testUnfold1.C

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/// \file
/// \ingroup tutorial_unfold
/// \notebook
///  Test program for the classes TUnfold and related.
///
///  1. Generate Monte Carlo and Data events
///      The events consist of
///        signal
///        background
///
///      The signal is a resonance. It is generated with a Breit-Wigner,
///      smeared by a Gaussian
///
///  2. Unfold the data. The result is:
///      The background level
///      The shape of the resonance, corrected for detector effects
///
///      Systematic errors from the MC shape variation are included
///      and propagated to the result
///
///  3. fit the unfolded distribution, including the correlation matrix
///
///  4. save six plots to a file `testUnfold1.ps`
/// ~~~
///        1  2  3
///        4  5  6
/// ~~~
///      1. 2d-plot of the matrix describing the migrations
///      2. generator-level distributions
///            - blue: unfolded data, total errors
///            - green: unfolded data, statistical errors
///            - red: generated data
///            - black: fit to green data points
///      3. detector level distributions
///            - blue: unfolded data, folded back through the matrix
///            - black: Monte Carlo (with wrong peal position)
///            - blue: data
///      4. global correlation coefficients
///      5. \f$ \chi^2 \f$ as a function of \f$ log(\tau) \f$
///           the star indicates the final choice of \f$ \tau \f$
///      6. the L curve
///
/// \macro_output
/// \macro_image
/// \macro_code
///
///  **Version 17.6, in parallel to changes in TUnfold**
///
/// #### History:
///  - Version 17.5, in parallel to changes in TUnfold
///  - Version 17.4, in parallel to changes in TUnfold
///  - Version 17.3, in parallel to changes in TUnfold
///  - Version 17.2, in parallel to changes in TUnfold
///  - Version 17.1, in parallel to changes in TUnfold
///  - Version 17.0, updated for using the classes TUnfoldDensity, TUnfoldBinning
///  - Version 16.1, parallel to changes in TUnfold
///  - Version 16.0, parallel to changes in TUnfold
///  - Version 15, with automated L-curve scan
///  - Version 14, with changes in TUnfoldSys.cxx
///  - Version 13, include test of systematic errors
///  - Version 12, catch error when defining the input
///  - Version 11,  print chi**2 and number of degrees of freedom
///  - Version 10,  with bug-fix in TUnfold.cxx
///  - Version 9,  with bug-fix in TUnfold.cxx and TUnfold.h
///  - Version 8,  with bug-fix in TUnfold.cxx and TUnfold.h
///  - Version 7,  with bug-fix in TUnfold.cxx and TUnfold.h
///  - Version 6a, fix problem with dynamic array allocation under windows
///  - Version 6, bug-fixes in TUnfold.C
///  - Version 5, replace main() by testUnfold1()
///  - Version 4, with bug-fix in TUnfold.C
///  - Version 3, with bug-fix in TUnfold.C
///  - Version 2, with changed ScanLcurve() arguments
///  - Version 1, remove L curve analysis, use ScanLcurve() method instead
///  - Version 0, L curve analysis included here
///
///  This file is part of TUnfold.
///
///  TUnfold is free software: you can redistribute it and/or modify
///  it under the terms of the GNU General Public License as published by
///  the Free Software Foundation, either version 3 of the License, or
///  (at your option) any later version.
///
///  TUnfold is distributed in the hope that it will be useful,
///  but WITHOUT ANY WARRANTY; without even the implied warranty of
///  MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
///  GNU General Public License for more details.
///
///  You should have received a copy of the GNU General Public License
///  along with TUnfold.  If not, see <http://www.gnu.org/licenses/>.
///
/// \author Stefan Schmitt DESY, 14.10.2008
 
 
#include <TError.h>
#include <TMath.h>
#include <TCanvas.h>
#include <TRandom3.h>
#include <TFitter.h>
#include <TF1.h>
#include <TStyle.h>
#include <TVector.h>
#include <TGraph.h>
 
#include "TUnfoldDensity.h"
 
// #define VERBOSE_LCURVE_SCAN
 
using namespace std;
 
