/*
OneSidedFrequentistUpperLimitWithBands
Author: Kyle Cranmer,
Contributions from Haichen Wang and Daniel Whiteson
date: Dec. 2010 - Feb. 2011
v1. Jan 28
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.
The first ~100 lines define a new test statistic, then the main macro starts.
You may want to control:
double confidenceLevel=0.95;
int nPointsToScan = 30;
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.
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.
This is done by hand for now, will later be part of the RooStats tools.
On a technical note, this technique is NOT the Feldman-Cousins technique,
because that is a 2-sided interval BY DEFINITION. However, like the
Feldman-Cousins technique this is a Neyman-Construction. For technical
reasons the easiest way to implement this right now is to use the
FeldmanCousins tool and then change the test statistic that it is using.
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 one-sided test statistic starts off very small because we
are only including downward fluctuations. You can see the threshold in these printouts:
[#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
controled, 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. This can be seen in the following
thought experiment. Say the model is
Pois(N | s + b) * G(b0|b,sigma)
where G(b0|b,sigma) is the external constraint and b0 is 100. If N is also 100
then the profiled value of b given s is going to be some trade off betwen 100-s and b0.
If sigma is \sqrt(N), then the profiled value of b is probably 100 - s/2 So for
s=60 we are going to have a profiled value of b~70. Now when we generate pseudo-experiments
for s=60, b=70 we will have N~130 and the average shat will be 30, not 60. In practice,
this is only an issue for values of s that are very excluded. For values of s near the 95%
limit this should not be a big effect. This can be avoided if the nominal values of the constraints also fluctuate, but that requires that those parameters are RooRealVars in the model.
This version does not deal with this issue, but it will be addressed in a future version.
*/
#include "TFile.h"
#include "TROOT.h"
#include "TH1F.h"
#include "TCanvas.h"
#include "TSystem.h"
#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"
using namespace RooFit;
using namespace RooStats;
/////////////////////////////////////////////////////////////////////////
// The actual macro
void OneSidedFrequentistUpperLimitWithBands(const char* infile = "",
const char* workspaceName = "combined",
const char* modelConfigName = "ModelConfig",
const char* dataName = "obsData"){
#ifdef __CINT__
cout << "DO NOT RUN WITH CINT: we are using a custom test statistic ";
cout << "which requires that this tutorial must be compiled ";
cout << "with ACLIC" << endl;
return;
#endif
double confidenceLevel=0.95;
int nPointsToScan = 30;
int nToyMC = 500;
/////////////////////////////////////////////////////////////
// First part is just to access a user-defined file
// or create the standard example file if it doesn't exist
////////////////////////////////////////////////////////////
const char* filename = "";
if (!strcmp(infile,""))
filename = "results/example_combined_GaussExample_model.root";
else
filename = infile;
// Check if example input file exists
TFile *file = TFile::Open(filename);
// if input file was specified byt not found, quit
if(!file && strcmp(infile,"")){
cout <<"file not found" << endl;
return;
}
// if default file not found, try to create it
if(!file ){
// 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;
}
// now try to access the file again
file = TFile::Open(filename);
if(!file){
// if it is still not there, then we can't continue
cout << "Not able to run hist2workspace to create example input" <<endl;
return;
}
/////////////////////////////////////////////////////////////
// Now get the data and workspace
////////////////////////////////////////////////////////////
// get the workspace out of the file
RooWorkspace* w = (RooWorkspace*) file->Get(workspaceName);
if(!w){
cout <<"workspace not found" << endl;
return;
}
// get the modelConfig out of the file
ModelConfig* mc = (ModelConfig*) w->obj(modelConfigName);
// get the modelConfig out of the file
RooAbsData* data = w->data(dataName);
// make sure ingredients are found
if(!data || !mc){
w->Print();
cout << "data or ModelConfig was not found" <<endl;
return;
}
/////////////////////////////////////////////////////////////
// Now get the POI for convenience
// you may want to adjust the range of your POI
////////////////////////////////////////////////////////////
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
FeldmanCousins fc(*data,*mc);
fc.SetConfidenceLevel(confidenceLevel);
// fc.AdditionalNToysFactor(0.25); // degrade/improve sampling that defines confidence belt
// fc.UseAdaptiveSampling(true); // speed it up a bit, don't use for expectd limits
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.
