rf403_weightedevts.C File Reference

Detailed Description

Data and categories: using weights in unbinned datasets

 
#include "RooRealVar.h"
#include "RooDataSet.h"
#include "RooDataHist.h"
#include "RooGaussian.h"
#include "RooFormulaVar.h"
#include "RooGenericPdf.h"
#include "RooPolynomial.h"
#include "RooMinimizer.h"
#include "TCanvas.h"
#include "TAxis.h"
#include "RooPlot.h"
#include "RooFitResult.h"
using namespace RooFit;
 
void rf403_weightedevts()
{
   // C r e a t e   o b s e r v a b l e   a n d   u n w e i g h t e d   d a t a s e t
   // -------------------------------------------------------------------------------
 
   // Declare observable
   RooRealVar x("x", "x", -10, 10);
   x.setBins(40);
 
   // Construction a uniform pdf
   RooPolynomial p0("px", "px", x);
 
   // Sample 1000 events from pdf
   RooDataSet *data = p0.generate(x, 1000);
 
   // C a l c u l a t e   w e i g h t   a n d   m a k e   d a t a s e t   w e i g h t e d
   // -----------------------------------------------------------------------------------
 
   // Construct formula to calculate (fake) weight for events
   RooFormulaVar wFunc("w", "event weight", "(x*x+10)", x);
 
   // Add column with variable w to previously generated dataset
   RooRealVar *w = (RooRealVar *)data->addColumn(wFunc);
 
   // Dataset d is now a dataset with two observable (x,w) with 1000 entries
   data->Print();
 
   // Instruct dataset wdata in interpret w as event weight rather than as observable
   RooDataSet wdata(data->GetName(), data->GetTitle(), data, *data->get(), 0, w->GetName());
 
   // Dataset d is now a dataset with one observable (x) with 1000 entries and a sum of weights of ~430K
   wdata.Print();
 
   // U n b i n n e d   M L   f i t   t o   w e i g h t e d   d a t a
   // ---------------------------------------------------------------
 
   // Construction quadratic polynomial pdf for fitting
   RooRealVar a0("a0", "a0", 1);
   RooRealVar a1("a1", "a1", 0, -1, 1);
   RooRealVar a2("a2", "a2", 1, 0, 10);
   RooPolynomial p2("p2", "p2", x, RooArgList(a0, a1, a2), 0);
 
   // Fit quadratic polynomial to weighted data
 
   // NOTE: A plain Maximum likelihood fit to weighted data does in general
   //       NOT result in correct error estimates, unless individual
   //       event weights represent Poisson statistics themselves.
   //
   // Fit with 'wrong' errors
   std::unique_ptr<RooFitResult> r_ml_wgt{p2.fitTo(wdata, Save(), PrintLevel(-1))};
 
   // A first order correction to estimated parameter errors in an
   // (unbinned) ML fit can be obtained by calculating the
   // covariance matrix as
   //
   //    V' = V C-1 V
   //
   // where V is the covariance matrix calculated from a fit
   // to -logL = - sum [ w_i log f(x_i) ] and C is the covariance
   // matrix calculated from -logL' = -sum [ w_i^2 log f(x_i) ]
   // (i.e. the weights are applied squared)
   //
   // A fit in this mode can be performed as follows:
 
   std::unique_ptr<RooFitResult> r_ml_wgt_corr{p2.fitTo(wdata, Save(), SumW2Error(true), PrintLevel(-1))};
 
   // P l o t   w e i g h e d   d a t a   a n d   f i t   r e s u l t
   // ---------------------------------------------------------------
 
   // Construct plot frame
   RooPlot *frame = x.frame(Title("Unbinned ML fit, binned chi^2 fit to weighted data"));
 
   // Plot data using sum-of-weights-squared error rather than Poisson errors
   wdata.plotOn(frame, DataError(RooAbsData::SumW2));
 
   // Overlay result of 2nd order polynomial fit to weighted data
   p2.plotOn(frame);
 
   // ML Fit of pdf to equivalent unweighted dataset
   // -----------------------------------------------------------------------------------------
 
   // Construct a pdf with the same shape as p0 after weighting
   RooGenericPdf genPdf("genPdf", "x*x+10", x);
 
