A hypothesis testing example based on number counting with background uncertainty.
A hypothesis testing example based on number counting with background uncertainty.
NOTE: This example is like HybridInstructional, but the model is more clearly generalizable to an analysis with shapes. There is a lot of flexibility for how one models a problem in RooFit/RooStats. Models come in a few common forms:
eg: P(m) ~ Sfs(m) + Bfb(m), with S+B events expected in this case the dataset has N rows corresponding to N events and the extended term is Pois(N|S+B)
eg: P(m) ~ sfs(m) + (1-s)fb(m), where s is a signal fraction in this case the dataset has N rows corresponding to N events and there is no extended term
and the counts are modeled directly (see HybridInstructional) eg: P(N) = Pois(N|S+B) in this case the dataset has 1 row corresponding to N events and the extended term is the PDF itself.
Here we convert the number counting form into the standard form by introducing a dummy discriminating variable m with a uniform distribution.
This example:
The basic setup here is that a main measurement has observed x events with an expectation of s+b. One can choose an ad hoc prior for the uncertainty on b, or try to base it on an auxiliary measurement. In this case, the auxiliary measurement (aka control measurement, sideband) is another counting experiment with measurement y and expectation tau*b. With an 'original prior' on b, called $ \eta(b) $ then one can obtain a posterior from the auxiliary measurement on b in the main measurement of x, which can then be treated in a hybrid Bayesian/Frequentist way. Additionally, one can try to treat the two measurements simultaneously, which is detailed in Part 6 of the tutorial.
This tutorial is related to the FourBin.C tutorial in the modeling, but focuses on hypothesis testing instead of interval estimation.
More background on this 'prototype problem' can be found in the following papers:
Evaluation of three methods for calculating statistical significance when incorporating a systematic uncertainty into a test of the background-only hypothesis for a Poisson process Authors: Robert D. Cousins, James T. Linnemann, Jordan Tucker http://arxiv.org/abs/physics/0702156 NIM A 595 (2008) 480--501
Statistical Challenges for Searches for New Physics at the LHC Author: Kyle Cranmer http://arxiv.org/abs/physics/0511028
Measures of Significance in HEP and Astrophysics Author: J. T. Linnemann http://arxiv.org/abs/physics/0312059
Author: Kyle Cranmer, Wouter Verkerke, and Sven Kreiss
This notebook tutorial was automatically generated with ROOTBOOK-izer from the macro found in the ROOT repository on Tuesday, March 19, 2024 at 07:18 PM.
%%cpp -d
#include "RooGlobalFunc.h"
#include "RooRealVar.h"
#include "RooProdPdf.h"
#include "RooWorkspace.h"
#include "RooDataSet.h"
#include "RooDataHist.h"
#include "TCanvas.h"
#include "TStopwatch.h"
#include "TH1.h"
#include "RooPlot.h"
#include "RooMsgService.h"
#include "RooStats/NumberCountingUtils.h"
#include "RooStats/HybridCalculator.h"
#include "RooStats/ToyMCSampler.h"
#include "RooStats/HypoTestPlot.h"
#include "RooStats/NumEventsTestStat.h"
#include "RooStats/ProfileLikelihoodTestStat.h"
#include "RooStats/SimpleLikelihoodRatioTestStat.h"
#include "RooStats/RatioOfProfiledLikelihoodsTestStat.h"
#include "RooStats/MaxLikelihoodEstimateTestStat.h"
using namespace RooFit;
using namespace RooStats;
A New Test Statistic Class for this example. It simply returns the sum of the values in a particular column of a dataset. You can ignore this class and focus on the macro below
class BinCountTestStat : public TestStatistic {
public:
BinCountTestStat(void) : fColumnName("tmp") {}
BinCountTestStat(string columnName) : fColumnName(columnName) {}
virtual Double_t Evaluate(RooAbsData &data, RooArgSet & /*nullPOI*/)
{
// This is the main method in the interface
Double_t value = 0.0;
for (int i = 0; i < data.numEntries(); i++) {
value += data.get(i)->getRealValue(fColumnName.c_str());
}
return value;
}
virtual const TString GetVarName() const { return fColumnName; }
private:
string fColumnName;
protected:
ClassDef(BinCountTestStat, 1)
};
ClassImp(BinCountTestStat)
Arguments are defined.
