TEfficiency Class Reference

Class to handle efficiency histograms.

• I. Overview
• II. Creating a TEfficiency object
  • Example 1
  • Example 2
• III. Filling with events
• IV. Statistic options
  • Frequentist methods
  • Bayesian methods
  • IV.1 Coverage probabilities for different methods
• V. Merging and combining TEfficiency objects
  • Example
  • V.1 When should I use merging?
    • Example
  • V.2 When should I use combining?
    • Example
• VI. Further operations
  • VI.1 Information about the internal histograms
  • VI.2 Fitting
  • VI.3 Draw a TEfficiency object

I. Overview

This class handles the calculation of efficiencies and their uncertainties. It provides several statistical methods for calculating frequentist and Bayesian confidence intervals as well as a function for combining several efficiencies.

Efficiencies have a lot of applications and meanings but in principle, they can be described by the fraction of good/passed events k out of sample containing N events. One is usually interested in the dependency of the efficiency on other (binned) variables. The number of passed and total events is therefore stored internally in two histograms (TEfficiency::fTotalHistogram and TEfficiency::fPassedHistogram). Then the efficiency, as well as its upper and lower error, can be calculated for each bin individually.

As the efficiency can be regarded as a parameter of a binomial distribution, the number of passed and total events must always be integer numbers. Therefore a filling with weights is not possible. However, you can assign a global weight to each TEfficiency object (TEfficiency::SetWeight). It is necessary to create one TEfficiency object for each weight if you investigate a process involving different weights. This procedure needs more effort but enables you to re-use the filled object in cases where you want to change one or more weights. This would not be possible if all events with different weights were filled in the same histogram.

II. Creating a TEfficiency object

If you start a new analysis, it is highly recommended to use the TEfficiency class from the beginning. You can then use one of the constructors for fixed or variable bin size and your desired dimension. These constructors append the created TEfficiency object to the current directory. So it will be written automatically to a file during the next TFile::Write command.

Example: create a two-dimensional TEfficiency object with

• name = "eff"
• title = "my efficiency"
• axis titles: x, y and LaTeX-formatted epsilon as a label for Z axis
• 10 bins with constant bin width (= 1) along X axis starting at 0 (lower edge from the first bin) up to 10 (upper edge of last bin)
• 20 bins with constant bin width (= 0.5) along Y axis starting at -5 (lower edge from the first bin) up to 5 (upper edge of last bin)

    TEfficiency* pEff = new TEfficiency("eff","my efficiency;x;y;#epsilon",10,0,10,20,-5,5);
  

  If you already have two histograms filled with the number of passed and total events, you will use the constructor TEfficiency(const TH1& passed,const TH1& total) to construct the TEfficiency object. The histograms "passed" and "total" have to fulfill the conditions mentioned in TEfficiency::CheckConsistency, otherwise the construction will fail. As the histograms already exist, the new TEfficiency is by default not attached to the current directory to avoid duplication of data. If you want to store the new object anyway, you can either write it directly by calling TObject::Write or attach it to a directory using TEfficiency::SetDirectory. This also applies to TEfficiency objects created by the copy constructor TEfficiency::TEfficiency(const TEfficiency& rEff).

Example 1

TEfficiency* pEff = 0;
TFile* pFile = new TFile("myfile.root","recreate");
 
//h_pass and h_total are valid and consistent histograms
if(TEfficiency::CheckConsistency(h_pass,h_total))
{
  pEff = new TEfficiency(h_pass,h_total);
  // this will write the TEfficiency object to "myfile.root"
  // AND pEff will be attached to the current directory
  pEff->Write();
}

Example 2

TEfficiency* pEff = 0;
TFile* pFile = new TFile("myfile.root","recreate");
 
//h_pass and h_total are valid and consistent histograms
if(TEfficiency::CheckConsistency(h_pass,h_total))
{
  pEff = new TEfficiency(h_pass,h_total);
  //this will attach the TEfficiency object to the current directory
  pEff->SetDirectory(gDirectory);
  //now all objects in gDirectory will be written to "myfile.root"
  pFile->Write();
}

In case you already have two filled histograms and you only want to plot them as a graph, you should rather use TGraphAsymmErrors::TGraphAsymmErrors(const TH1* pass,const TH1* total,Option_t* opt) to create a graph object.

III. Filling with events

You can fill the TEfficiency object by calling the TEfficiency::Fill(Bool_t bPassed,Double_t x,Double_t y,Double_t z) method. The "bPassed" boolean flag indicates whether the current event is good (both histograms are filled) or not (only TEfficiency::fTotalHistogram is filled). The x, y and z variables determine the bin which is filled. For lower dimensions, the z- or even the y-value may be omitted.

{
   //canvas only needed for this documentation
   TCanvas* c1 = new TCanvas("example","",600,400);
   c1->SetFillStyle(1001);
   c1->SetFillColor(kWhite);
 
   //create one-dimensional TEfficiency object with fixed bin size
   TEfficiency* pEff = new TEfficiency("eff","my efficiency;x;#epsilon",20,0,10);
   TRandom3 rand3;
 
   bool bPassed;
   double x;
   for(int i=0; i<10000; ++i)
   {
      //simulate events with variable under investigation
      x = rand3.Uniform(10);
      //check selection: bPassed = DoesEventPassSelection(x)
      bPassed = rand3.Rndm() < TMath::Gaus(x,5,4);
      pEff->Fill(bPassed,x);
   }
 
   pEff->Draw("AP");
}

You can also set the number of passed or total events for a bin directly by using the TEfficiency::SetPassedEvents or TEfficiency::SetTotalEvents method.

IV. Statistic options

The calculation of the estimated efficiency depends on the chosen statistic option. Let k denotes the number of passed events and N the number of total events.

