Rs_Bernstein Correction

Example of the BernsteinCorrection utility in RooStats.

The idea is that one has a distribution coming either from data or Monte Carlo (called "reality" in the macro) and a nominal model that is not sufficiently flexible to take into account the real distribution. One wants to take into account the systematic associated with this imperfect modeling by augmenting the nominal model with some correction term (in this case a polynomial). The BernsteinCorrection utility will import into your workspace a corrected model given by nominal(x) * poly_N(x), where poly_N is an n-th order polynomial in the Bernstein basis. The degree N of the polynomial is chosen by specifying the tolerance one has in adding an extra term to the polynomial. The Bernstein basis is nice because it only has positive-definite terms and works well with PDFs. Finally, the macro makes a plot of:

  • the data (drawn from 'reality'),
  • the best fit of the nominal model (blue)
  • and the best fit corrected model.

Author: Kyle Cranmer
This notebook tutorial was automatically generated with ROOTBOOK-izer from the macro found in the ROOT repository on Thursday, June 30, 2022 at 09:59 AM.

In [ ]:
%%cpp -d
#include "RooDataSet.h"
#include "RooRealVar.h"
#include "RooConstVar.h"
#include "RooBernstein.h"
#include "TCanvas.h"
#include "RooAbsPdf.h"
#include "RooFitResult.h"
#include "RooPlot.h"
#include <string>
#include <vector>
#include <stdio.h>
#include <sstream>
#include <iostream>

#include "RooProdPdf.h"
#include "RooAddPdf.h"
#include "RooGaussian.h"
#include "RooProfileLL.h"
#include "RooWorkspace.h"

#include "RooStats/BernsteinCorrection.h"

Use this order for safety on library loading

In [ ]:
%%cpp -d
// This is a workaround to make sure the namespace is used inside functions
using namespace RooFit;
using namespace RooStats;

Set range of observable

In [ ]:
Double_t lowRange = -1, highRange = 5;

Make a roorealvar for the observable

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RooRealVar x("x", "x", lowRange, highRange);

True model

In [ ]:
RooGaussian narrow("narrow", "", x, RooConst(0.), RooConst(.8));
RooGaussian wide("wide", "", x, RooConst(0.), RooConst(2.));
RooAddPdf reality("reality", "", RooArgList(narrow, wide), RooConst(0.8));

RooDataSet *data = reality.generate(x, 1000);

Nominal model

In [ ]:
RooRealVar sigma("sigma", "", 1., 0, 10);
RooGaussian nominal("nominal", "", x, RooConst(0.), sigma);

RooWorkspace *wks = new RooWorkspace("myWorksspace");

wks->import(*data, Rename("data"));
wks->import(nominal);

if (TClass::GetClass("ROOT::Minuit2::Minuit2Minimizer")) {
   // use Minuit2 if ROOT was built with support for it:
   ROOT::Math::MinimizerOptions::SetDefaultMinimizer("Minuit2");
}

The tolerance sets the probability to add an unnecessary term. lower tolerance will add fewer terms, while higher tolerance will add more terms and provide a more flexible function.

In [ ]:
Double_t tolerance = 0.05;
BernsteinCorrection bernsteinCorrection(tolerance);
Int_t degree = bernsteinCorrection.ImportCorrectedPdf(wks, "nominal", "x", "data");

if (degree < 0) {
   Error("rs_bernsteinCorrection", "Bernstein correction failed ! ");
   return;
}

cout << " Correction based on Bernstein Poly of degree " << degree << endl;

RooPlot *frame = x.frame();
data->plotOn(frame);

Plot the best fit nominal model in blue

In [ ]:
TString minimType = ROOT::Math::MinimizerOptions::DefaultMinimizerType();
nominal.fitTo(*data, PrintLevel(0), Minimizer(minimType));
nominal.plotOn(frame);

Plot the best fit corrected model in red

In [ ]:
RooAbsPdf *corrected = wks->pdf("corrected");
if (!corrected)
   return;

Fit corrected model

In [ ]:
corrected->fitTo(*data, PrintLevel(0), Minimizer(minimType));
corrected->plotOn(frame, LineColor(kRed));

Plot the correction term (* norm constant) in dashed green should make norm constant just be 1, not depend on binning of data

In [ ]:
RooAbsPdf *poly = wks->pdf("poly");
if (poly)
   poly->plotOn(frame, LineColor(kGreen), LineStyle(kDashed));

This is a switch to check the sampling distribution of -2 log LR for two comparisons: the first is for n-1 vs. n degree polynomial corrections the second is for n vs. n+1 degree polynomial corrections Here we choose n to be the one chosen by the tolerance criterion above, eg. n = "degree" in the code. Setting this to true is takes about 10 min.

In [ ]:
bool checkSamplingDist = true;
int numToyMC = 20; // increase this value for sensible results

TCanvas *c1 = new TCanvas();
if (checkSamplingDist) {
   c1->Divide(1, 2);
   c1->cd(1);
}
frame->Draw();
gPad->Update();

if (checkSamplingDist) {
   // check sampling dist
   ROOT::Math::MinimizerOptions::SetDefaultPrintLevel(-1);
   TH1F *samplingDist = new TH1F("samplingDist", "", 20, 0, 10);
   TH1F *samplingDistExtra = new TH1F("samplingDistExtra", "", 20, 0, 10);
   bernsteinCorrection.CreateQSamplingDist(wks, "nominal", "x", "data", samplingDist, samplingDistExtra, degree,
                                           numToyMC);

   c1->cd(2);
   samplingDistExtra->SetLineColor(kRed);
   samplingDistExtra->Draw();
   samplingDist->Draw("same");
}

Draw all canvases

In [ ]:
%jsroot on
gROOT->GetListOfCanvases()->Draw()