int ntoys = 6000;
double nToysRatio = 20; // ratio Ntoys Null/ntoys ALT
This tutorial has 6 parts Table of Contents Setup
Special cases 2. NOT RELEVANT HERE 3. Use RooStats analytic solution for this problem RooStats HybridCalculator -- can be generalized 4. RooStats ToyMC version of 2. & 3. 5. RooStats ToyMC with an equivalent test statistic 6. RooStats ToyMC with simultaneous control & main measurement
Part 4 takes ~4 min without PROOF. Part 5 takes about ~2 min with PROOF on 4 cores. Of course, everything looks nicer with more toys, which takes longer.
TStopwatch t;
t.Start();
TCanvas *c = new TCanvas;
c->Divide(2, 2);
Make model for prototype on/off problem Pois(x | s+b) * Pois(y | tau b ) for Z_Gamma, use uniform prior on b.
RooWorkspace *w = new RooWorkspace("w");
replace the pdf in 'number counting form' w->factory("Poisson::px(x[150,0,500],sum::splusb(s[0,0,100],b[100,0,300]))"); with one in standard form. Now x is encoded in event count
w->factory("Uniform::f(m[0,1])"); // m is a dummy discriminating variable
w->factory("ExtendPdf::px(f,sum::splusb(s[0,0,100],b[100,0.1,300]))");
w->factory("Poisson::py(y[100,0.1,500],prod::taub(tau[1.],b))");
w->factory("PROD::model(px,py)");
w->factory("Uniform::prior_b(b)");
We will control the output level in a few places to avoid verbose progress messages. We start by keeping track of the current threshold on messages.
RooFit::MsgLevel msglevel = RooMsgService::instance().globalKillBelow();
Use PROOF-lite on multi-core machines
ProofConfig *pc = NULL;
uncomment below if you want to use PROOF
pc = new ProofConfig(*w, 4, "workers=4", kFALSE); // machine with 4 cores
pc = new ProofConfig(*w, 2, "workers=2", kFALSE); // machine with 2 cores
In this special case, the integrals are known analytically and they are implemented in RooStats::NumberCountingUtils
analytic Z_Bi
double p_Bi = NumberCountingUtils::BinomialWithTauObsP(150, 100, 1);
double Z_Bi = NumberCountingUtils::BinomialWithTauObsZ(150, 100, 1);
cout << "-----------------------------------------" << endl;
cout << "Part 3" << endl;
std::cout << "Z_Bi p-value (analytic): " << p_Bi << std::endl;
std::cout << "Z_Bi significance (analytic): " << Z_Bi << std::endl;
t.Stop();
t.Print();
t.Reset();
t.Start();
----------------------------------------- Part 3 Z_Bi p-value (analytic): 0.00094165 Z_Bi significance (analytic): 3.10804 Real time 0:00:01, CP time 0.620
Now we demonstrate the RooStats HybridCalculator.
Like all RooStats calculators it needs the data and a ModelConfig for the relevant hypotheses. Since we are doing hypothesis testing we need a ModelConfig for the null (background only) and the alternate (signal+background) hypotheses. We also need to specify the PDF, the parameters of interest, and the observables. Furthermore, since the parameter of interest is floating, we need to specify which values of the parameter corresponds to the null and alternate (eg. s=0 and s=50)
define some sets of variables obs={x} and poi={s} note here, x is the only observable in the main measurement and y is treated as a separate measurement, which is used to produce the prior that will be used in this calculation to randomize the nuisance parameters.