Frequentist methods

The expectation value of the number of passed events is given by the true efficiency times the total number of events. One can estimate the efficiency by replacing the expected number of passed events by the observed number of passed events.

\[ k = \epsilon \times N \Rightarrow \hat{\varepsilon} = \frac{k}{N} \]

Bayesian methods

In Bayesian statistics a likelihood-function (how probable is it to get the observed data assuming a true efficiency) and a prior probability (what is the probability that a certain true efficiency is actually realised) are used to determine a posterior probability by using Bayes theorem. At the moment, only beta distributions (have 2 free parameters) are supported as prior probabilities.

\begin{eqnarray*} P(\epsilon | k ; N) &=& \frac{1}{norm} \times P(k | \epsilon ; N) \times Prior(\epsilon) \\ P(k | \epsilon ; N) &=& Binomial(N,k) \times \epsilon^{k} \times (1 - \epsilon)^{N - k} ...\ binomial\ distribution \\ Prior(\epsilon) &=& \frac{1}{B(\alpha,\beta)} \times \epsilon ^{\alpha - 1} \times (1 - \epsilon)^{\beta - 1} \equiv Beta(\epsilon; \alpha,\beta) \\ \Rightarrow P(\epsilon | k ; N) &=& \frac{1}{norm'} \times \epsilon^{k + \alpha - 1} \times (1 - \epsilon)^{N - k + \beta - 1} \equiv Beta(\epsilon; k + \alpha, N - k + \beta) \end{eqnarray*}

By default the expectation value of this posterior distribution is used as an estimator for the efficiency:

\[ \hat{\varepsilon} = \frac{k + \alpha}{N + \alpha + \beta} \]

Optionally the mode can also be used as a value for the estimated efficiency. This can be done by calling SetBit(kPosteriorMode) or TEfficiency::SetPosteriorMode. In this case, the estimated efficiency is:

\[ \hat{\varepsilon} = \frac{k + \alpha -1}{N + \alpha + \beta - 2} \]

In the case of a uniform prior distribution, B(x,1,1), the posterior mode is k/n, equivalent to the frequentist estimate (the maximum likelihood value).

The statistic options also specify which confidence interval is used for calculating the uncertainties of the efficiency. The following properties define the error calculation:

• fConfLevel: desired confidence level: 0 < fConfLevel < 1 (TEfficiency::GetConfidenceLevel / TEfficiency::SetConfidenceLevel)
• fStatisticOption defines which method is used to calculate the boundaries of the confidence interval (TEfficiency::SetStatisticOption)
• fBeta_alpha, fBeta_beta: parameters for the prior distribution which is only used in the bayesian case (TEfficiency::GetBetaAlpha / TEfficiency::GetBetaBeta / TEfficiency::SetBetaAlpha / TEfficiency::SetBetaBeta)
• kIsBayesian: flag whether bayesian statistics are used or not (TEfficiency::UsesBayesianStat)
• kShortestInterval: flag whether shortest interval (instead of central one) are used in case of Bayesian statistics (TEfficiency::UsesShortestInterval). Normally shortest interval should be used in combination with the mode (see TEfficiency::UsesPosteriorMode)
• fWeight: global weight for this TEfficiency object which is used during combining or merging with other TEfficiency objects(TEfficiency::GetWeight / TEfficiency::SetWeight)

In the following table, the implemented confidence intervals are listed with their corresponding statistic option. For more details on the calculation, please have a look at the mentioned functions.

name	statistic option	function	kIsBayesian	parameters
Clopper-Pearson	kFCP	TEfficiency::ClopperPearson	false	total events, passed events, confidence level
normal approximation	kFNormal	TEfficiency::Normal	false	total events, passed events, confidence level
Wilson	kFWilson	TEfficiency::Wilson	false	total events, passed events, confidence level
Agresti-Coull	kFAC	TEfficiency::AgrestiCoull	false	total events, passed events. confidence level
Feldman-Cousins	kFFC	TEfficiency::FeldmanCousins	false	total events, passed events, confidence level
Mid-P Lancaster	kMidP	TEfficiency::MidPInterval	false	total events, passed events, confidence level
Jeffrey	kBJeffrey	TEfficiency::Bayesian	true	total events, passed events, confidence level, fBeta_alpha = 0.5, fBeta_beta = 0.5
Uniform prior	kBUniform	TEfficiency::Bayesian	true	total events, passed events, confidence level, fBeta_alpha = 1, fBeta_beta = 1
custom prior	kBBayesian	TEfficiency::Bayesian	true	total events, passed events, confidence level, fBeta_alpha, fBeta_beta

The following example demonstrates the effect of different statistic options and confidence levels.

{
   //canvas only needed for the documentation
   TCanvas* c1 = new TCanvas("c1","",600,400);
   c1->Divide(2);
   c1->SetFillStyle(1001);
   c1->SetFillColor(kWhite);
 
   //create one-dimensional TEfficiency object with fixed bin size
   TEfficiency* pEff = new TEfficiency("eff","different confidence levels;x;#epsilon",20,0,10);
   TRandom3 rand3;
 
   bool bPassed;
   double x;
   for(int i=0; i<1000; ++i)
   {
      //simulate events with variable under investigation
      x = rand3.Uniform(10);
      //check selection: bPassed = DoesEventPassSelection(x)
      bPassed = rand3.Rndm() < TMath::Gaus(x,5,4);
      pEff->Fill(bPassed,x);
   }
 
   //set style attributes
   pEff->SetFillStyle(3004);
   pEff->SetFillColor(kRed);
 
   //copy current TEfficiency object and set new confidence level
   TEfficiency* pCopy = new TEfficiency(*pEff);
   pCopy->SetConfidenceLevel(0.90);
 
   //set style attributes
   pCopy->SetFillStyle(3005);
   pCopy->SetFillColor(kBlue);
 
   c1->cd(1);
 
   //add legend
   TLegend* leg1 = new TLegend(0.3,0.1,0.7,0.5);
   leg1->AddEntry(pEff,"95%","F");
   leg1->AddEntry(pCopy,"68.3%","F");
 
   pEff->Draw("A4");
   pCopy->Draw("same4");
   leg1->Draw("same");
 
   //use same confidence level but different statistic methods
   TEfficiency* pEff2 = new TEfficiency(*pEff);
   TEfficiency* pCopy2 = new TEfficiency(*pEff);
 
   pEff2->SetStatisticOption(TEfficiency::kFNormal);
   pCopy2->SetStatisticOption(TEfficiency::kFAC);
 
   pEff2->SetTitle("different statistic options;x;#epsilon");
 
   //set style attributes
   pCopy2->SetFillStyle(3005);
   pCopy2->SetFillColor(kBlue);
 
   c1->cd(2);
 
   //add legend
   TLegend* leg2 = new TLegend(0.3,0.1,0.7,0.5);
   leg2->AddEntry(pEff2,"kFNormal","F");
   leg2->AddEntry(pCopy2,"kFAC","F");
 
   pEff2->Draw("a4");
   pCopy2->Draw("same4");
   leg2->Draw("same");
}

The prior probability of the efficiency in Bayesian statistics can be given in terms of a beta distribution. The beta distribution has two positive shape parameters. The resulting priors for different combinations of these shape parameters are shown in the plot below.

