Two Sided Frequentist Upper Limit With Bands

This is a standard demo that can be used with any ROOT file prepared in the standard way. You specify:

  • name for input ROOT file
  • name of workspace inside ROOT file that holds model and data
  • name of ModelConfig that specifies details for calculator tools
  • name of dataset

With default parameters the macro will attempt to run the standard hist2workspace example and read the ROOT file that it produces.

You may want to control:

double confidenceLevel=0.95;
  double additionalToysFac = 1.;
  int nPointsToScan = 12;
  int nToyMC = 200;

This uses a modified version of the profile likelihood ratio as a test statistic for upper limits (eg. test stat = 0 if muhat>mu).

Based on the observed data, one defines a set of parameter points to be tested based on the value of the parameter of interest and the conditional MLE (eg. profiled) values of the nuisance parameters.

At each parameter point, pseudo-experiments are generated using this fixed reference model and then the test statistic is evaluated. The auxiliary measurements (global observables) associated with the constraint terms in nuisance parameters are also fluctuated in the process of generating the pseudo-experiments in a frequentist manner forming an 'unconditional ensemble'. One could form a 'conditional' ensemble in which these auxiliary measurements are fixed. Note that the nuisance parameters are not randomized, which is a Bayesian procedure. Note, the nuisance parameters are floating in the fits. For each point, the threshold that defines the 95% acceptance region is found. This forms a "Confidence Belt".

After constructing the confidence belt, one can find the confidence interval for any particular dataset by finding the intersection of the observed test statistic and the confidence belt. First this is done on the observed data to get an observed 1-sided upper limt.

Finally, there expected limit and bands (from background-only) are formed by generating background-only data and finding the upper limit. The background-only is defined as such that the nuisance parameters are fixed to their best fit value based on the data with the signal rate fixed to 0. The bands are done by hand for now, will later be part of the RooStats tools.

On a technical note, this technique IS the generalization of Feldman-Cousins with nuisance parameters.

Building the confidence belt can be computationally expensive. Once it is built, one could save it to a file and use it in a separate step.

We can use PROOF to speed things along in parallel, however, the test statistic has to be installed on the workers so either turn off PROOF or include the modified test statistic in your $ROOTSYS/roofit/roostats/inc directory, add the additional line to the LinkDef.h file, and recompile root.

Note, if you have a boundary on the parameter of interest (eg. cross-section) the threshold on the two-sided test statistic starts off at moderate values and plateaus.

[#0] PROGRESS:Generation -- generated toys: 500 / 999 NeymanConstruction: Prog: 12/50 total MC = 39 this test stat = 0 SigXsecOverSM=0.69 alpha_syst1=0.136515 alpha_syst3=0.425415 beta_syst2=1.08496 [-1e+30, 0.011215] in interval = 1

this tells you the values of the parameters being used to generate the pseudo-experiments and the threshold in this case is 0.011215. One would expect for 95% that the threshold would be ~1.35 once the cross-section is far enough away from 0 that it is essentially unaffected by the boundary. As one reaches the last points in the scan, the theshold starts to get artificially high. This is because the range of the parameter in the fit is the same as the range in the scan. In the future, these should be independently controlled, but they are not now. As a result the ~50% of pseudo-experiments that have an upward fluctuation end up with muhat = muMax. Because of this, the upper range of the parameter should be well above the expected upper limit... but not too high or one will need a very large value of nPointsToScan to resolve the relevant region. This can be improved, but this is the first version of this script.

Important note: when the model includes external constraint terms, like a Gaussian constraint to a nuisance parameter centered around some nominal value there is a subtlety. The asymptotic results are all based on the assumption that all the measurements fluctuate... including the nominal values from auxiliary measurements. If these do not fluctuate, this corresponds to an "conditional ensemble". The result is that the distribution of the test statistic can become very non-chi^2. This results in thresholds that become very large.