w->defineSet("obs", "m");
w->defineSet("poi", "s");
create a toy dataset with the x=150 RooDataSet data = new RooDataSet("d", "d", w->set("obs")); data->add(*w->set("obs"));
std::unique_ptr<RooDataSet> data{w->pdf("px")->generate(*w->set("obs"), 150)};
input_line_66:2:2: warning: 'data' shadows a declaration with the same name in the 'std' namespace; use '::data' to reference this declaration std::unique_ptr<RooDataSet> data{w->pdf("px")->generate(*w->set("obs"), 150)}; ^
create the null (background-only) ModelConfig with s=0
ModelConfig b_model("B_model", w);
b_model.SetPdf(*w->pdf("px"));
b_model.SetObservables(*w->set("obs"));
b_model.SetParametersOfInterest(*w->set("poi"));
w->var("s")->setVal(0.0); // important!
b_model.SetSnapshot(*w->set("poi"));
create the alternate (signal+background) ModelConfig with s=50
ModelConfig sb_model("S+B_model", w);
sb_model.SetPdf(*w->pdf("px"));
sb_model.SetObservables(*w->set("obs"));
sb_model.SetParametersOfInterest(*w->set("poi"));
w->var("s")->setVal(50.0); // important!
sb_model.SetSnapshot(*w->set("poi"));
To make an equivalent calculation we need to use x as the test statistic. This is not a built-in test statistic in RooStats so we define it above. The new class inherits from the RooStats::TestStatistic interface, and simply returns the value of x in the dataset.
NumEventsTestStat eventCount(*w->pdf("px"));
The prior used for the hybrid calculator is the posterior from the auxiliary measurement y. The model for the aux. measurement is Pois(y|taub), thus the likelihood function is proportional to (has the form of) a Gamma distribution. if the 'original prior' $\eta(b)$ is uniform, then from Bayes's theorem we have the posterior: $\pi(b) = Pois(y|taub) * \eta(b)$ If $\eta(b)$ is flat, then we arrive at a Gamma distribution. Since RooFit will normalize the PDF we can actually supply py=Pois(y,tau*b) that will be equivalent to multiplying by a uniform.
Alternatively, we could explicitly use a gamma distribution:
w->factory("Gamma::gamma(b,sum::temp(y,1),1,0)");
or we can use some other ad hoc prior that do not naturally follow from the known form of the auxiliary measurement. The common choice is the equivalent Gaussian:
w->factory("Gaussian::gauss_prior(b,y, expr::sqrty('sqrt(y)',y))");
this corresponds to the "Z_N" calculation.
or one could use the analogous log-normal prior
w->factory("Lognormal::lognorm_prior(b,y, expr::kappa('1+1./sqrt(y)',y))");
Ideally, the HybridCalculator would be able to inspect the full model Pois(x | s+b) * Pois(y | tau b ) and be given the original prior $\eta(b)$ to form $\pi(b) = Pois(y|tau*b) * \eta(b)$. This is not yet implemented because in the general case it is not easy to identify the terms in the PDF that correspond to the auxiliary measurement. So for now, it must be set explicitly with:
the name "ForcePriorNuisance" was chosen because we anticipate this to be auto-detected, but will leave the option open to force to a different prior for the nuisance parameters.
HybridCalculator hc1(*data, sb_model, b_model);
ToyMCSampler *toymcs1 = (ToyMCSampler *)hc1.GetTestStatSampler();
input_line_82:2:24: error: reference to 'data' is ambiguous HybridCalculator hc1(*data, sb_model, b_model); ^ input_line_66:2:30: note: candidate found by name lookup is 'data' std::unique_ptr<RooDataSet> data{w->pdf("px")->generate(*w->set("obs"), 150)}; ^ /usr/include/c++/9/bits/range_access.h:318:5: note: candidate found by name lookup is 'std::data' data(initializer_list<_Tp> __il) noexcept ^ /usr/include/c++/9/bits/range_access.h:289:5: note: candidate found by name lookup is 'std::data' data(_Container& __cont) noexcept(noexcept(__cont.data())) ^ /usr/include/c++/9/bits/range_access.h:299:5: note: candidate found by name lookup is 'std::data' data(const _Container& __cont) noexcept(noexcept(__cont.data())) ^ /usr/include/c++/9/bits/range_access.h:309:5: note: candidate found by name lookup is 'std::data' data(_Tp (&__array)[_Nm]) noexcept ^
toymcs1->SetNEventsPerToy(1); // because the model is in number counting form
toymcs1->SetTestStatistic(&eventCount); // set the test statistic
input_line_84:2:3: error: use of undeclared identifier 'toymcs1' (toymcs1->SetTestStatistic(&((*(NumEventsTestStat*)0x7fc3e8043000)))) ^ Error in <HandleInterpreterException>: Error evaluating expression (toymcs1->SetTestStatistic(&((*(NumEventsTestStat*)0x7fc3e8043000)))) Execution of your code was aborted.
