Why is the pull of the coeff. of RooPolynomial bad?

ROOT
1

Hi,

Actually, I met this problem some time ago. You know, I just want to use RooPolynomial (1st polynomial) to describe the background, and I use a Gaussian for the signal. I guess this is the very simple usage of RooFit. But there is one thing I could not understand. If I do a toy Monte Carlo with this PDF, the pull of the coefficient would be bad. Please see the attached plot.

I guess that it is cased by the wrong step size? But I failed to find how I could set the step size for the floating variable in RooFit. Could you please give some hints here? It seems to me that setError doesn’t help.

BTW: this is done in Root 5.17/04, but I think the root version doesn’t matter.

Thanks a lot.

Cheers, Jibo

void mass() 
{
  gSystem->Load("libRooFit");
  using namespace RooFit;
  
  RooRealVar *m  = new RooRealVar( "m", "invariant mass", 5000, 5600, "MeV/c^{2}");
  
  RooArgSet* observables = new RooArgSet( *m );
  
  /// sig mass
  RooRealVar *SigMass    = new RooRealVar("SigMass","B  mass", 5279, 5000, 5600, "MeV/c^{2}" );
  RooRealVar *SigMassRes = new RooRealVar("SigMassRes","B mass resolution", 25, 0., 60, "MeV/c^{2}");
  RooAbsPdf  *SigMassPdf = new RooGaussian("SigMassPdf","MassPDF of signal",*m,*SigMass,*SigMassRes);
  
  /// bkg mass
  RooRealVar *BkgMassSlope = new RooRealVar("BkgMassSlope","Slope of mass bkg", -9.7194e-05, -1.0, 1.0 );
  
  RooAbsPdf  *BkgMassPdf = new RooPolynomial("BkgMassPdf","BkgMassPdf",*m, *BkgMassSlope );
  
  RooRealVar *FracSig = new RooRealVar("FracSig","fraction of signal", 0.11, 0, 1.0); //1.2802e-01
  
  RooAbsPdf  *MassPdf = new RooAddPdf("MassPdf","mass pdf",
                                      RooArgList(*SigMassPdf,*BkgMassPdf),
                                      RooArgList(*FracSig) );

  BkgMassSlope->setError( 1.0e-8 ) ;

  RooMCStudy *mgr = new RooMCStudy(*MassPdf, *observables, FitModel(*MassPdf), FitOptions( InitialHesse(true), Minos(false) ) );

  mgr->generateAndFit( 1000, 3000 );

  TCanvas *c = new TCanvas();
  c->Divide(2,2);

  c->cd(1);
  RooPlot* frame1 = mgr->plotParam(*BkgMassSlope);
  frame1->Draw();

  c->cd(2);
  RooPlot* frame2 = mgr->plotPull(*BkgMassSlope,-5,5,40,kTRUE);
  frame2->Draw();

  c->cd(3);
  mgr->plotNLL()->Draw();

  c->cd(4);
  mgr->plotError(*BkgMassSlope)->Draw();
 
}

badpull.gif1020×667 28.5 KB

2

Hi,

The bias occurrs because a sizeable fraction of the toy fits do not converge properly because you run into problems with a fitted slope value that causes the background p.d.f to become negative inside the domain of m.

When events with negative p.d.f values occur the likelihood becomes unphysical and RooFit puts up a ‘barrier’ to force MINUIT to retreat to an allowed region of the parameter space.

This effectively makes toy experiments that should have very negative slope values (= very positive pull values) migrate to less positive pull values, hence the skewed pull distribution you observed. If you increase the per-experiment statistics by a factor of 10 or so you see that the problem will go away. (I recommend the Binned() option of RooMCStudy here to speed up the fitting by changing from unbinned to a binned
likelihood calculation, which should make much difference at high statistics)

I note (perhaps superfluously) that problem this not a software feature but it is inherent to the problem definition, which is precisely a good reason why it is prudent to run such toy MC studies.

You can improve this by e.g. setting a more stringent limit on the slope parameter range so that negative p.d.f values cannot occur, or choose a different parameterization for the background.

Wouter