// ProfileLikelihoodTestStatModified onesided(*mc->GetPdf());
// fc.GetTestStatSampler()->SetTestStatistic(&onesided);
// ((ToyMCSampler*) fc.GetTestStatSampler())->SetGenerateBinned(true);
ToyMCSampler* toymcsampler = (ToyMCSampler*) fc.GetTestStatSampler();
ProfileLikelihoodTestStat* testStat = dynamic_cast<ProfileLikelihoodTestStat*>(toymcsampler->GetTestStatistic());
testStat->SetOneSided(true);
// 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.
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.
ProofConfig pc(*w, 4, "workers=4");
if(mc->GetGlobalObservables()){
cout << "will use global observables for unconditional ensemble"<<endl;
mc->GetGlobalObservables()->Print();
toymcsampler->SetGlobalObservables(*mc->GetGlobalObservables());
}
toymcsampler->SetProofConfig(&pc); // enable proof
// Now get the interval
PointSetInterval* interval = fc.GetInterval();
ConfidenceBelt* belt = fc.GetConfidenceBelt();
// print out the iterval on the first Parameter of Interest
cout << "\n95% interval on " <<firstPOI->GetName()<<" is : ["<<
interval->LowerLimit(*firstPOI) << ", "<<
interval->UpperLimit(*firstPOI) <<"] "<<endl;
// get observed UL and value of test statistic evaluated there
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
RooDataSet* parameterScan = (RooDataSet*) fc.GetPointsToScan();
RooArgSet* tmpPoint;
// make a histogram of parameter vs. threshold
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
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-constriant
////////////////////////////////////////////////////////////
// First: find parameter point for mu=0, with conditional MLEs for nuisance parameters
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
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
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
// RooDataSet* globalData = mc->GetPdf()->generate(*mc->GetGlobalObservables(),1);
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 values;
delete one;
} else {
//try fix for sim pdf
TIterator* iter = simPdf->indexCat().typeIterator() ;
RooCatType* tt = NULL;
while((tt=(RooCatType*) iter->Next())) {
// Get pdf associated with state from simpdf
RooAbsPdf* pdftmp = simPdf->getPdf(tt->GetName()) ;
// Generate only global variables defined by the pdf associated with this state
RooArgSet* globtmp = pdftmp->getObservables(*mc->GetGlobalObservables()) ;
RooDataSet* tmp = pdftmp->generate(*globtmp,1) ;
// Transfer values to output placeholder
*globtmp = *tmp->get(0) ;
// Cleanup
delete globtmp ;
delete tmp ;
}
}
// globalData->Print("v");
// unconditionalObs = *globalData->get();
// mc->GetGlobalObservables()->Print("v");
// delete globalData;
// cout << "toy data = " << endl;
// toyData->get()->Print("v");
// 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;
}
}
/*
// loop over points in belt to find upper limit for this toy data
double thisUL = 0;
for(Int_t i=0; i<histOfThresholds->GetNbinsX(); ++i){
tmpPoint = (RooArgSet*) parameterScan->get(i)->clone("temp");
cout <<"---------------- "<<i<<endl;
tmpPoint->Print("v");
cout << "from hist " << histOfThresholds->GetBinCenter(i+1) <<endl;
double arMax = histOfThresholds->GetBinContent(i+1);
// cout << " threhold from Hist = aMax " << arMax<<endl;
// double arMax2 = belt->GetAcceptanceRegionMax(*tmpPoint);
// cout << "from scan arMax2 = "<< arMax2 << endl; // not the same due to TH1F not TH1D
// cout << "scan - hist" << arMax2-arMax << endl;
firstPOI->setVal( histOfThresholds->GetBinCenter(i+1));
// 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;
// NOTE: need to add a small epsilon term for single precision vs. double precision
if(thisTS<=arMax + 1e-7){
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("one-sided_upper_limit_output.pdf");
// if you want to see a plot of the sampling distribution for a particular scan point:
/*
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
Double_t* bins = histOfUL->GetIntegral();
TH1F* cumulative = (TH1F*) histOfUL->Clone("cumulative");
cumulative->SetContent(bins);
double band2sigDown, band1sigDown, bandMedian, band1sigUp,band2sigUp;
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 Constriant)]" << endl;
cout << "median of band " << bandMedian << endl;
cout << "+1 sigma band " << band1sigUp << endl;
cout << "+2 sigma band " << band2sigUp << endl;
// print out the iterval on the first Parameter of Interest
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;
}