toymcs1->SetGenerateBinned();
hc1.SetToys(ntoys, ntoys / nToysRatio);
hc1.ForcePriorNuisanceAlt(*w->pdf("py"));
hc1.ForcePriorNuisanceNull(*w->pdf("py"));
input_line_86:2:3: error: use of undeclared identifier 'hc1' (hc1.SetToys(((*(int*)0x7fc3f00ab000)), ((*(int*)0x7fc3f00ab000)) / ((*(double*)0x7fc3f00a8000)))) ^ Error in <HandleInterpreterException>: Error evaluating expression (hc1.SetToys(((*(int*)0x7fc3f00ab000)), ((*(int*)0x7fc3f00ab000)) / ((*(double*)0x7fc3f00a8000)))) Execution of your code was aborted.
if you wanted to use the ad hoc Gaussian prior instead
hc1.ForcePriorNuisanceAlt(*w->pdf("gauss_prior"));
hc1.ForcePriorNuisanceNull(*w->pdf("gauss_prior"));
if you wanted to use the ad hoc log-normal prior instead
hc1.ForcePriorNuisanceAlt(*w->pdf("lognorm_prior"));
hc1.ForcePriorNuisanceNull(*w->pdf("lognorm_prior"));
enable proof proof not enabled for this test statistic if(pc) toymcs1->SetProofConfig(pc);
these lines save current msg level and then kill any messages below ERROR
RooMsgService::instance().setGlobalKillBelow(RooFit::ERROR);
Get the result
HypoTestResult *r1 = hc1.GetHypoTest();
RooMsgService::instance().setGlobalKillBelow(msglevel); // set it back
cout << "-----------------------------------------" << endl;
cout << "Part 4" << endl;
r1->Print();
t.Stop();
t.Print();
t.Reset();
t.Start();
c->cd(2);
HypoTestPlot *p1 = new HypoTestPlot(*r1, 30); // 30 bins, TS is discrete
p1->Draw();
return; // keep the running time sort by default
input_line_89:2:3: error: use of undeclared identifier 'hc1' (hc1.GetHypoTest()) ^ Error in <HandleInterpreterException>: Error evaluating expression (hc1.GetHypoTest()) Execution of your code was aborted.
SimpleLikelihoodRatioTestStat slrts(*b_model.GetPdf(), *sb_model.GetPdf());
slrts.SetNullParameters(*b_model.GetSnapshot());
slrts.SetAltParameters(*sb_model.GetSnapshot());
HYBRID CALCULATOR
HybridCalculator hc2(*data, sb_model, b_model);
ToyMCSampler *toymcs2 = (ToyMCSampler *)hc2.GetTestStatSampler();
input_line_91:2:24: error: reference to 'data' is ambiguous HybridCalculator hc2(*data, sb_model, b_model); ^ input_line_66:2:30: note: candidate found by name lookup is 'data' std::unique_ptr<RooDataSet> data{w->pdf("px")->generate(*w->set("obs"), 150)}; ^ /usr/include/c++/9/bits/range_access.h:318:5: note: candidate found by name lookup is 'std::data' data(initializer_list<_Tp> __il) noexcept ^ /usr/include/c++/9/bits/range_access.h:289:5: note: candidate found by name lookup is 'std::data' data(_Container& __cont) noexcept(noexcept(__cont.data())) ^ /usr/include/c++/9/bits/range_access.h:299:5: note: candidate found by name lookup is 'std::data' data(const _Container& __cont) noexcept(noexcept(__cont.data())) ^ /usr/include/c++/9/bits/range_access.h:309:5: note: candidate found by name lookup is 'std::data' data(_Tp (&__array)[_Nm]) noexcept ^
toymcs2->SetNEventsPerToy(1);
toymcs2->SetTestStatistic(&slrts);
input_line_93:2:3: error: use of undeclared identifier 'toymcs2' (toymcs2->SetTestStatistic(&((*(SimpleLikelihoodRatioTestStat*)0x7fc3e8015000)))) ^ Error in <HandleInterpreterException>: Error evaluating expression (toymcs2->SetTestStatistic(&((*(SimpleLikelihoodRatioTestStat*)0x7fc3e8015000)))) Execution of your code was aborted.
