on,off,onbar,offbar,mcCons)");
  wspace->defineSet("obs","non,noff,nonbar,noffbar,rhonom");

  wspace->factory("Uniform::prior_poi({s})");
  wspace->factory("Uniform::prior_nuis({b,bbar,tau, rho})");
  wspace->factory("PROD::prior(prior_poi,prior_nuis)"); 

  ///////////////////////////////////////////
  // Control some interesting variations
  // define parameers of interest
  // for 1-d plots
  wspace->defineSet("poi","s");
  wspace->defineSet("nuis","b,tau,rho,bbar");
  // for 2-d plots to inspect correlations:
  //  wspace->defineSet("poi","s,rho");

  // test simpler cases where parameters are known.
  //  wspace->var("tau")->setConstant();
  //  wspace->var("rho")->setConstant();
  //  wspace->var("b")->setConstant();
  //  wspace->var("bbar")->setConstant();

  // inspect workspace
  //  wspace->Print();

  ////////////////////////////////////////////////////////////
  // Generate toy data
  // generate toy data assuming current value of the parameters
  // import into workspace. 
  // add Verbose() to see how it's being generated
  RooDataSet* data =   wspace->pdf("model")->generate(*wspace->set("obs"),1);
  //  data->Print("v");
  wspace->import(*data);

  /////////////////////////////////////////////////////
  // Now the statistical tests
  // model config
  ModelConfig* modelConfig = new ModelConfig("FourBins");
  modelConfig->SetWorkspace(*wspace);
  modelConfig->SetPdf(*wspace->pdf("model"));
  modelConfig->SetPriorPdf(*wspace->pdf("prior"));
  modelConfig->SetParametersOfInterest(*wspace->set("poi"));
  modelConfig->SetNuisanceParameters(*wspace->set("nuis"));
  wspace->import(*modelConfig);
  wspace->writeToFile("FourBin.root");

  //////////////////////////////////////////////////
  // If you want to see the covariance matrix uncomment
  //  wspace->pdf("model")->fitTo(*data);

  // use ProfileLikelihood
  ProfileLikelihoodCalculator plc(*data, *modelConfig);
  plc.SetConfidenceLevel(0.95);
  LikelihoodInterval* plInt = plc.GetInterval();
  RooFit::MsgLevel msglevel = RooMsgService::instance().globalKillBelow();
  RooMsgService::instance().setGlobalKillBelow(RooFit::FATAL);
  plInt->LowerLimit( *wspace->var("s") ); // get ugly print out of the way. Fix.
  RooMsgService::instance().setGlobalKillBelow(msglevel);

  // use FeldmaCousins (takes ~20 min)  
  FeldmanCousins fc(*data, *modelConfig);
  fc.SetConfidenceLevel(0.95);
  //number counting: dataset always has 1 entry with N events observed
  fc.FluctuateNumDataEntries(false); 
  fc.UseAdaptiveSampling(true);
  fc.SetNBins(40);
  PointSetInterval* fcInt = NULL;
  if(doFeldmanCousins){ // takes 7 minutes
    fcInt = (PointSetInterval*) fc.GetInterval(); // fix cast
  }


  // use BayesianCalculator (only 1-d parameter of interest, slow for this problem)  
  BayesianCalculator bc(*data, *modelConfig);
  bc.SetConfidenceLevel(0.95);
  SimpleInterval* bInt = NULL;
  if(doBayesian && wspace->set("poi")->getSize() == 1)   {
    bInt = bc.GetInterval();
  } else{
    cout << "Bayesian Calc. only supports on parameter of interest" << endl;
  }


  // use MCMCCalculator  (takes about 1 min)
  // Want an efficient proposal function, so derive it from covariance
  // matrix of fit
  RooFitResult* fit = wspace->pdf("model")->fitTo(*data,Save());
  ProposalHelper ph;
  ph.SetVariables((RooArgSet&)fit->floatParsFinal());
  ph.SetCovMatrix(fit->covarianceMatrix());
  ph.SetUpdateProposalParameters(kTRUE); // auto-create mean vars and add mappings
  ph.SetCacheSize(100);
  ProposalFunction* pf = ph.GetProposalFunction();

  MCMCCalculator mc(*data, *modelConfig);
  mc.SetConfidenceLevel(0.95);
  mc.SetProposalFunction(*pf);
  mc.SetNumBurnInSteps(500); // first N steps to be ignored as burn-in
  mc.SetNumIters(50000);
  mc.SetLeftSideTailFraction(0.5); // make a central interval
  MCMCInterval* mcInt = NULL;
  if(doMCMC)
    mcInt = mc.GetInterval();

  //////////////////////////////////////
  // Make some  plots
  TCanvas* c1 = (TCanvas*) gROOT->Get("c1");  
  if(!c1)
    c1 = new TCanvas("c1");

  if(doBayesian && doMCMC){
    c1->Divide(3);
    c1->cd(1);
  }
  else if (doBayesian || doMCMC){
    c1->Divide(2);
    c1->cd(1);
  }

  LikelihoodIntervalPlot* lrplot = new LikelihoodIntervalPlot(plInt);
  lrplot->Draw();

  if(doBayesian && wspace->set("poi")->getSize() == 1)   {
    c1->cd(2);
    // the plot takes a long time and print lots of error
    // using a scan it is better
    bc.SetScanOfPosterior(20);
    RooPlot* bplot = bc.GetPosteriorPlot();
    bplot->Draw();
  } 

  if(doMCMC){
    if(doBayesian && wspace->set("poi")->getSize() == 1) 
      c1->cd(3);
    else 
      c1->cd(2);
    MCMCIntervalPlot mcPlot(*mcInt); 
    mcPlot.Draw();
  }

  ////////////////////////////////////
  // querry intervals
  cout << "Profile Likelihood interval on s = [" << 
    plInt->LowerLimit( *wspace->var("s") ) << ", " <<
    plInt->UpperLimit( *wspace->var("s") ) << "]" << endl; 
  //Profile Likelihood interval on s = [12.1902, 88.6871]

   
  if(doBayesian && wspace->set("poi")->getSize() == 1)   {
    cout << "Bayesian interval on s = [" << 
      bInt->LowerLimit( ) << ", " <<
      bInt->UpperLimit( ) << "]" << endl;
  }  
  
  if(doFeldmanCousins){    
    cout << "Feldman Cousins interval on s = [" << 
      fcInt->LowerLimit( *wspace->var("s") ) << ", " <<
      fcInt->UpperLimit( *wspace->var("s") ) << "]" << endl;
    //Feldman Cousins interval on s = [18.75 +/- 2.45, 83.75 +/- 2.45]
  }

  if(doMCMC){
    cout << "MCMC interval on s = [" << 
      mcInt->LowerLimit(*wspace->var("s") ) << ", " <<
      mcInt->UpperLimit(*wspace->var("s") ) << "]" << endl;
    //MCMC interval on s = [15.7628, 84.7266]

  }

  t.Print();
   

}

---