Author: Kyle Cranmer,Contributions from Aaron Armbruster, Haoshuang Ji, Haichen Wang and Daniel Whiteson
This notebook tutorial was automatically generated with ROOTBOOK-izer from the macro found in the ROOT repository on Friday, September 18, 2020 at 10:04 AM.

In [ ]:
%%cpp -d
#include "TFile.h"
#include "TROOT.h"
#include "TH1F.h"
#include "TCanvas.h"
#include "TSystem.h"
#include <iostream>

#include "RooWorkspace.h"
#include "RooSimultaneous.h"
#include "RooAbsData.h"

#include "RooStats/ModelConfig.h"
#include "RooStats/FeldmanCousins.h"
#include "RooStats/ToyMCSampler.h"
#include "RooStats/PointSetInterval.h"
#include "RooStats/ConfidenceBelt.h"

#include "RooStats/RooStatsUtils.h"
#include "RooStats/ProfileLikelihoodTestStat.h"
In [ ]:
%%cpp -d
// This is a workaround to make sure the namespace is used inside functions
using namespace RooFit;
using namespace RooStats;
In [ ]:
using namespace std;

bool useProof = false; // flag to control whether to use Proof
int nworkers = 0;      // number of workers (default use all available cores)

Arguments are defined.

In [ ]:
const char *infile = "";
const char *workspaceName = "combined";
const char *modelConfigName = "ModelConfig";
const char *dataName = "obsData";
In [ ]:
double confidenceLevel = 0.95;

Degrade/improve number of pseudo-experiments used to define the confidence belt. value of 1 corresponds to default number of toys in the tail, which is 50/(1-confidenceLevel)

In [ ]:
double additionalToysFac = 0.5;
int nPointsToScan = 20; // number of steps in the parameter of interest
int nToyMC = 200;       // number of toys used to define the expected limit and band

First part is just to access a user-defined file or create the standard example file if it doesn't exist

In [ ]:
const char *filename = "";
if (!strcmp(infile, "")) {
   filename = "results/example_combined_GaussExample_model.root";
   bool fileExist = !gSystem->AccessPathName(filename); // note opposite return code
   // if file does not exists generate with histfactory
   if (!fileExist) {
#ifdef _WIN32
      cout << "HistFactory file cannot be generated on Windows - exit" << endl;
      return;
#endif
      // Normally this would be run on the command line
      cout << "will run standard hist2workspace example" << endl;
      gROOT->ProcessLine(".! prepareHistFactory .");
      gROOT->ProcessLine(".! hist2workspace config/example.xml");
      cout << "\n\n---------------------" << endl;
      cout << "Done creating example input" << endl;
      cout << "---------------------\n\n" << endl;
   }

} else
   filename = infile;

Try to open the file

In [ ]:
TFile *file = TFile::Open(filename);

If input file was specified byt not found, quit

In [ ]:
if (!file) {
   cout << "StandardRooStatsDemoMacro: Input file " << filename << " is not found" << endl;
   return;
}

Now get the data and workspace

Get the workspace out of the file

In [ ]:
RooWorkspace *w = (RooWorkspace *)file->Get(workspaceName);
if (!w) {
   cout << "workspace not found" << endl;
   return;
}

Get the modelconfig out of the file

In [ ]:
ModelConfig *mc = (ModelConfig *)w->obj(modelConfigName);

Get the modelconfig out of the file

In [ ]:
RooAbsData *data = w->data(dataName);

Make sure ingredients are found

In [ ]:
if (!data || !mc) {
   w->Print();
   cout << "data or ModelConfig was not found" << endl;
   return;
}

cout << "Found data and ModelConfig:" << endl;
mc->Print();

Now get the POI for convenience you may want to adjust the range of your POI

In [ ]:
RooRealVar *firstPOI = (RooRealVar *)mc->GetParametersOfInterest()->first();
/*  firstPOI->setMin(0);*/
/*  firstPOI->setMax(10);*/

create and use the FeldmanCousins tool to find and plot the 95% confidence interval on the parameter of interest as specified in the model config REMEMBER, we will change the test statistic so this is NOT a Feldman-Cousins interval