toymcs2->SetGenerateBinned();
hc2.SetToys(ntoys, ntoys / nToysRatio);
hc2.ForcePriorNuisanceAlt(*w->pdf("py"));
hc2.ForcePriorNuisanceNull(*w->pdf("py"));
input_line_95:2:3: error: use of undeclared identifier 'hc2' (hc2.SetToys(((*(int*)0x7fc3f00ab000)), ((*(int*)0x7fc3f00ab000)) / ((*(double*)0x7fc3f00a8000)))) ^ Error in <HandleInterpreterException>: Error evaluating expression (hc2.SetToys(((*(int*)0x7fc3f00ab000)), ((*(int*)0x7fc3f00ab000)) / ((*(double*)0x7fc3f00a8000)))) Execution of your code was aborted.
if you wanted to use the ad hoc Gaussian prior instead
hc2.ForcePriorNuisanceAlt(*w->pdf("gauss_prior"));
hc2.ForcePriorNuisanceNull(*w->pdf("gauss_prior"));
if you wanted to use the ad hoc log-normal prior instead
hc2.ForcePriorNuisanceAlt(*w->pdf("lognorm_prior"));
hc2.ForcePriorNuisanceNull(*w->pdf("lognorm_prior"));
enable proof
if (pc)
toymcs2->SetProofConfig(pc);
input_line_97:2:3: error: use of undeclared identifier 'toymcs2' (toymcs2->SetProofConfig(((*(ProofConfig **)0x7fc3f0047000)))) ^ Error in <HandleInterpreterException>: Error evaluating expression (toymcs2->SetProofConfig(((*(ProofConfig **)0x7fc3f0047000)))) Execution of your code was aborted.
these lines save current msg level and then kill any messages below ERROR
RooMsgService::instance().setGlobalKillBelow(RooFit::ERROR);
Get the result
HypoTestResult *r2 = hc2.GetHypoTest();
cout << "-----------------------------------------" << endl;
cout << "Part 5" << endl;
r2->Print();
t.Stop();
t.Print();
t.Reset();
t.Start();
RooMsgService::instance().setGlobalKillBelow(msglevel);
c->cd(3);
HypoTestPlot *p2 = new HypoTestPlot(*r2, 30); // 30 bins
p2->Draw();
return; // so standard tutorial runs faster
input_line_100:2:3: error: use of undeclared identifier 'hc2' (hc2.GetHypoTest()) ^ Error in <HandleInterpreterException>: Error evaluating expression (hc2.GetHypoTest()) Execution of your code was aborted.
Part 3 Z_Bi p-value (analytic): 0.00094165 Z_Bi significance (analytic): 3.10804 Real time 0:00:00, CP time 0.610
Results HybridCalculator_result:
Real time 0:04:43, CP time 283.780
Part 5
Results HybridCalculator_result:
Real time 0:02:22, CP time 0.990
LEPStatToolsForLHC https://plone4.fnal.gov:4430/P0/phystat/packages/0703002 Uses Gaussian prior CL_b = 6.218476e-04, Significance = 3.228665 sigma
Asymptotic From the value of the profile likelihood ratio (5.0338) The significance can be estimated using Wilks's theorem significance = sqrt(2*profileLR) = 3.1729 sigma
Draw all canvases
%jsroot on
gROOT->GetListOfCanvases()->Draw()