{
   //canvas only needed for the documentation
   TCanvas* c1 = new TCanvas("c1","",600,400);
   c1->SetFillStyle(1001);
   c1->SetFillColor(kWhite);
 
   //create different beta distributions
   TF1* f1 = new TF1("f1","TMath::BetaDist(x,1,1)",0,1);
   f1->SetLineColor(kBlue);
   TF1* f2 = new TF1("f2","TMath::BetaDist(x,0.5,0.5)",0,1);
   f2->SetLineColor(kRed);
   TF1* f3 = new TF1("f3","TMath::BetaDist(x,1,5)",0,1);
   f3->SetLineColor(kGreen+3);
   f3->SetTitle("Beta distributions as priors;#epsilon;P(#epsilon)");
   TF1* f4 = new TF1("f4","TMath::BetaDist(x,4,3)",0,1);
   f4->SetLineColor(kViolet);
 
   //add legend
   TLegend* leg = new TLegend(0.25,0.5,0.85,0.89);
   leg->SetFillColor(kWhite);
   leg->SetFillStyle(1001);
   leg->AddEntry(f1,"a=1, b=1","L");
   leg->AddEntry(f2,"a=0.5, b=0.5","L");
   leg->AddEntry(f3,"a=1, b=5","L");
   leg->AddEntry(f4,"a=4, b=3","L");
 
   f3->Draw();
   f1->Draw("same");
   f2->Draw("Same");
   f4->Draw("same");
   leg->Draw("same");
}

IV.1 Coverage probabilities for different methods

The following pictures illustrate the actual coverage probability for the different values of the true efficiency and the total number of events when a confidence level of 95% is desired.

Normal Approximation

Wilson

Agresti Coull

Clopper Pearson

Bayesian with Uniform Prior

Bayesian with Jeffrey Prior

The average (over all possible true efficiencies) coverage probability for different number of total events is shown in the next picture.

Average Coverage

V. Merging and combining TEfficiency objects

In many applications, the efficiency should be calculated for an inhomogeneous sample in the sense that it contains events with different weights. In order to be able to determine the correct overall efficiency, it is necessary to use for each subsample (= all events with the same weight) a different TEfficiency object. After finishing your analysis you can then construct the overall efficiency with its uncertainty.

This procedure has the advantage that you can change the weight of one subsample easily without rerunning the whole analysis. On the other hand, more effort is needed to handle several TEfficiency objects instead of one histogram. In the case of many different or even continuously distributed weights, this approach becomes cumbersome. One possibility to overcome this problem is the usage of binned weights.

Example

In particle physics weights arises from the fact that you want to normalise your results to a certain reference value. A very common formula for calculating weights is

\begin{eqnarray*} w &=& \frac{\sigma L}{N_{gen} \epsilon_{trig}} \\ &-& \sigma ...\ cross\ section \\ &-& L ...\ luminosity \\ &-& N_{gen}\ ... number\ of\ generated\ events \\ &-& \epsilon_{trig}\ ...\ (known)\ trigger\ efficiency \\ \end{eqnarray*}

The reason for different weights can therefore be:

• different processes
• other integrated luminosity
• varying trigger efficiency
• different sample sizes
• ...
• or even combination of them

Depending on the actual meaning of different weights in your case, you should either merge or combine them to get the overall efficiency.

V.1 When should I use merging?

If the weights are artificial and do not represent real alternative hypotheses, you should merge the different TEfficiency objects. That means especially for the Bayesian case that the prior probability should be the same for all merged TEfficiency objects. The merging can be done by invoking one of the following operations:

• eff1.Add(eff2)
• eff1 += eff2
• eff1 = eff1 + eff2

The result of the merging is stored in the TEfficiency object which is marked bold above. The contents of the internal histograms of both TEfficiency objects are added and a new weight is assigned. The statistic options are not changed.

\[ \frac{1}{w_{new}} = \frac{1}{w_{1}} + \frac{1}{w_{2}} \]

Example:

If you use two samples with different numbers of generated events for the same process and you want to normalise both to the same integrated luminosity and trigger efficiency, the different weights then arise just from the fact that you have different numbers of events. The TEfficiency objects should be merged because the samples do not represent true alternatives. You expect the same result as if you would have a big sample with all events in it.

\[ w_{1} = \frac{\sigma L}{\epsilon N_{1}}, w_{2} = \frac{\sigma L}{\epsilon N_{2}} \Rightarrow w_{new} = \frac{\sigma L}{\epsilon (N_{1} + N_{2})} = \frac{1}{\frac{1}{w_{1}} + \frac{1}{w_{2}}} \]

V.2 When should I use combining?

You should combine TEfficiency objects whenever the weights represent alternatives processes for the efficiency. As the combination of two TEfficiency objects is not always consistent with the representation by two internal histograms, the result is not stored in a TEfficiency object but a TGraphAsymmErrors is returned which shows the estimated combined efficiency and its uncertainty for each bin. At the moment the combination method TEfficiency::Combine only supports a combination of 1-dimensional efficiencies in a Bayesian approach.

For calculating the combined efficiency and its uncertainty for each bin only Bayesian statistics is used. No frequentists methods are presently supported for computing the combined efficiency and its confidence interval. In the case of the Bayesian statistics, a combined posterior is constructed taking into account the weight of each TEfficiency object. The same prior is used for all the TEfficiency objects.

\begin{eqnarray*} P_{comb}(\epsilon | {w_{i}}, {k_{i}} , {N_{i}}) = \frac{1}{norm} \prod_{i}{L(k_{i} | N_{i}, \epsilon)}^{w_{i}} \Pi( \epsilon )\\ L(k_{i} | N_{i}, \epsilon)\ is\ the\ likelihood\ function\ for\ the\ sample\ i\ (a\ Binomial\ distribution)\\ \Pi( \epsilon)\ is\ the\ prior,\ a\ beta\ distribution\ B(\epsilon, \alpha, \beta).\\ The\ resulting\ combined\ posterior\ is \\ P_{comb}(\epsilon |{w_{i}}; {k_{i}}; {N_{i}}) = B(\epsilon, \sum_{i}{ w_{i} k_{i}} + \alpha, \sum_{i}{ w_{i}(n_{i}-k_{i})}+\beta) \\ \hat{\varepsilon} = \int_{0}^{1} \epsilon \times P_{comb}(\epsilon | {k_{i}} , {N_{i}}) d\epsilon \\ confidence\ level = 1 - \alpha \\ \frac{\alpha}{2} = \int_{0}^{\epsilon_{low}} P_{comb}(\epsilon | {k_{i}} , {N_{i}}) d\epsilon ...\ defines\ lower\ boundary \\ 1- \frac{\alpha}{2} = \int_{0}^{\epsilon_{up}} P_{comb}(\epsilon | {k_{i}} , {N_{i}}) d\epsilon ...\ defines\ upper\ boundary \end{eqnarray*}