In [ ]:
FeldmanCousins fc(*data, *mc);
fc.SetConfidenceLevel(confidenceLevel);
fc.AdditionalNToysFactor(additionalToysFac); // improve sampling that defines confidence belt

Fc.useadaptivesampling(true); // speed it up a bit, but don't use for expected limits

In [ ]:
fc.SetNBins(nPointsToScan); // set how many points per parameter of interest to scan
fc.CreateConfBelt(true);    // save the information in the belt for plotting

Feldman-Cousins is a unified limit by definition but the tool takes care of a few things for us like which values of the nuisance parameters should be used to generate toys. so let's just change the test statistic and realize this is no longer "Feldman-Cousins" but is a fully frequentist Neyman-Construction. fc.GetTestStatSampler()->SetTestStatistic(&onesided); ((ToyMCSampler*) fc.GetTestStatSampler())->SetGenerateBinned(true);

In [ ]:
ToyMCSampler *toymcsampler = (ToyMCSampler *)fc.GetTestStatSampler();
ProfileLikelihoodTestStat *testStat = dynamic_cast<ProfileLikelihoodTestStat *>(toymcsampler->GetTestStatistic());

Since this tool needs to throw toy mc the pdf needs to be extended or the tool needs to know how many entries in a dataset per pseudo experiment. In the 'number counting form' where the entries in the dataset are counts, and not values of discriminating variables, the datasets typically only have one entry and the PDF is not extended.

In [ ]:
if (!mc->GetPdf()->canBeExtended()) {
   if (data->numEntries() == 1)
      fc.FluctuateNumDataEntries(false);
   else
      cout << "Not sure what to do about this model" << endl;
}

We can use proof to speed things along in parallel However, the test statistic has to be installed on the workers so either turn off PROOF or include the modified test statistic in your $ROOTSYS/roofit/roostats/inc directory, add the additional line to the LinkDef.h file, and recompile root.

In [ ]:
if (useProof) {
   ProofConfig pc(*w, nworkers, "", false);
   toymcsampler->SetProofConfig(&pc); // enable proof
}

if (mc->GetGlobalObservables()) {
   cout << "will use global observables for unconditional ensemble" << endl;
   mc->GetGlobalObservables()->Print();
   toymcsampler->SetGlobalObservables(*mc->GetGlobalObservables());
}

Now get the interval

In [ ]:
PointSetInterval *interval = fc.GetInterval();
ConfidenceBelt *belt = fc.GetConfidenceBelt();

Print out the interval on the first parameter of interest

In [ ]:
cout << "\n95% interval on " << firstPOI->GetName() << " is : [" << interval->LowerLimit(*firstPOI) << ", "
     << interval->UpperLimit(*firstPOI) << "] " << endl;

Get observed ul and value of test statistic evaluated there

In [ ]:
RooArgSet tmpPOI(*firstPOI);
double observedUL = interval->UpperLimit(*firstPOI);
firstPOI->setVal(observedUL);
double obsTSatObsUL = fc.GetTestStatSampler()->EvaluateTestStatistic(*data, tmpPOI);

Ask the calculator which points were scanned

In [ ]:
RooDataSet *parameterScan = (RooDataSet *)fc.GetPointsToScan();
RooArgSet *tmpPoint;

Make a histogram of parameter vs. threshold

In [ ]:
TH1F *histOfThresholds =
   new TH1F("histOfThresholds", "", parameterScan->numEntries(), firstPOI->getMin(), firstPOI->getMax());
histOfThresholds->GetXaxis()->SetTitle(firstPOI->GetName());
histOfThresholds->GetYaxis()->SetTitle("Threshold");

Loop through the points that were tested and ask confidence belt what the upper/lower thresholds were. For FeldmanCousins, the lower cut off is always 0