TRandom *rnd=0;
 
TH2 *gHistInvEMatrix;
 
TVirtualFitter *gFitter=0;
 
void chisquare_corr(Int_t &npar, Double_t * /*gin */, Double_t &f, Double_t *u, Int_t /*flag */) {
  //  Minimization function for H1s using a Chisquare method
  //  only one-dimensional histograms are supported
  //  Correlated errors are taken from an external inverse covariance matrix
  //  stored in a 2-dimensional histogram
  Double_t x;
  TH1 *hfit = (TH1*)gFitter->GetObjectFit();
  TF1 *f1   = (TF1*)gFitter->GetUserFunc();
 
  f1->InitArgs(&x,u);
  npar = f1->GetNpar();
  f = 0;
 
  Int_t npfit = 0;
  Int_t nPoints=hfit->GetNbinsX();
  Double_t *df=new Double_t[nPoints];
  for (Int_t i=0;i<nPoints;i++) {
    x     = hfit->GetBinCenter(i+1);
    TF1::RejectPoint(kFALSE);
    df[i] = f1->EvalPar(&x,u)-hfit->GetBinContent(i+1);
    if (TF1::RejectedPoint()) df[i]=0.0;
    else npfit++;
  }
  for (Int_t i=0;i<nPoints;i++) {
    for (Int_t j=0;j<nPoints;j++) {
      f += df[i]*df[j]*gHistInvEMatrix->GetBinContent(i+1,j+1);
    }
  }
  delete[] df;
  f1->SetNumberFitPoints(npfit);
}
 
Double_t bw_func(Double_t *x,Double_t *par) {
  Double_t dm=x[0]-par[1];
  return par[0]/(dm*dm+par[2]*par[2]);
}
 
 
// generate an event
// output:
//  negative mass: background event
//  positive mass: signal event
Double_t GenerateEvent(Double_t bgr, // relative fraction of background
                       Double_t mass, // peak position
                       Double_t gamma) // peak width
{
  Double_t t;
  if(rnd->Rndm()>bgr) {
    // generate signal event
    // with positive mass
    do {
      do {
        t=rnd->Rndm();
      } while(t>=1.0);
      t=TMath::Tan((t-0.5)*TMath::Pi())*gamma+mass;
    } while(t<=0.0);
    return t;
  } else {
    // generate background event
    // generate events following a power-law distribution
    //   f(E) = K * TMath::power((E0+E),N0)
    static Double_t const E0=2.4;
    static Double_t const N0=2.9;
    do {
      do {
        t=rnd->Rndm();
      } while(t>=1.0);
      // the mass is returned negative
      // In our example a convenient way to indicate it is a background event.
      t= -(TMath::Power(1.-t,1./(1.-N0))-1.0)*E0;
    } while(t>=0.0);
    return t;
  }
}
 
// smear the event to detector level
// input:
//   mass on generator level (mTrue>0 !)
// output:
//   mass on detector level
Double_t DetectorEvent(Double_t mTrue) {
  // smear by double-gaussian
  static Double_t frac=0.1;
  static Double_t wideBias=0.03;
  static Double_t wideSigma=0.5;
  static Double_t smallBias=0.0;
  static Double_t smallSigma=0.1;
  if(rnd->Rndm()>frac) {
    return rnd->Gaus(mTrue+smallBias,smallSigma);
  } else {
    return rnd->Gaus(mTrue+wideBias,wideSigma);
  }
}
 
int testUnfold1()
{
  // switch on histogram errors
  TH1::SetDefaultSumw2();
 
  // show fit result
  gStyle->SetOptFit(1111);
 
  // random generator
  rnd=new TRandom3();
 
  // data and MC luminosity, cross-section
  Double_t const luminosityData=100000;
  Double_t const luminosityMC=1000000;
  Double_t const crossSection=1.0;
 
  Int_t const nDet=250;
  Int_t const nGen=100;
  Double_t const xminDet=0.0;
  Double_t const xmaxDet=10.0;
  Double_t const xminGen=0.0;
  Double_t const xmaxGen=10.0;
 