Example:

If you use cuts to select electrons which can originate from two different processes, you can determine the selection efficiency for each process. The overall selection efficiency is then the combined efficiency. The weights to be used in the combination should be the probability that an electron comes from the corresponding process.

\[ p_{1} = \frac{\sigma_{1}}{\sigma_{1} + \sigma_{2}} = \frac{N_{1}w_{1}}{N_{1}w_{1} + N_{2}w_{2}}\\ p_{2} = \frac{\sigma_{2}}{\sigma_{1} + \sigma_{2}} = \frac{N_{2}w_{2}}{N_{1}w_{1} + N_{2}w_{2}} \]

VI. Further operations

VI.1 Information about the internal histograms

The methods TEfficiency::GetPassedHistogram and TEfficiency::GetTotalHistogram return a constant pointer to the internal histograms. They can be used to obtain information about the internal histograms (e.g., the binning, number of passed / total events in a bin, mean values...). One can obtain a clone of the internal histograms by calling TEfficiency::GetCopyPassedHisto or TEfficiency::GetCopyTotalHisto. The returned histograms are completely independent from the current TEfficiency object. By default, they are not attached to a directory to avoid the duplication of data and the user is responsible for deleting them.

//open a root file which contains a TEfficiency object
TFile* pFile = new TFile("myfile.root","update");
 
//get TEfficiency object with name "my_eff"
TEfficiency* pEff = (TEfficiency*)pFile->Get("my_eff");
 
//get clone of total histogram
TH1* clone = pEff->GetCopyTotalHisto();
 
//change clone...
//save changes of clone directly
clone->Write();
//or append it to the current directory and write the file
//clone->SetDirectory(gDirectory);
//pFile->Write();
 
//delete histogram object
delete clone;
clone = 0;

It is also possible to set the internal total or passed histogram by using the methods TEfficiency::SetPassedHistogram or TEfficiency::SetTotalHistogram.

In order to ensure the validity of the TEfficiency object, the consistency of the new histogram and the stored histogram is checked. It might be impossible sometimes to change the histograms in a consistent way. Therefore one can force the replacement by passing the "f" option. Then the user has to ensure that the other internal histogram is replaced as well and that the TEfficiency object is in a valid state.

VI.2 Fitting

The efficiency can be fitted using the TEfficiency::Fit function which internally uses the TBinomialEfficiencyFitter::Fit method. As this method is using a maximum-likelihood-fit, it is necessary to initialise the given fit function with reasonable start values. The resulting fit function is attached to the list of associated functions and will be drawn automatically during the next TEfficiency::Draw command. The list of associated function can be modified by using the pointer returned by TEfficiency::GetListOfFunctions.

{
   //canvas only needed for this documentation
   TCanvas* c1 = new TCanvas("example","",600,400);
   c1->SetFillStyle(1001);
   c1->SetFillColor(kWhite);
 
   //create one-dimensional TEfficiency object with fixed bin size
   TEfficiency* pEff = new TEfficiency("eff","my efficiency;x;#epsilon",20,0,10);
   TRandom3 rand3;
 
   bool bPassed;
   double x;
   for (int i=0; i<10000; ++i) {
      //simulate events with variable under investigation
      x = rand3.Uniform(10);
      //check selection: bPassed = DoesEventPassSelection(x)
      bPassed = rand3.Rndm() < TMath::Gaus(x,5,4);
      pEff->Fill(bPassed,x);
   }
 
   //create a function for fitting and do the fit
   TF1* f1 = new TF1("f1","gaus",0,10);
   f1->SetParameters(1,5,2);
   pEff->Fit(f1);
 
   //create a threshold function
   TF1* f2 = new TF1("thres","0.8",0,10);
   f2->SetLineColor(kRed);
   //add it to the list of functions
   //use add first because the parameters of the last function will be displayed
   pEff->GetListOfFunctions()->AddFirst(f2);
 
   pEff->Draw("AP");
}

VI.3 Draw a TEfficiency object

A TEfficiency object can be drawn by calling the usual TEfficiency::Draw method. At the moment drawing is only supported for 1- and 2-dimensional TEfficiency objects. In the 1-dimensional case, you can use the same options as for the TGraphAsymmErrors::Draw method. For 2-dimensional TEfficiency objects, you can pass the same options as for a TH2::Draw object.

Definition at line 27 of file TEfficiency.h.

Public Types
enum	EStatOption {   kFCP = 0 , kFNormal , kFWilson , kFAC ,   kFFC , kBJeffrey , kBUniform , kBBayesian ,   kMidP }
	Enumeration type for different statistic options for calculating confidence intervals kF* ... frequentist methods; kB* ... bayesian methods. More...
Public Types inherited from TObject
enum	{   kIsOnHeap = 0x01000000 , kNotDeleted = 0x02000000 , kZombie = 0x04000000 , kInconsistent = 0x08000000 ,   kBitMask = 0x00ffffff }
enum	{ kSingleKey = (1ULL << ( 0 )) , kOverwrite = (1ULL << ( 1 )) , kWriteDelete = (1ULL << ( 2 )) }
enum	EDeprecatedStatusBits { kObjInCanvas = (1ULL << ( 3 )) }
enum	EStatusBits {   kCanDelete = (1ULL << ( 0 )) , kMustCleanup = (1ULL << ( 3 )) , kIsReferenced = (1ULL << ( 4 )) , kHasUUID = (1ULL << ( 5 )) ,   kCannotPick = (1ULL << ( 6 )) , kNoContextMenu = (1ULL << ( 8 )) , kInvalidObject = (1ULL << ( 13 )) }

