rs401d_FeldmanCousins.C: 'Neutrino Oscillation Example from Feldman & Cousins'

/////////////////////////////////////////////////////////////////////////
//
// 'Neutrino Oscillation Example from Feldman & Cousins'
// author: Kyle Cranmer
// date March 2009
//
// This tutorial shows a more complex example using the FeldmanCousins utility
// to create a confidence interval for a toy neutrino oscillation experiment.
// The example attempts to faithfully reproduce the toy example described in Feldman & Cousins' 
// original paper, Phys.Rev.D57:3873-3889,1998. 
//
// to run it:
// .x tutorials/roostats/rs401d_FeldmanCousins.C+
/////////////////////////////////////////////////////////////////////////

#ifndef __CINT__
#include "RooGlobalFunc.h"
#endif
#include "RooStats/ConfInterval.h"
#include "RooStats/FeldmanCousins.h"
#include "RooStats/ProfileLikelihoodCalculator.h"
#include "RooStats/MCMCCalculator.h"
#include "RooStats/UniformProposal.h"
#include "RooStats/LikelihoodIntervalPlot.h"

#include "RooDataSet.h"
#include "RooDataHist.h"
#include "RooRealVar.h"
#include "RooConstVar.h"
#include "RooAddition.h"
#include "RooProduct.h"
#include "RooProdPdf.h"
#include "RooAddPdf.h"

#include "TROOT.h"
#include "RooPolynomial.h"
#include "RooRandom.h"

#include "RooNLLVar.h"
#include "RooProfileLL.h"

#include "RooPlot.h"

#include "TCanvas.h"
#include "TH1F.h"
#include "TH2F.h"
#include "TTree.h"
#include "TMarker.h"
#include "TStopwatch.h"

#include <iostream>

// PDF class created for this macro
#include "NuMuToNuE_Oscillation.h"
#include "NuMuToNuE_Oscillation.cxx" // so that it can be executed directly

// use this order for safety on library loading
using namespace RooFit ;
using namespace RooStats ;


void rs401d_FeldmanCousins(bool doFeldmanCousins=false, bool doMCMC = true)
{

  // to time the macro
  TStopwatch t;
  t.Start();


  /*
    Taken from Feldman & Cousins paper, Phys.Rev.D57:3873-3889,1998. 
    e-Print: physics/9711021 (see page 13.)

    Quantum mechanics dictates that the probability of such a transformation is given by the formula 
    P (νµ → ν e ) = sin^2 (2θ) sin^2 (1.27 ∆m^2 L /E )
    where P is the probability for a νµ to transform into a νe , L is the distance in km between 
    the creation of the neutrino from meson decay and its interaction in the detector, E is the 
    neutrino energy in GeV, and ∆m^2 = |m^2− m^2 | in (eV/c^2 )^2 . 

    To demonstrate how this works in practice, and how it compares to alternative approaches 
    that have been used, we consider a toy model of a typical neutrino oscillation experiment. 
    The toy model is defined by the following parameters: Mesons are assumed to decay to 
    neutrinos uniformly in a region 600 m to 1000 m from the detector. The expected background 
    from conventional νe interactions and misidentified νµ interactions is assumed to be 100 
    events in each of 5 energy bins which span the region from 10 to 60 GeV. We assume that 
    the νµ flux is such that if P (νµ → ν e ) = 0.01 averaged over any bin, then that bin would 
    have an expected additional contribution of 100 events due to νµ → ν e oscillations. 
   */

  // Make signal model model
  RooRealVar E("E","", 15,10,60,"GeV");
  RooRealVar L("L","", .800,.600, 1.0,"km"); // need these units in formula
  RooRealVar deltaMSq("deltaMSq","#Delta m^{2}",40,1,300,"eV/c^{2}");
  RooRealVar sinSq2theta("sinSq2theta","sin^{2}(2#theta)", .006,.0,.02);
  //RooRealVar deltaMSq("deltaMSq","#Delta m^{2}",40,20,70,"eV/c^{2}");
  //  RooRealVar sinSq2theta("sinSq2theta","sin^{2}(2#theta)", .006,.001,.01);
  // PDF for oscillation only describes deltaMSq dependence, sinSq2theta goes into sigNorm
  // 1) The code for this PDF was created by issuing these commands
  //    root [0] RooClassFactory x
  //    root [1] x.makePdf("NuMuToNuE_Oscillation","L,E,deltaMSq","","pow(sin(1.27*deltaMSq*L/E),2)")
  NuMuToNuE_Oscillation PnmuTone("PnmuTone","P(#nu_{#mu} #rightarrow #nu_{e}",L,E,deltaMSq);

  // only E is observable, so create the signal model by integrating out L 
  RooAbsPdf* sigModel = PnmuTone.createProjection(L);

  // create   \int dE' dL' P(E',L' | \Delta m^2). 
  // Given RooFit will renormalize the PDF in the range of the observables, 
  // the average probability to oscillate in the experiment's acceptance
  // needs to be incorporated into the extended term in the likelihood.
  // Do this by creating a RooAbsReal representing the integral and divide by 
  // the area in the E-L plane.
  // The integral should be over "primed" observables, so we need
  // an independent copy of PnmuTone not to interfere with the original.