   // Sample a dataset with the same number of events as data
   RooDataSet *data2 = genPdf.generate(x, 1000);
 
   // Sample a dataset with the same number of weights as data
   RooDataSet *data3 = genPdf.generate(x, 43000);
 
   // Fit the 2nd order polynomial to both unweighted datasets and save the results for comparison
   std::unique_ptr<RooFitResult> r_ml_unw10{p2.fitTo(*data2, Save(), PrintLevel(-1))};
   std::unique_ptr<RooFitResult> r_ml_unw43{p2.fitTo(*data3, Save(), PrintLevel(-1))};
 
   // C h i 2   f i t   o f   p d f   t o   b i n n e d   w e i g h t e d   d a t a s e t
   // ------------------------------------------------------------------------------------
 
   // Construct binned clone of unbinned weighted dataset
   RooDataHist *binnedData = wdata.binnedClone();
   binnedData->Print("v");
 
   // Perform chi2 fit to binned weighted dataset using sum-of-weights errors
   //
   // NB: Within the usual approximations of a chi2 fit, a chi2 fit to weighted
   // data using sum-of-weights-squared errors does give correct error
   // estimates
   std::unique_ptr<RooAbsReal> chi2{p2.createChi2(*binnedData, DataError(RooAbsData::SumW2))};
   RooMinimizer m(*chi2);
   m.migrad();
   m.hesse();
 
   // Plot chi^2 fit result on frame as well
   std::unique_ptr<RooFitResult> r_chi2_wgt{m.save()};
   p2.plotOn(frame, LineStyle(kDashed), LineColor(kRed));
 
   // C o m p a r e   f i t   r e s u l t s   o f   c h i 2 , M L   f i t s   t o   ( u n ) w e i g h t e d   d a t a
   // ---------------------------------------------------------------------------------------------------------------
 
   // Note that ML fit on 1Kevt of weighted data is closer to result of ML fit on 43Kevt of unweighted data
   // than to 1Kevt of unweighted data, whereas the reference chi^2 fit with SumW2 error gives a result closer to
   // that of an unbinned ML fit to 1Kevt of unweighted data.
 
   cout << "==> ML Fit results on 1K unweighted events" << endl;
   r_ml_unw10->Print();
   cout << "==> ML Fit results on 43K unweighted events" << endl;
   r_ml_unw43->Print();
   cout << "==> ML Fit results on 1K weighted events with a summed weight of 43K" << endl;
   r_ml_wgt->Print();
   cout << "==> Corrected ML Fit results on 1K weighted events with a summed weight of 43K" << endl;
   r_ml_wgt_corr->Print();
   cout << "==> Chi2 Fit results on 1K weighted events with a summed weight of 43K" << endl;
   r_chi2_wgt->Print();
 
   new TCanvas("rf403_weightedevts", "rf403_weightedevts", 600, 600);
   gPad->SetLeftMargin(0.15);
   frame->GetYaxis()->SetTitleOffset(1.8);
   frame->Draw();
}