Public Member Functions
	TEfficiency ()
	Default constructor.
	TEfficiency (const char *name, const char *title, Int_t nbins, const Double_t *xbins)
	Create 1-dimensional TEfficiency object with variable bin size.
	TEfficiency (const char *name, const char *title, Int_t nbins, Double_t xlow, Double_t xup)
	Create 1-dimensional TEfficiency object with fixed bins size.
	TEfficiency (const char *name, const char *title, Int_t nbinsx, const Double_t *xbins, Int_t nbinsy, const Double_t *ybins)
	Create 2-dimensional TEfficiency object with variable bin size.
	TEfficiency (const char *name, const char *title, Int_t nbinsx, const Double_t *xbins, Int_t nbinsy, const Double_t *ybins, Int_t nbinsz, const Double_t *zbins)
	Create 3-dimensional TEfficiency object with variable bin size.
	TEfficiency (const char *name, const char *title, Int_t nbinsx, Double_t xlow, Double_t xup, Int_t nbinsy, Double_t ylow, Double_t yup)
	Create 2-dimensional TEfficiency object with fixed bin size.
	TEfficiency (const char *name, const char *title, Int_t nbinsx, Double_t xlow, Double_t xup, Int_t nbinsy, Double_t ylow, Double_t yup, Int_t nbinsz, Double_t zlow, Double_t zup)
	Create 3-dimensional TEfficiency object with fixed bin size.
	TEfficiency (const TEfficiency &heff)
	Copy constructor.
	TEfficiency (const TH1 &passed, const TH1 &total)
	Constructor using two existing histograms as input.
	~TEfficiency () override
	default destructor
void	Add (const TEfficiency &rEff)
void	Browse (TBrowser *) override
	Browse object. May be overridden for another default action.
TGraphAsymmErrors *	CreateGraph (Option_t *opt="") const
	Create the graph used be painted (for dim=1 TEfficiency) The return object is managed by the caller.
TH2 *	CreateHistogram (Option_t *opt="") const
	Create the histogram used to be painted (for dim=2 TEfficiency) The return object is managed by the caller.
Int_t	DistancetoPrimitive (Int_t px, Int_t py) override
	Compute distance from point px,py to a graph.
void	Draw (Option_t *opt="") override
	Draws the current TEfficiency object.
void	ExecuteEvent (Int_t event, Int_t px, Int_t py) override
	Execute action corresponding to one event.
void	Fill (Bool_t bPassed, Double_t x, Double_t y=0, Double_t z=0)
	This function is used for filling the two histograms.
void	FillWeighted (Bool_t bPassed, Double_t weight, Double_t x, Double_t y=0, Double_t z=0)
	This function is used for filling the two histograms with a weight.
Int_t	FindFixBin (Double_t x, Double_t y=0, Double_t z=0) const
	Returns the global bin number containing the given values.
TFitResultPtr	Fit (TF1 *f1, Option_t *opt="")
	Fits the efficiency using the TBinomialEfficiencyFitter class.
Double_t	GetBetaAlpha (Int_t bin=-1) const
Double_t	GetBetaBeta (Int_t bin=-1) const
Double_t	GetConfidenceLevel () const
TH1 *	GetCopyPassedHisto () const
	Returns a cloned version of fPassedHistogram.
TH1 *	GetCopyTotalHisto () const
	Returns a cloned version of fTotalHistogram.
Int_t	GetDimension () const
	returns the dimension of the current TEfficiency object
TDirectory *	GetDirectory () const
Double_t	GetEfficiency (Int_t bin) const
	Returns the efficiency in the given global bin.
Double_t	GetEfficiencyErrorLow (Int_t bin) const
	Returns the lower error on the efficiency in the given global bin.
Double_t	GetEfficiencyErrorUp (Int_t bin) const
	Returns the upper error on the efficiency in the given global bin.
Int_t	GetGlobalBin (Int_t binx, Int_t biny=0, Int_t binz=0) const
	Returns the global bin number which can be used as argument for the following functions:
TList *	GetListOfFunctions ()
TGraphAsymmErrors *	GetPaintedGraph () const
TH2 *	GetPaintedHistogram () const
const TH1 *	GetPassedHistogram () const
EStatOption	GetStatisticOption () const
const TH1 *	GetTotalHistogram () const
Double_t	GetWeight () const
TClass *	IsA () const override
Long64_t	Merge (TCollection *list)
	Merges the TEfficiency objects in the given list to the given TEfficiency object using the operator+=(TEfficiency&)
TEfficiency &	operator+= (const TEfficiency &rhs)
	Adds the histograms of another TEfficiency object to current histograms.
TEfficiency &	operator= (const TEfficiency &rhs)
	Assignment operator.
void	Paint (Option_t *opt) override
	Paints this TEfficiency object.
void	SavePrimitive (std::ostream &out, Option_t *opt="") override
	Have histograms fixed bins along each axis?
void	SetBetaAlpha (Double_t alpha)
	Sets the shape parameter α.
void	SetBetaBeta (Double_t beta)
	Sets the shape parameter β.
void	SetBetaBinParameters (Int_t bin, Double_t alpha, Double_t beta)
	Sets different shape parameter α and β for the prior distribution for each bin.
Bool_t	SetBins (Int_t nx, const Double_t *xBins)
	Set the bins for the underlined passed and total histograms If the class have been already filled the previous contents will be lost.
Bool_t	SetBins (Int_t nx, const Double_t *xBins, Int_t ny, const Double_t *yBins)
	Set the bins for the underlined passed and total histograms If the class have been already filled the previous contents will be lost.
Bool_t	SetBins (Int_t nx, const Double_t *xBins, Int_t ny, const Double_t *yBins, Int_t nz, const Double_t *zBins)
	Set the bins for the underlined passed and total histograms If the class have been already filled the previous contents will be lost.
Bool_t	SetBins (Int_t nx, Double_t xmin, Double_t xmax)
	Set the bins for the underlined passed and total histograms If the class have been already filled the previous contents will be lost.
Bool_t	SetBins (Int_t nx, Double_t xmin, Double_t xmax, Int_t ny, Double_t ymin, Double_t ymax)
	Set the bins for the underlined passed and total histograms If the class have been already filled the previous contents will be lost.
Bool_t	SetBins (Int_t nx, Double_t xmin, Double_t xmax, Int_t ny, Double_t ymin, Double_t ymax, Int_t nz, Double_t zmin, Double_t zmax)
	Set the bins for the underlined passed and total histograms If the class have been already filled the previous contents will be lost.
void	SetCentralInterval (Bool_t on=true)
void	SetConfidenceLevel (Double_t level)
	Sets the confidence level (0 < level < 1) The default value is 1-sigma :~ 0.683.
void	SetDirectory (TDirectory *dir)
	Sets the directory holding this TEfficiency object.
void	SetName (const char *name) override
	Sets the name.
Bool_t	SetPassedEvents (Int_t bin, Int_t events)
	Sets the number of passed events in the given global bin.
Bool_t	SetPassedHistogram (const TH1 &rPassed, Option_t *opt)
	Sets the histogram containing the passed events.
void	SetPosteriorAverage (Bool_t on=true)
void	SetPosteriorMode (Bool_t on=true)
void	SetShortestInterval (Bool_t on=true)
void	SetStatisticOption (EStatOption option)
	Sets the statistic option which affects the calculation of the confidence interval.
void	SetTitle (const char *title) override
	Sets the title.
Bool_t	SetTotalEvents (Int_t bin, Int_t events)
	Sets the number of total events in the given global bin.
Bool_t	SetTotalHistogram (const TH1 &rTotal, Option_t *opt)
	Sets the histogram containing all events.
void	SetUseWeightedEvents (Bool_t on=kTRUE)
void	SetWeight (Double_t weight)
	Sets the global weight for this TEfficiency object.
void	Streamer (TBuffer &) override
	Stream an object of class TObject.
void	StreamerNVirtual (TBuffer &ClassDef_StreamerNVirtual_b)
Bool_t	UsesBayesianStat () const
Bool_t	UsesCentralInterval () const
Bool_t	UsesPosteriorAverage () const
Bool_t	UsesPosteriorMode () const
Bool_t	UsesShortestInterval () const
Bool_t	UsesWeights () const
Public Member Functions inherited from TNamed
	TNamed ()
	TNamed (const char *name, const char *title)
	TNamed (const TNamed &named)
	TNamed copy ctor.
	TNamed (const TString &name, const TString &title)
virtual	~TNamed ()
	TNamed destructor.
void	Clear (Option_t *option="") override
	Set name and title to empty strings ("").
TObject *	Clone (const char *newname="") const override
	Make a clone of an object using the Streamer facility.
Int_t	Compare (const TObject *obj) const override
	Compare two TNamed objects.
void	Copy (TObject &named) const override
	Copy this to obj.
virtual void	FillBuffer (char *&buffer)
	Encode TNamed into output buffer.
const char *	GetName () const override
	Returns name of object.
const char *	GetTitle () const override
	Returns title of object.
ULong_t	Hash () const override
	Return hash value for this object.
TClass *	IsA () const override
Bool_t	IsSortable () const override
void	ls (Option_t *option="") const override
	List TNamed name and title.
TNamed &	operator= (const TNamed &rhs)
	TNamed assignment operator.
void	Print (Option_t *option="") const override
	Print TNamed name and title.
virtual void	SetNameTitle (const char *name, const char *title)
	Set all the TNamed parameters (name and title).
virtual Int_t	Sizeof () const
	Return size of the TNamed part of the TObject.
void	Streamer (TBuffer &) override
	Stream an object of class TObject.
void	StreamerNVirtual (TBuffer &ClassDef_StreamerNVirtual_b)
Public Member Functions inherited from TObject
	TObject ()
	TObject constructor.
	TObject (const TObject &object)
	TObject copy ctor.
virtual	~TObject ()
	TObject destructor.
void	AbstractMethod (const char *method) const