  //============================================
  // generate MC distribution
  //
  TH1D *histMgenMC=new TH1D("MgenMC",";mass(gen)",nGen,xminGen,xmaxGen);
  TH1D *histMdetMC=new TH1D("MdetMC",";mass(det)",nDet,xminDet,xmaxDet);
  TH2D *histMdetGenMC=new TH2D("MdetgenMC",";mass(det);mass(gen)",
                               nDet,xminDet,xmaxDet,nGen,xminGen,xmaxGen);
  Int_t neventMC=rnd->Poisson(luminosityMC*crossSection);
  for(Int_t i=0;i<neventMC;i++) {
    Double_t mGen=GenerateEvent(0.3, // relative fraction of background
                                4.0, // peak position in MC
                                0.2); // peak width in MC
    Double_t mDet=DetectorEvent(TMath::Abs(mGen));
    // the generated mass is negative for background
    // and positive for signal
    // so it will be filled in the underflow bin
    // this is very convenient for the unfolding:
    // the unfolded result will contain the number of background
    // events in the underflow bin
 
    // generated MC distribution (for comparison only)
    histMgenMC->Fill(mGen,luminosityData/luminosityMC);
    // reconstructed MC distribution (for comparison only)
    histMdetMC->Fill(mDet,luminosityData/luminosityMC);
 
    // matrix describing how the generator input migrates to the
    // reconstructed level. Unfolding input.
    // NOTE on underflow/overflow bins:
    //  (1) the detector level under/overflow bins are used for
    //       normalisation ("efficiency" correction)
    //       in our toy example, these bins are populated from tails
    //       of the initial MC distribution.
    //  (2) the generator level underflow/overflow bins are
    //       unfolded. In this example:
    //       underflow bin: background events reconstructed in the detector
    //       overflow bin: signal events generated at masses > xmaxDet
    // for the unfolded result these bins will be filled
    //  -> the background normalisation will be contained in the underflow bin
    histMdetGenMC->Fill(mDet,mGen,luminosityData/luminosityMC);
  }
 
  //============================================
  // generate alternative MC
  // this will be used to derive a systematic error due to MC
  // parameter uncertainties
  TH2D *histMdetGenSysMC=new TH2D("MdetgenSysMC",";mass(det);mass(gen)",
                                  nDet,xminDet,xmaxDet,nGen,xminGen,xmaxGen);
  neventMC=rnd->Poisson(luminosityMC*crossSection);
  for(Int_t i=0;i<neventMC;i++) {
    Double_t mGen=GenerateEvent
       (0.5, // relative fraction of background
        3.6, // peak position in MC with systematic shift
        0.15); // peak width in MC
    Double_t mDet=DetectorEvent(TMath::Abs(mGen));
    histMdetGenSysMC->Fill(mDet,mGen,luminosityData/luminosityMC);
  }
 
  //============================================
  // generate data distribution
  //
  TH1D *histMgenData=new TH1D("MgenData",";mass(gen)",nGen,xminGen,xmaxGen);
  TH1D *histMdetData=new TH1D("MdetData",";mass(det)",nDet,xminDet,xmaxDet);
  Int_t neventData=rnd->Poisson(luminosityData*crossSection);
  for(Int_t i=0;i<neventData;i++) {
    Double_t mGen=GenerateEvent(0.4, // relative fraction of background
                                3.8, // peak position in data
                                0.15); // peak width in data
    Double_t mDet=DetectorEvent(TMath::Abs(mGen));
    // generated data mass for comparison plots
    // for real data, we do not have this histogram
    histMgenData->Fill(mGen);
 
    // reconstructed mass, unfolding input
    histMdetData->Fill(mDet);
  }
 
  //=========================================================================
  // divide by bin width to get density distributions
  TH1D *histDensityGenData=new TH1D("DensityGenData",";mass(gen)",
                                    nGen,xminGen,xmaxGen);
  TH1D *histDensityGenMC=new TH1D("DensityGenMC",";mass(gen)",
                                    nGen,xminGen,xmaxGen);
  for(Int_t i=1;i<=nGen;i++) {
     histDensityGenData->SetBinContent(i,histMgenData->GetBinContent(i)/
                                       histMgenData->GetBinWidth(i));
     histDensityGenMC->SetBinContent(i,histMgenMC->GetBinContent(i)/
                                       histMgenMC->GetBinWidth(i));
  }
 
