# Rs 4 0 1D_ Feldman Cousins¶

Neutrino Oscillation Example from Feldman & Cousins

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.

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 09:52 AM.

In [ ]:
%%cpp -d
#include "RooGlobalFunc.h"
#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 "RooStats/MCMCIntervalPlot.h"
#include "RooStats/MCMCInterval.h"

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

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

In [ ]:
TString tutDir = gROOT->GetTutorialDir();
TString headerDir = TString::Format("#include \"%s/roostats/NuMuToNuE_Oscillation.h\"", tutDir.Data());
TString impDir = TString::Format("#include \"%s/roostats/NuMuToNuE_Oscillation.cxx\"", tutDir.Data());
gROOT->ProcessLine(impDir);


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


Arguments are defined.

In [ ]:
bool doFeldmanCousins = false;
bool doMCMC = true;


To time the macro

In [ ]:
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 $P (\nu\mu \rightarrow \nu e ) = sin^2 (2\theta) sin^2 (1.27 \Delta m^2 L /E )$ where P is the probability for a $\nu\mu$ to transform into a $\nu e$ , 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 $\Delta m^2 = |m^2 - m^2 |$ 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 $\nu e$ interactions and misidentified $\nu\mu$ 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 $\nu\mu$ flux is such that if $P (\nu\mu \rightarrow \nu e ) = 0.01$ averaged over any bin, then that bin would have an expected additional contribution of 100 events due to $\nu\mu \rightarrow \nu e$ oscillations.

Make signal model model

In [ ]:
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  RooClassFactory x root  x.makePdf("NuMuToNuE_Oscillation","L,E,deltaMSq","","pow(sin(1.27deltaMSqL/E),2)")

In [ ]:
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

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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

In [ ]:
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 clear to include a cross section term that is not explicitly referred to in the text, eg. number 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

In [ ]:
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()));


$signorm = maxeventstot \cdot \int de dl \frac{p_{oscillate\ in\ experiment}}{area} \cdot {sin}^2(2\theta)$

In [ ]:
RooProduct sigNorm("sigNorm", "", RooArgSet(maxEventsTot, *intProbToOscInExp, inverseArea, sinSq2theta));


Bkg = 5 bins * 100 events / bin

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RooConstVar bkgNorm("bkgNorm", "normalization for background", 500);


Flat background (0th order polynomial, so no arguments for coefficients)

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RooPolynomial bkgEShape("bkgEShape", "flat bkg shape", E);


Total model

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


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

Turn off some messages

In [ ]:
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

In [ ]:
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")

In [ ]:
RooRandom::randomGenerator()->SetSeed(3);


Create a toy dataset

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RooDataSet *data = model.generate(RooArgSet(E), nEventsData);


make some plots

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


Plot the pdf

In [ ]:
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

In [ ]:
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

In [ ]:
dataCanvas->cd(3);
std::unique_ptr<RooAbsReal> nll{model.createNLL(*data, Extended(true))};
RooProfileLL pll("pll", "", *nll, RooArgSet(deltaMSq, sinSq2theta));


Th1* hhh = nll.createhistogram("hhh",sinsq2theta,binning(40),yvar(deltamsq,binning(40))) ;

In [ ]:
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 set the distribution creator, which encodes the test statistic

In [ ]:
RooArgSet parameters(deltaMSq, sinSq2theta);
RooWorkspace *w = new RooWorkspace();

ModelConfig modelConfig;
modelConfig.SetWorkspace(*w);
modelConfig.SetPdf(model);
modelConfig.SetParametersOfInterest(parameters);

RooStats::FeldmanCousins fc(*data, modelConfig);
fc.SetTestSize(.1); // set size of test
fc.SetNBins(10); // number of points to test per parameter


Use the feldman-cousins tool

In [ ]:
ConfInterval *interval = 0;
if (doFeldmanCousins)
interval = fc.GetInterval();


show use of ProfileLikeihoodCalculator utility in RooStats

In [ ]:
RooStats::ProfileLikelihoodCalculator plc(*data, modelConfig);
plc.SetTestSize(.1);

ConfInterval *plcInterval = plc.GetInterval();


show use of MCMCCalculator utility in RooStats

In [ ]:
MCMCInterval *mcInt = NULL;

if (doMCMC) {
// turn some messages back on
RooMsgService::instance().setStreamStatus(0, kTRUE);
RooMsgService::instance().setStreamStatus(1, kTRUE);

TStopwatch mcmcWatch;
mcmcWatch.Start();

RooArgList axisList(deltaMSq, sinSq2theta);
MCMCCalculator mc(*data, modelConfig);
mc.SetNumIters(5000);
mc.SetNumBurnInSteps(100);
mc.SetUseKeys(true);
mc.SetTestSize(.1);
mc.SetAxes(axisList); // set which is x and y axis in posterior histogram
// mc.SetNumBins(50);
mcInt = (MCMCInterval *)mc.GetInterval();

mcmcWatch.Stop();
mcmcWatch.Print();
}


make plot of resulting interval

In [ ]:
dataCanvas->cd(4);


First plot a small dot for every point tested

In [ ]:
if (doFeldmanCousins) {
RooDataHist *parameterScan = (RooDataHist *)fc.GetPointsToScan();
TH2F *hist = parameterScan->createHistogram(deltaMSq,sinSq2theta, 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");
}
}

MCMCIntervalPlot *mcPlot = NULL;
if (mcInt) {
cout << "MCMC actual confidence level: " << mcInt->GetActualConfidenceLevel() << endl;
mcPlot = new MCMCIntervalPlot(*mcInt);
mcPlot->SetLineColor(kMagenta);
mcPlot->Draw();
}
dataCanvas->Update();

LikelihoodIntervalPlot plotInt((LikelihoodInterval *)plcInterval);
plotInt.SetTitle("90% Confidence Intervals");
if (mcInt)
plotInt.Draw("same");
else
plotInt.Draw();
dataCanvas->Update();


/ print timing info

In [ ]:
t.Stop();
t.Print();


Draw all canvases

In [ ]:
gROOT->GetListOfCanvases()->Draw()