	Use this method to implement an "abstract" method that you don't want to leave purely abstract.
virtual void	AppendPad (Option_t *option="")
	Append graphics object to current pad.
ULong_t	CheckedHash ()
	Check and record whether this class has a consistent Hash/RecursiveRemove setup (*) and then return the regular Hash value for this object.
virtual const char *	ClassName () const
	Returns name of class to which the object belongs.
virtual void	Delete (Option_t *option="")
	Delete this object.
virtual void	DrawClass () const
	Draw class inheritance tree of the class to which this object belongs.
virtual TObject *	DrawClone (Option_t *option="") const
	Draw a clone of this object in the current selected pad with: gROOT->SetSelectedPad(c1).
virtual void	Dump () const
	Dump contents of object on stdout.
virtual void	Error (const char *method, const char *msgfmt,...) const
	Issue error message.
virtual void	Execute (const char *method, const char *params, Int_t *error=nullptr)
	Execute method on this object with the given parameter string, e.g.
virtual void	Execute (TMethod *method, TObjArray *params, Int_t *error=nullptr)
	Execute method on this object with parameters stored in the TObjArray.
virtual void	Fatal (const char *method, const char *msgfmt,...) const
	Issue fatal error message.
virtual TObject *	FindObject (const char *name) const
	Must be redefined in derived classes.
virtual TObject *	FindObject (const TObject *obj) const
	Must be redefined in derived classes.
virtual Option_t *	GetDrawOption () const
	Get option used by the graphics system to draw this object.
virtual const char *	GetIconName () const
	Returns mime type name of object.
virtual char *	GetObjectInfo (Int_t px, Int_t py) const
	Returns string containing info about the object at position (px,py).
virtual Option_t *	GetOption () const
virtual UInt_t	GetUniqueID () const
	Return the unique object id.
virtual Bool_t	HandleTimer (TTimer *timer)
	Execute action in response of a timer timing out.
Bool_t	HasInconsistentHash () const
	Return true is the type of this object is known to have an inconsistent setup for Hash and RecursiveRemove (i.e.
virtual void	Info (const char *method, const char *msgfmt,...) const
	Issue info message.
virtual Bool_t	InheritsFrom (const char *classname) const
	Returns kTRUE if object inherits from class "classname".
virtual Bool_t	InheritsFrom (const TClass *cl) const
	Returns kTRUE if object inherits from TClass cl.
virtual void	Inspect () const
	Dump contents of this object in a graphics canvas.
void	InvertBit (UInt_t f)
Bool_t	IsDestructed () const
	IsDestructed.
virtual Bool_t	IsEqual (const TObject *obj) const
	Default equal comparison (objects are equal if they have the same address in memory).
virtual Bool_t	IsFolder () const
	Returns kTRUE in case object contains browsable objects (like containers or lists of other objects).
R__ALWAYS_INLINE Bool_t	IsOnHeap () const
R__ALWAYS_INLINE Bool_t	IsZombie () const
void	MayNotUse (const char *method) const
	Use this method to signal that a method (defined in a base class) may not be called in a derived class (in principle against good design since a child class should not provide less functionality than its parent, however, sometimes it is necessary).
virtual Bool_t	Notify ()
	This method must be overridden to handle object notification.
void	Obsolete (const char *method, const char *asOfVers, const char *removedFromVers) const
	Use this method to declare a method obsolete.
void	operator delete (void *ptr)
	Operator delete.
void	operator delete[] (void *ptr)
	Operator delete [].
void *	operator new (size_t sz)
void *	operator new (size_t sz, void *vp)
void *	operator new[] (size_t sz)
void *	operator new[] (size_t sz, void *vp)
TObject &	operator= (const TObject &rhs)
	TObject assignment operator.
virtual void	Pop ()
	Pop on object drawn in a pad to the top of the display list.
virtual Int_t	Read (const char *name)
	Read contents of object with specified name from the current directory.
virtual void	RecursiveRemove (TObject *obj)
	Recursively remove this object from a list.
void	ResetBit (UInt_t f)
virtual void	SaveAs (const char *filename="", Option_t *option="") const
	Save this object in the file specified by filename.
void	SetBit (UInt_t f)
void	SetBit (UInt_t f, Bool_t set)
	Set or unset the user status bits as specified in f.
virtual void	SetDrawOption (Option_t *option="")
	Set drawing option for object.
virtual void	SetUniqueID (UInt_t uid)
	Set the unique object id.
void	StreamerNVirtual (TBuffer &ClassDef_StreamerNVirtual_b)
virtual void	SysError (const char *method, const char *msgfmt,...) const
	Issue system error message.
R__ALWAYS_INLINE Bool_t	TestBit (UInt_t f) const
Int_t	TestBits (UInt_t f) const
virtual void	UseCurrentStyle ()
	Set current style settings in this object This function is called when either TCanvas::UseCurrentStyle or TROOT::ForceStyle have been invoked.
virtual void	Warning (const char *method, const char *msgfmt,...) const
	Issue warning message.
virtual Int_t	Write (const char *name=nullptr, Int_t option=0, Int_t bufsize=0)
	Write this object to the current directory.
virtual Int_t	Write (const char *name=nullptr, Int_t option=0, Int_t bufsize=0) const
	Write this object to the current directory.
Public Member Functions inherited from TAttLine
	TAttLine ()
	AttLine default constructor.
	TAttLine (Color_t lcolor, Style_t lstyle, Width_t lwidth)
	AttLine normal constructor.
virtual	~TAttLine ()
	AttLine destructor.
void	Copy (TAttLine &attline) const
	Copy this line attributes to a new TAttLine.
Int_t	DistancetoLine (Int_t px, Int_t py, Double_t xp1, Double_t yp1, Double_t xp2, Double_t yp2)
	Compute distance from point px,py to a line.
virtual Color_t	GetLineColor () const
	Return the line color.
virtual Style_t	GetLineStyle () const
	Return the line style.
virtual Width_t	GetLineWidth () const
	Return the line width.
virtual void	Modify ()
	Change current line attributes if necessary.
virtual void	ResetAttLine (Option_t *option="")
	Reset this line attributes to default values.
virtual void	SaveLineAttributes (std::ostream &out, const char *name, Int_t coldef=1, Int_t stydef=1, Int_t widdef=1)
	Save line attributes as C++ statement(s) on output stream out.
virtual void	SetLineAttributes ()
	Invoke the DialogCanvas Line attributes.
virtual void	SetLineColor (Color_t lcolor)
	Set the line color.
virtual void	SetLineColorAlpha (Color_t lcolor, Float_t lalpha)
	Set a transparent line color.
virtual void	SetLineStyle (Style_t lstyle)
	Set the line style.
virtual void	SetLineWidth (Width_t lwidth)
	Set the line width.
void	StreamerNVirtual (TBuffer &ClassDef_StreamerNVirtual_b)
Public Member Functions inherited from TAttFill
	TAttFill ()
	AttFill default constructor.
	TAttFill (Color_t fcolor, Style_t fstyle)
	AttFill normal constructor.
virtual	~TAttFill ()
	AttFill destructor.
void	Copy (TAttFill &attfill) const
	Copy this fill attributes to a new TAttFill.
virtual Color_t	GetFillColor () const
	Return the fill area color.
virtual Style_t	GetFillStyle () const
	Return the fill area style.
virtual Bool_t	IsTransparent () const
virtual void	Modify ()
	Change current fill area attributes if necessary.
virtual void	ResetAttFill (Option_t *option="")
	Reset this fill attributes to default values.
virtual void	SaveFillAttributes (std::ostream &out, const char *name, Int_t coldef=1, Int_t stydef=1001)
	Save fill attributes as C++ statement(s) on output stream out.
virtual void	SetFillAttributes ()
	Invoke the DialogCanvas Fill attributes.
virtual void	SetFillColor (Color_t fcolor)
	Set the fill area color.
virtual void	SetFillColorAlpha (Color_t fcolor, Float_t falpha)
	Set a transparent fill color.
virtual void	SetFillStyle (Style_t fstyle)
	Set the fill area style.
void	StreamerNVirtual (TBuffer &ClassDef_StreamerNVirtual_b)
Public Member Functions inherited from TAttMarker
	TAttMarker ()
	TAttMarker default constructor.
	TAttMarker (Color_t color, Style_t style, Size_t msize)
	TAttMarker normal constructor.
virtual	~TAttMarker ()
	TAttMarker destructor.
void	Copy (TAttMarker &attmarker) const
	Copy this marker attributes to a new TAttMarker.
virtual Color_t	GetMarkerColor () const
	Return the marker color.
virtual Size_t	GetMarkerSize () const
	Return the marker size.
virtual Style_t	GetMarkerStyle () const
	Return the marker style.
virtual void	Modify ()
	Change current marker attributes if necessary.
virtual void	ResetAttMarker (Option_t *toption="")
	Reset this marker attributes to the default values.
virtual void	SaveMarkerAttributes (std::ostream &out, const char *name, Int_t coldef=1, Int_t stydef=1, Int_t sizdef=1)
	Save line attributes as C++ statement(s) on output stream out.
virtual void	SetMarkerAttributes ()
	Invoke the DialogCanvas Marker attributes.
virtual void	SetMarkerColor (Color_t mcolor=1)
	Set the marker color.
virtual void	SetMarkerColorAlpha (Color_t mcolor, Float_t malpha)
	Set a transparent marker color.
virtual void	SetMarkerSize (Size_t msize=1)
	Set the marker size.
virtual void	SetMarkerStyle (Style_t mstyle=1)
	Set the marker style.
void	StreamerNVirtual (TBuffer &ClassDef_StreamerNVirtual_b)