  // Independent copy for Integral
  RooRealVar EPrime("EPrime","", 15,10,60,"GeV");
  RooRealVar LPrime("LPrime","", .800,.600, 1.0,"km"); // need these units in formula
  NuMuToNuE_Oscillation PnmuTonePrime("PnmuTonePrime","P(#nu_{#mu} #rightarrow #nu_{e}",
				      LPrime,EPrime,deltaMSq);
  RooAbsReal* intProbToOscInExp = PnmuTonePrime.createIntegral(RooArgSet(EPrime,LPrime));

  // Getting the flux is a bit tricky.  It is more celear to include a cross section term that is not
  // explicitly refered to in the text, eg.
  // # events in bin = flux * cross-section for nu_e interaction in E bin * average prob nu_mu osc. to nu_e in bin
  // let maxEventsInBin = flux * cross-section for nu_e interaction in E bin 
  // maxEventsInBin * 1% chance per bin =  100 events / bin
  // therefore maxEventsInBin = 10,000.
  // for 5 bins, this means maxEventsTot = 50,000   
  RooConstVar maxEventsTot("maxEventsTot","maximum number of sinal events",50000);
  RooConstVar inverseArea("inverseArea","1/(#Delta E #Delta L)",
			   1./(EPrime.getMax()-EPrime.getMin())/(LPrime.getMax()-LPrime.getMin()));

  // sigNorm = maxEventsTot * (\int dE dL prob to oscillate in experiment / Area) * sin^2(2\theta)
  RooProduct sigNorm("sigNorm", "", RooArgSet(maxEventsTot, *intProbToOscInExp, inverseArea, sinSq2theta));
  // bkg = 5 bins * 100 events / bin
  RooConstVar bkgNorm("bkgNorm","normalization for background",500);

  // flat background (0th order polynomial, so no arguments for coefficients)
  RooPolynomial bkgEShape("bkgEShape","flat bkg shape", E);

  // total model
  RooAddPdf model("model","",RooArgList(*sigModel,bkgEShape),
		  RooArgList(sigNorm,bkgNorm));

  // for debugging, check model tree
  //  model.printCompactTree();
  //  model.graphVizTree("model.dot");


  // turn off some messages
  RooMsgService::instance().setStreamStatus(0,kFALSE);
  RooMsgService::instance().setStreamStatus(1,kFALSE);
  RooMsgService::instance().setStreamStatus(2,kFALSE);


  //////////////////////////////////////////////
  // n events in data to data, simply sum of sig+bkg
  Int_t nEventsData = bkgNorm.getVal()+sigNorm.getVal(); 
  cout << "generate toy data with nEvents = " << nEventsData << endl;
  // adjust random seed to get a toy dataset similar to one in paper. 
  // Found by trial and error (3 trials, so not very "fine tuned")
  RooRandom::randomGenerator()->SetSeed(3); 
  // create a toy dataset
  RooDataSet* data = model.generate(RooArgSet(E), nEventsData);
  

  /////////////////////////////////////////////
  // make some plots
  TCanvas* dataCanvas = new TCanvas("dataCanvas");
  dataCanvas->Divide(2,2);

  // plot the PDF
  dataCanvas->cd(1);
  TH1* hh = PnmuTone.createHistogram("hh",E,Binning(40),YVar(L,Binning(40)),Scaling(kFALSE)) ;
  hh->SetLineColor(kBlue) ;
  hh->SetTitle("True Signal Model");
  hh->Draw("surf");

  // plot the data with the best fit
  dataCanvas->cd(2);
  RooPlot* Eframe = E.frame();
  data->plotOn(Eframe);
  model.fitTo(*data, Extended());
  model.plotOn(Eframe);
  model.plotOn(Eframe,Components(*sigModel),LineColor(kRed));
  model.plotOn(Eframe,Components(bkgEShape),LineColor(kGreen));
  model.plotOn(Eframe);
  Eframe->SetTitle("toy data with best fit model (and sig+bkg components)");
  Eframe->Draw();