In [ ]:
for (Int_t i = 0; i < parameterScan->numEntries(); ++i) {
   tmpPoint = (RooArgSet *)parameterScan->get(i)->clone("temp");
   // cout <<"get threshold"<<endl;
   double arMax = belt->GetAcceptanceRegionMax(*tmpPoint);
   double poiVal = tmpPoint->getRealValue(firstPOI->GetName());
   histOfThresholds->Fill(poiVal, arMax);
}
TCanvas *c1 = new TCanvas();
c1->Divide(2);
c1->cd(1);
histOfThresholds->SetMinimum(0);
histOfThresholds->Draw();
c1->cd(2);

Now we generate the expected bands and power-constraint

First: find parameter point for mu=0, with conditional mles for nuisance parameters

In [ ]:
RooAbsReal *nll = mc->GetPdf()->createNLL(*data);
RooAbsReal *profile = nll->createProfile(*mc->GetParametersOfInterest());
firstPOI->setVal(0.);
profile->getVal(); // this will do fit and set nuisance parameters to profiled values
RooArgSet *poiAndNuisance = new RooArgSet();
if (mc->GetNuisanceParameters())
   poiAndNuisance->add(*mc->GetNuisanceParameters());
poiAndNuisance->add(*mc->GetParametersOfInterest());
w->saveSnapshot("paramsToGenerateData", *poiAndNuisance);
RooArgSet *paramsToGenerateData = (RooArgSet *)poiAndNuisance->snapshot();
cout << "\nWill use these parameter points to generate pseudo data for bkg only" << endl;
paramsToGenerateData->Print("v");

RooArgSet unconditionalObs;
unconditionalObs.add(*mc->GetObservables());
unconditionalObs.add(*mc->GetGlobalObservables()); // comment this out for the original conditional ensemble

double CLb = 0;
double CLbinclusive = 0;

Now we generate background only and find distribution of upper limits

In [ ]:
TH1F *histOfUL = new TH1F("histOfUL", "", 100, 0, firstPOI->getMax());
histOfUL->GetXaxis()->SetTitle("Upper Limit (background only)");
histOfUL->GetYaxis()->SetTitle("Entries");
for (int imc = 0; imc < nToyMC; ++imc) {

   // set parameters back to values for generating pseudo data
   //    cout << "\n get current nuis, set vals, print again" << endl;
   w->loadSnapshot("paramsToGenerateData");
   //    poiAndNuisance->Print("v");

   RooDataSet *toyData = 0;
   // now generate a toy dataset for the main measurement
   if (!mc->GetPdf()->canBeExtended()) {
      if (data->numEntries() == 1)
         toyData = mc->GetPdf()->generate(*mc->GetObservables(), 1);
      else
         cout << "Not sure what to do about this model" << endl;
   } else {
      //      cout << "generating extended dataset"<<endl;
      toyData = mc->GetPdf()->generate(*mc->GetObservables(), Extended());
   }

   // generate global observables
   // need to be careful for simpdf.
   // In ROOT 5.28 there is a problem with generating global observables
   // with a simultaneous PDF.  In 5.29 there is a solution with
   // RooSimultaneous::generateSimGlobal, but this may change to
   // the standard generate interface in 5.30.

   RooSimultaneous *simPdf = dynamic_cast<RooSimultaneous *>(mc->GetPdf());
   if (!simPdf) {
      RooDataSet *one = mc->GetPdf()->generate(*mc->GetGlobalObservables(), 1);
      const RooArgSet *values = one->get();
      RooArgSet *allVars = mc->GetPdf()->getVariables();
      *allVars = *values;
      delete allVars;
      delete one;
   } else {
      RooDataSet *one = simPdf->generateSimGlobal(*mc->GetGlobalObservables(), 1);
      const RooArgSet *values = one->get();
      RooArgSet *allVars = mc->GetPdf()->getVariables();
      *allVars = *values;
      delete allVars;
      delete one;
   }