Static Public Member Functions
static Double_t	AgrestiCoull (Double_t total, Double_t passed, Double_t level, Bool_t bUpper)
	Calculates the boundaries for the frequentist Agresti-Coull interval.
static Double_t	Bayesian (Double_t total, Double_t passed, Double_t level, Double_t alpha, Double_t beta, Bool_t bUpper, Bool_t bShortest=false)
	Calculates the boundaries for a Bayesian confidence interval (shortest or central interval depending on the option)
static Double_t	BetaCentralInterval (Double_t level, Double_t alpha, Double_t beta, Bool_t bUpper)
	Calculates the boundaries for a central confidence interval for a Beta distribution.
static Double_t	BetaMean (Double_t alpha, Double_t beta)
	Compute the mean (average) of the beta distribution.
static Double_t	BetaMode (Double_t alpha, Double_t beta)
	Compute the mode of the beta distribution.
static Bool_t	BetaShortestInterval (Double_t level, Double_t alpha, Double_t beta, Double_t &lower, Double_t &upper)
	Calculates the boundaries for a shortest confidence interval for a Beta distribution.
static Bool_t	CheckBinning (const TH1 &pass, const TH1 &total)
	Checks binning for each axis.
static Bool_t	CheckConsistency (const TH1 &pass, const TH1 &total, Option_t *opt="")
	Checks the consistence of the given histograms.
static Bool_t	CheckEntries (const TH1 &pass, const TH1 &total, Option_t *opt="")
	Checks whether bin contents are compatible with binomial statistics.
static Bool_t	CheckWeights (const TH1 &pass, const TH1 &total)
	Check if both histogram are weighted.
static TClass *	Class ()
static const char *	Class_Name ()
static constexpr Version_t	Class_Version ()
static Double_t	ClopperPearson (Double_t total, Double_t passed, Double_t level, Bool_t bUpper)
	Calculates the boundaries for the frequentist Clopper-Pearson interval.
static Double_t	Combine (Double_t &up, Double_t &low, Int_t n, const Int_t *pass, const Int_t *total, Double_t alpha, Double_t beta, Double_t level=0.683, const Double_t *w=nullptr, Option_t *opt="")
static TGraphAsymmErrors *	Combine (TCollection *pList, Option_t *opt="", Int_t n=0, const Double_t *w=nullptr)
	Combines a list of 1-dimensional TEfficiency objects.
static const char *	DeclFileName ()
static Double_t	FeldmanCousins (Double_t total, Double_t passed, Double_t level, Bool_t bUpper)
	Calculates the boundaries for the frequentist Feldman-Cousins interval.
static Bool_t	FeldmanCousinsInterval (Double_t total, Double_t passed, Double_t level, Double_t &lower, Double_t &upper)
	Calculates the interval boundaries using the frequentist methods of Feldman-Cousins.
static Double_t	MidPInterval (Double_t total, Double_t passed, Double_t level, Bool_t bUpper)
	Calculates the boundaries using the mid-P binomial interval (Lancaster method) from B.
static Double_t	Normal (Double_t total, Double_t passed, Double_t level, Bool_t bUpper)
	Returns the confidence limits for the efficiency supposing that the efficiency follows a normal distribution with the rms below.
static Double_t	Wilson (Double_t total, Double_t passed, Double_t level, Bool_t bUpper)
	Calculates the boundaries for the frequentist Wilson interval.
Static Public Member Functions inherited from TNamed
static TClass *	Class ()
static const char *	Class_Name ()
static constexpr Version_t	Class_Version ()
static const char *	DeclFileName ()
Static Public Member Functions inherited from TObject
static TClass *	Class ()
static const char *	Class_Name ()
static constexpr Version_t	Class_Version ()
static const char *	DeclFileName ()
static Longptr_t	GetDtorOnly ()
	Return destructor only flag.
static Bool_t	GetObjectStat ()
	Get status of object stat flag.
static void	SetDtorOnly (void *obj)
	Set destructor only flag.
static void	SetObjectStat (Bool_t stat)
	Turn on/off tracking of objects in the TObjectTable.
Static Public Member Functions inherited from TAttLine
static TClass *	Class ()
static const char *	Class_Name ()
static constexpr Version_t	Class_Version ()
static const char *	DeclFileName ()
Static Public Member Functions inherited from TAttFill
static TClass *	Class ()
static const char *	Class_Name ()
static constexpr Version_t	Class_Version ()
static const char *	DeclFileName ()
Static Public Member Functions inherited from TAttMarker
static TClass *	Class ()
static const char *	Class_Name ()
static constexpr Version_t	Class_Version ()
static const char *	DeclFileName ()
static Width_t	GetMarkerLineWidth (Style_t style)
	Internal helper function that returns the line width of the given marker style (0 = filled marker)
static Style_t	GetMarkerStyleBase (Style_t style)
	Internal helper function that returns the corresponding marker style with line width 1 for the given style.