  //=========================================================================
  // set up the unfolding
  // define migration matrix
  TUnfoldDensity unfold(histMdetGenMC,TUnfold::kHistMapOutputVert);
 
  // define input and bias scheme
  // do not use the bias, because MC peak may be at the wrong place
  // watch out for error codes returned by the SetInput method
  // errors larger or equal 10000 are fatal:
  // the data points specified as input are not sufficient to constrain the
  // unfolding process
  if(unfold.SetInput(histMdetData)>=10000) {
    std::cout<<"Unfolding result may be wrong\n";
  }
 
  //========================================================================
  // the unfolding is done here
  //
  // scan L curve and find best point
  Int_t nScan=30;
  // use automatic L-curve scan: start with taumin=taumax=0.0
  Double_t tauMin=0.0;
  Double_t tauMax=0.0;
  Int_t iBest;
  TSpline *logTauX,*logTauY;
  TGraph *lCurve;
 
  // if required, report Info messages (for debugging the L-curve scan)
#ifdef VERBOSE_LCURVE_SCAN
  Int_t oldinfo=gErrorIgnoreLevel;
  gErrorIgnoreLevel=kInfo;
#endif
  // this method scans the parameter tau and finds the kink in the L curve
  // finally, the unfolding is done for the best choice of tau
  iBest=unfold.ScanLcurve(nScan,tauMin,tauMax,&lCurve,&logTauX,&logTauY);
 
  // if required, switch to previous log-level
#ifdef VERBOSE_LCURVE_SCAN
  gErrorIgnoreLevel=oldinfo;
#endif
 
  //==========================================================================
  // define a correlated systematic error
  // for example, assume there is a 10% correlated error for all reconstructed
  // masses larger than 7
  Double_t SYS_ERROR1_MSTART=6;
  Double_t SYS_ERROR1_SIZE=0.1;
  TH2D *histMdetGenSys1=new TH2D("Mdetgensys1",";mass(det);mass(gen)",
                                 nDet,xminDet,xmaxDet,nGen,xminGen,xmaxGen);
  for(Int_t i=0;i<=nDet+1;i++) {
     if(histMdetData->GetBinCenter(i)>=SYS_ERROR1_MSTART) {
        for(Int_t j=0;j<=nGen+1;j++) {
           histMdetGenSys1->SetBinContent(i,j,SYS_ERROR1_SIZE);
        }
     }
  }
  unfold.AddSysError(histMdetGenSysMC,"SYSERROR_MC",TUnfold::kHistMapOutputVert,
                     TUnfoldSys::kSysErrModeMatrix);
  unfold.AddSysError(histMdetGenSys1,"SYSERROR1",TUnfold::kHistMapOutputVert,
                     TUnfoldSys::kSysErrModeRelative);
 
  //==========================================================================
  // print some results
  //
  std::cout<<"tau="<<unfold.GetTau()<<"\n";
  std::cout<<"chi**2="<<unfold.GetChi2A()<<"+"<<unfold.GetChi2L()
           <<" / "<<unfold.GetNdf()<<"\n";
  std::cout<<"chi**2(sys)="<<unfold.GetChi2Sys()<<"\n";
 
 
  //==========================================================================
  // create graphs with one point to visualize the best choice of tau
  //
  Double_t t[1],x[1],y[1];
  logTauX->GetKnot(iBest,t[0],x[0]);
  logTauY->GetKnot(iBest,t[0],y[0]);
  TGraph *bestLcurve=new TGraph(1,x,y);
  TGraph *bestLogTauLogChi2=new TGraph(1,t,x);
 
  //==========================================================================
  // retrieve results into histograms
 
  // get unfolded distribution
  TH1 *histMunfold=unfold.GetOutput("Unfolded");
 