 
RooDataSet::pxData[x,w] = 1000 entries
RooDataSet::pxData[x,weight:w] = 1000 entries (43238.9 weighted)
[#0] WARNING:InputArguments -- RooAbsPdf::fitTo(p2) WARNING: a likelihood fit is requested of what appears to be weighted data.
       While the estimated values of the parameters will always be calculated taking the weights into account,
       there are multiple ways to estimate the errors of the parameters. You are advised to make an 
       explicit choice for the error calculation:
           - Either provide SumW2Error(true), to calculate a sum-of-weights-corrected HESSE error matrix
             (error will be proportional to the number of events in MC).
           - Or provide SumW2Error(false), to return errors from original HESSE error matrix
             (which will be proportional to the sum of the weights, i.e., a dataset with <sum of weights> events).
           - Or provide AsymptoticError(true), to use the asymptotically correct expression
             (for details see https://arxiv.org/abs/1911.01303).
[#1] INFO:Minimization -- RooAbsMinimizerFcn::setOptimizeConst: activating const optimization
[#1] INFO:Minimization -- RooAbsMinimizerFcn::setOptimizeConst: deactivating const optimization
[#1] INFO:Minimization -- RooAbsMinimizerFcn::setOptimizeConst: activating const optimization
[#1] INFO:Fitting -- RooAbsPdf::fitTo(p2) Calculating sum-of-weights-squared correction matrix for covariance matrix
[#1] INFO:Minimization -- RooAbsMinimizerFcn::setOptimizeConst: deactivating const optimization
[#1] INFO:NumericIntegration -- RooRealIntegral::init(genPdf_Int[x]) using numeric integrator RooIntegrator1D to calculate Int(x)
[#1] INFO:NumericIntegration -- RooRealIntegral::init(genPdf_Int[x]) using numeric integrator RooIntegrator1D to calculate Int(x)
[#1] INFO:NumericIntegration -- RooRealIntegral::init(genPdf_Int[x]) using numeric integrator RooIntegrator1D to calculate Int(x)
[#1] INFO:NumericIntegration -- RooRealIntegral::init(genPdf_Int[x]) using numeric integrator RooIntegrator1D to calculate Int(x)
[#1] INFO:Minimization -- RooAbsMinimizerFcn::setOptimizeConst: activating const optimization
[#1] INFO:Minimization -- RooAbsMinimizerFcn::setOptimizeConst: deactivating const optimization
[#1] INFO:Minimization -- RooAbsMinimizerFcn::setOptimizeConst: activating const optimization
[#1] INFO:Minimization -- RooAbsMinimizerFcn::setOptimizeConst: deactivating const optimization
DataStore pxData_binned (Generated From px_binned)
  Contains 40 entries
  Observables: 
    1)  x = 9.75  L(-10 - 10) B(40)  "x"
Binned Dataset pxData_binned (Generated From px_binned)
  Contains 40 bins with a total weight of 43238.9
  Observables:     1)  x = 9.75  L(-10 - 10) B(40)  "x"
 **********
 **   54 **SET NOGRAD
 **********
 PARAMETER DEFINITIONS:
    NO.   NAME         VALUE      STEP SIZE      LIMITS
     1 a1          -1.13263e-03  4.02065e-03   -1.00000e+00  1.00000e+00
     2 a2           9.75516e-02  2.36614e-03    0.00000e+00  1.00000e+01
 **********
 **   55 **SET ERR           1
 **********
 **********
 **   56 **SET PRINT           1
 **********
 **********
 **   57 **SET STR           1
 **********
 NOW USING STRATEGY  1: TRY TO BALANCE SPEED AGAINST RELIABILITY
 **********
 **   58 **MIGRAD        1000           1
 **********
 FIRST CALL TO USER FUNCTION AT NEW START POINT, WITH IFLAG=4.
 START MIGRAD MINIMIZATION.  STRATEGY  1.  CONVERGENCE WHEN EDM .LT. 1.00e-03
 FCN=32.3464 FROM MIGRAD    STATUS=INITIATE        6 CALLS           7 TOTAL
                     EDM= unknown      STRATEGY= 1      NO ERROR MATRIX       
  EXT PARAMETER               CURRENT GUESS       STEP         FIRST   
  NO.   NAME      VALUE            ERROR          SIZE      DERIVATIVE 
   1  a1          -1.13263e-03   4.02065e-03   4.02066e-03   2.74413e+01
   2  a2           9.75516e-02   2.36614e-03   2.40760e-03  -2.04371e+02
 MIGRAD MINIMIZATION HAS CONVERGED.
 MIGRAD WILL VERIFY CONVERGENCE AND ERROR MATRIX.
 COVARIANCE MATRIX CALCULATED SUCCESSFULLY
 FCN=31.3747 FROM MIGRAD    STATUS=CONVERGED      28 CALLS          29 TOTAL
                     EDM=8.52339e-08    STRATEGY= 1      ERROR MATRIX ACCURATE 
  EXT PARAMETER                                   STEP         FIRST   
  NO.   NAME      VALUE            ERROR          SIZE      DERIVATIVE 
   1  a1          -9.98900e-03   2.62975e-02   7.27538e-05   4.94399e-03
   2  a2           1.06373e-01   1.01849e-02   2.73457e-05  -3.91519e-02
 EXTERNAL ERROR MATRIX.    NDIM=  25    NPAR=  2    ERR DEF=1
  6.917e-04 -6.529e-06 
 -6.529e-06  1.037e-04 
 PARAMETER  CORRELATION COEFFICIENTS  
       NO.  GLOBAL      1      2
        1  0.02437   1.000 -0.024
        2  0.02437  -0.024  1.000
 **********
 **   59 **SET ERR           1
 **********
 **********
 **   60 **SET PRINT           1
 **********
 **********
 **   61 **HESSE        1000
 **********
 COVARIANCE MATRIX CALCULATED SUCCESSFULLY
 FCN=31.3747 FROM HESSE     STATUS=OK             10 CALLS          39 TOTAL
                     EDM=8.52317e-08    STRATEGY= 1      ERROR MATRIX ACCURATE 
  EXT PARAMETER                                INTERNAL      INTERNAL  
  NO.   NAME      VALUE            ERROR       STEP SIZE       VALUE   
   1  a1          -9.98900e-03   2.62976e-02   1.45508e-05  -9.98917e-03
   2  a2           1.06373e-01   1.01850e-02   5.46914e-06  -1.36415e+00
 EXTERNAL ERROR MATRIX.    NDIM=  25    NPAR=  2    ERR DEF=1
  6.917e-04 -6.575e-06 
 -6.575e-06  1.037e-04 
 PARAMETER  CORRELATION COEFFICIENTS  
       NO.  GLOBAL      1      2
        1  0.02455   1.000 -0.025
        2  0.02455  -0.025  1.000
==> ML Fit results on 1K unweighted events
 