   // get test stat at observed UL in observed data
   firstPOI->setVal(observedUL);
   double toyTSatObsUL = fc.GetTestStatSampler()->EvaluateTestStatistic(*toyData, tmpPOI);
   //    toyData->get()->Print("v");
   //    cout <<"obsTSatObsUL " <<obsTSatObsUL << "toyTS " << toyTSatObsUL << endl;
   if (obsTSatObsUL < toyTSatObsUL) // not sure about <= part yet
      CLb += (1.) / nToyMC;
   if (obsTSatObsUL <= toyTSatObsUL) // not sure about <= part yet
      CLbinclusive += (1.) / nToyMC;

   // loop over points in belt to find upper limit for this toy data
   double thisUL = 0;
   for (Int_t i = 0; i < parameterScan->numEntries(); ++i) {
      tmpPoint = (RooArgSet *)parameterScan->get(i)->clone("temp");
      double arMax = belt->GetAcceptanceRegionMax(*tmpPoint);
      firstPOI->setVal(tmpPoint->getRealValue(firstPOI->GetName()));
      //   double thisTS = profile->getVal();
      double thisTS = fc.GetTestStatSampler()->EvaluateTestStatistic(*toyData, tmpPOI);

      //   cout << "poi = " << firstPOI->getVal()
      // << " max is " << arMax << " this profile = " << thisTS << endl;
      //      cout << "thisTS = " << thisTS<<endl;
      if (thisTS <= arMax) {
         thisUL = firstPOI->getVal();
      } else {
         break;
      }
   }

   histOfUL->Fill(thisUL);

   // for few events, data is often the same, and UL is often the same
   //    cout << "thisUL = " << thisUL<<endl;

   delete toyData;
}
histOfUL->Draw();
c1->SaveAs("two-sided_upper_limit_output.pdf");

If you want to see a plot of the sampling distribution for a particular scan point:

In [ ]:
/*
SamplingDistPlot sampPlot;
int indexInScan = 0;
tmpPoint = (RooArgSet*) parameterScan->get(indexInScan)->clone("temp");
firstPOI->setVal( tmpPoint->getRealValue(firstPOI->GetName()) );
toymcsampler->SetParametersForTestStat(tmpPOI);
SamplingDistribution* samp = toymcsampler->GetSamplingDistribution(*tmpPoint);
sampPlot.AddSamplingDistribution(samp);
sampPlot.Draw();
   */

Now find bands and power constraint

In [ ]:
Double_t *bins = histOfUL->GetIntegral();
TH1F *cumulative = (TH1F *)histOfUL->Clone("cumulative");
cumulative->SetContent(bins);
double band2sigDown = 0, band1sigDown = 0, bandMedian = 0, band1sigUp = 0, band2sigUp = 0;
for (int i = 1; i <= cumulative->GetNbinsX(); ++i) {
   if (bins[i] < RooStats::SignificanceToPValue(2))
      band2sigDown = cumulative->GetBinCenter(i);
   if (bins[i] < RooStats::SignificanceToPValue(1))
      band1sigDown = cumulative->GetBinCenter(i);
   if (bins[i] < 0.5)
      bandMedian = cumulative->GetBinCenter(i);
   if (bins[i] < RooStats::SignificanceToPValue(-1))
      band1sigUp = cumulative->GetBinCenter(i);
   if (bins[i] < RooStats::SignificanceToPValue(-2))
      band2sigUp = cumulative->GetBinCenter(i);
}
cout << "-2 sigma  band " << band2sigDown << endl;
cout << "-1 sigma  band " << band1sigDown << " [Power Constraint)]" << endl;
cout << "median of band " << bandMedian << endl;
cout << "+1 sigma  band " << band1sigUp << endl;
cout << "+2 sigma  band " << band2sigUp << endl;

Print out the interval on the first parameter of interest

In [ ]:
cout << "\nobserved 95% upper-limit " << interval->UpperLimit(*firstPOI) << endl;
cout << "CLb strict [P(toy>obs|0)] for observed 95% upper-limit " << CLb << endl;
cout << "CLb inclusive [P(toy>=obs|0)] for observed 95% upper-limit " << CLbinclusive << endl;

delete profile;
delete nll;

Draw all canvases

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