  // get unfolding result, folded back
  TH1 *histMdetFold=unfold.GetFoldedOutput("FoldedBack");
 
  // get error matrix (input distribution [stat] errors only)
  // TH2D *histEmatData=unfold.GetEmatrix("EmatData");
 
  // get total error matrix:
  //   migration matrix uncorrelated and correlated systematic errors
  //   added in quadrature to the data statistical errors
  TH2 *histEmatTotal=unfold.GetEmatrixTotal("EmatTotal");
 
  // create data histogram with the total errors
  TH1D *histTotalError=
     new TH1D("TotalError",";mass(gen)",nGen,xminGen,xmaxGen);
  for(Int_t bin=1;bin<=nGen;bin++) {
    histTotalError->SetBinContent(bin,histMunfold->GetBinContent(bin));
    histTotalError->SetBinError
       (bin,TMath::Sqrt(histEmatTotal->GetBinContent(bin,bin)));
  }
 
  // get global correlation coefficients
  // for this calculation one has to specify whether the
  // underflow/overflow bins are included or not
  // default: include all bins
  // here: exclude underflow and overflow bins
  TH1 *histRhoi=unfold.GetRhoItotal("rho_I",
                                    0, // use default title
                                    0, // all distributions
                                    "*[UO]", // discard underflow and overflow bins on all axes
                                    kTRUE, // use original binning
                                    &gHistInvEMatrix // store inverse of error matrix
                                    );
 
  //======================================================================
  // fit Breit-Wigner shape to unfolded data, using the full error matrix
  // here we use a "user" chi**2 function to take into account
  // the full covariance matrix
 
  gFitter=TVirtualFitter::Fitter(histMunfold);
  gFitter->SetFCN(chisquare_corr);
 
  TF1 *bw=new TF1("bw",bw_func,xminGen,xmaxGen,3);
  bw->SetParameter(0,1000.);
  bw->SetParameter(1,3.8);
  bw->SetParameter(2,0.2);
 
  // for (wrong!) fitting without correlations, drop the option "U"
  // here.
  histMunfold->Fit(bw,"UE");
 
  //=====================================================================
  // plot some histograms
  TCanvas output;
  output.Divide(3,2);
 
  // Show the matrix which connects input and output
  // There are overflow bins at the bottom, not shown in the plot
  // These contain the background shape.
  // The overflow bins to the left and right contain
  // events which are not reconstructed. These are necessary for proper MC
  // normalisation
  output.cd(1);
  histMdetGenMC->Draw("BOX");
 
  // draw generator-level distribution:
  //   data (red) [for real data this is not available]
  //   MC input (black) [with completely wrong peak position and shape]
  //   unfolded data (blue)
  output.cd(2);
  histTotalError->SetLineColor(kBlue);
  histTotalError->Draw("E");
  histMunfold->SetLineColor(kGreen);
  histMunfold->Draw("SAME E1");
  histDensityGenData->SetLineColor(kRed);
  histDensityGenData->Draw("SAME");
  histDensityGenMC->Draw("SAME HIST");
 
  // show detector level distributions
  //    data (red)
  //    MC (black) [with completely wrong peak position and shape]
  //    unfolded data (blue)
  output.cd(3);
  histMdetFold->SetLineColor(kBlue);
  histMdetFold->Draw();
  histMdetMC->Draw("SAME HIST");
 
  TH1 *histInput=unfold.GetInput("Minput",";mass(det)");
 
  histInput->SetLineColor(kRed);
  histInput->Draw("SAME");
 
  // show correlation coefficients
  output.cd(4);
  histRhoi->Draw();
 
  // show tau as a function of chi**2
  output.cd(5);
  logTauX->Draw();
  bestLogTauLogChi2->SetMarkerColor(kRed);
  bestLogTauLogChi2->Draw("*");
 
  // show the L curve
  output.cd(6);
  lCurve->Draw("AL");
  bestLcurve->SetMarkerColor(kRed);
  bestLcurve->Draw("*");
 
  output.SaveAs("testUnfold1.ps");
 
  return 0;
}