  RooFitResult: minimized FCN value: 2766.49, estimated distance to minimum: 7.63714e-07
                covariance matrix quality: Full, accurate covariance matrix
                Status : MINIMIZE=0 HESSE=0 
 
    Floating Parameter    FinalValue +/-  Error   
  --------------------  --------------------------
                    a1    8.9440e-03 +/-  2.69e-02
                    a2    1.0129e-01 +/-  1.67e-02
 
==> ML Fit results on 43K unweighted events
 
  RooFitResult: minimized FCN value: 118892, estimated distance to minimum: 2.21254e-06
                covariance matrix quality: Full, accurate covariance matrix
                Status : MINIMIZE=0 HESSE=0 
 
    Floating Parameter    FinalValue +/-  Error   
  --------------------  --------------------------
                    a1   -1.1326e-03 +/-  4.02e-03
                    a2    9.7552e-02 +/-  2.37e-03
 
==> ML Fit results on 1K weighted events with a summed weight of 43K
 
  RooFitResult: minimized FCN value: 119682, estimated distance to minimum: 1.28662e-07
                covariance matrix quality: Full, accurate covariance matrix
                Status : MINIMIZE=0 HESSE=0 
 
    Floating Parameter    FinalValue +/-  Error   
  --------------------  --------------------------
                    a1   -4.8560e-03 +/-  4.03e-03
                    a2    9.8651e-02 +/-  2.41e-03
 
==> Corrected ML Fit results on 1K weighted events with a summed weight of 43K
 
  RooFitResult: minimized FCN value: 6.84247e+06, estimated distance to minimum: 158914
                covariance matrix quality: Full, accurate covariance matrix
                Status : MINIMIZE=0 HESSE=0 HESSE=0 
 
    Floating Parameter    FinalValue +/-  Error   
  --------------------  --------------------------
                    a1   -4.8555e-03 +/-  3.00e-02
                    a2    9.8652e-02 +/-  2.99e-02
 
==> Chi2 Fit results on 1K weighted events with a summed weight of 43K
 
  RooFitResult: minimized FCN value: 31.3747, estimated distance to minimum: 8.52317e-08
                covariance matrix quality: Full, accurate covariance matrix
                Status : MIGRAD=0 HESSE=0 
 
    Floating Parameter    FinalValue +/-  Error   
  --------------------  --------------------------
                    a1   -9.9890e-03 +/-  2.63e-02
                    a2    1.0637e-01 +/-  1.02e-02
 

Date

July 2008

Author

Wouter Verkerke

Definition in file rf403_weightedevts.C.