  // plot the likelihood function
  dataCanvas->cd(3);
  RooNLLVar nll("nll", "nll", model, *data, Extended());
  RooProfileLL pll("pll", "", nll, RooArgSet(deltaMSq, sinSq2theta));
  //  TH1* hhh = nll.createHistogram("hhh",sinSq2theta,Binning(40),YVar(deltaMSq,Binning(40))) ;
  TH1* hhh = pll.createHistogram("hhh",sinSq2theta,Binning(40),YVar(deltaMSq,Binning(40)),Scaling(kFALSE)) ;
  hhh->SetLineColor(kBlue) ;
  hhh->SetTitle("Likelihood Function");
  hhh->Draw("surf");

  dataCanvas->Update();



  //////////////////////////////////////////////////////////
  //////// show use of Feldman-Cousins utility in RooStats
  RooStats::FeldmanCousins fc;
  // set the distribution creator, which encodes the test statistic
  RooArgSet parameters(deltaMSq, sinSq2theta);
  fc.SetPdf(model);
  fc.SetParameters(parameters);
  fc.SetTestSize(.1); // set size of test
  fc.SetData(*data);
  fc.UseAdaptiveSampling(true);
  fc.SetNBins(30); // number of points to test per parameter

  // use the Feldman-Cousins tool
  ConfInterval* interval = 0;
  if(doFeldmanCousins)
    interval = fc.GetInterval();


  ///////////////////////////////////////////////////////////////////
  ///////// show use of ProfileLikeihoodCalculator utility in RooStats
  RooStats::ProfileLikelihoodCalculator plc;
  plc.SetPdf(model);
  plc.SetParameters(parameters);
  plc.SetTestSize(.1);
  plc.SetData(*data);
  
  ConfInterval* plcInterval = plc.GetInterval();

  ///////////////////////////////////////////////////////////////////
  ///////// show use of MCMCCalculator utility in RooStats
  ConfInterval* mcmcInterval = NULL;

  if (doMCMC) {
      TStopwatch mcmcWatch;
      mcmcWatch.Start();
      cout << "========================= MCMC =========================" << endl;
      UniformProposal up;

      RooArgList axisList(sinSq2theta, deltaMSq);
      MCMCCalculator mc(*data, model, parameters, up, 10000);
      mc.SetTestSize(.1);
      mc.SetAxes(axisList); // set which is x and y axis in posterior histogram
      mcmcInterval = mc.GetInterval();

      mcmcWatch.Stop();
      mcmcWatch.Print();
      cout << "======================= END MCMC =======================" << endl;
  }
  ////////////////////////////////////////////
  // make plot of resulting interval

  dataCanvas->cd(4);
  LikelihoodIntervalPlot plotInt((LikelihoodInterval*)plcInterval);
  plotInt.SetTitle("90% Confidence Intervals");
  plotInt.Draw();
  dataCanvas->Update();

  // first plot a small dot for every point tested
  RooDataHist* parameterScan = (RooDataHist*) fc.GetPointsToScan();
  TH2F* hist = (TH2F*) parameterScan->createHistogram("sinSq2theta:deltaMSq",30,30);
  //  hist->Draw();
  TH2F* forContour = (TH2F*)hist->Clone();

  // now loop through the points and put a marker if it's in the interval
  RooArgSet* tmpPoint;
  // loop over points to test
  for(Int_t i=0; i<parameterScan->numEntries(); ++i){
     // get a parameter point from the list of points to test.
    tmpPoint = (RooArgSet*) parameterScan->get(i)->clone("temp");

    if (interval){
      if (interval->IsInInterval( *tmpPoint ) ) {
	forContour->SetBinContent( hist->FindBin(tmpPoint->getRealValue("sinSq2theta"), 
						  tmpPoint->getRealValue("deltaMSq")),	 1);
      }else{
	forContour->SetBinContent( hist->FindBin(tmpPoint->getRealValue("sinSq2theta"), 
						  tmpPoint->getRealValue("deltaMSq")),	 0);
      }
    }


    delete tmpPoint;
  }

  if (interval){
    Double_t level=0.5;
    forContour->SetContour(1,&level);
    forContour->SetLineWidth(2);
    forContour->SetLineColor(kRed);
    forContour->Draw("cont2,same");
  }


  if (mcmcInterval){
    // most of this code is just to switch x/y axes in the histogram
    TH1* posterior = ((MCMCInterval*)mcmcInterval)->GetPosteriorHist();

    Double_t mcmclevel = ((MCMCInterval*)mcmcInterval)->GetCutoff();
    posterior->SetContour(1,&mcmclevel);
    posterior->SetLineColor(kMagenta);
    posterior->SetLineWidth(2);
    posterior->Draw("cont2,same");
  
  }
  dataCanvas->Update();
  
  /// print timing info
  t.Stop();
  t.Print();
    

}