Protected Types
enum	EStatusBits {   kIsBayesian = (1ULL << ( 14 )) , kPosteriorMode = (1ULL << ( 15 )) , kShortestInterval = (1ULL << ( 16 )) , kUseBinPrior = (1ULL << ( 17 )) ,   kUseWeights = (1ULL << ( 18 )) }
Protected Types inherited from TObject
enum	{ kOnlyPrepStep = (1ULL << ( 3 )) }

Protected Member Functions
void	Build (const char *name, const char *title)
	Building standard data structure of a TEfficiency object.
void	FillGraph (TGraphAsymmErrors *graph, Option_t *opt) const
	Fill the graph to be painted with information from TEfficiency Internal method called by TEfficiency::Paint or TEfficiency::CreateGraph.
void	FillHistogram (TH2 *h2) const
	Fill the 2d histogram to be painted with information from TEfficiency 2D Internal method called by TEfficiency::Paint or TEfficiency::CreatePaintingGraph.
Protected Member Functions inherited from TObject
virtual void	DoError (int level, const char *location, const char *fmt, va_list va) const
	Interface to ErrorHandler (protected).
void	MakeZombie ()

Protected Attributes
Double_t	fBeta_alpha
	Global parameter for prior beta distribution (default = 1)
Double_t	fBeta_beta
	Global parameter for prior beta distribution (default = 1)
std::vector< std::pair< Double_t, Double_t > >	fBeta_bin_params
	Parameter for prior beta distribution different bin by bin (default vector is empty)
Double_t(*	fBoundary )(Double_t, Double_t, Double_t, Bool_t)
	! Pointer to a method calculating the boundaries of confidence intervals
Double_t	fConfLevel
	Confidence level (default = 0.683, 1 sigma)
TDirectory *	fDirectory
	! Pointer to directory holding this TEfficiency object
TList *	fFunctions
	->Pointer to list of functions
TGraphAsymmErrors *	fPaintGraph
	! Temporary graph for painting
TH2 *	fPaintHisto
	! Temporary histogram for painting
TH1 *	fPassedHistogram
	Histogram for events which passed certain criteria.
EStatOption	fStatisticOption
	Defines how the confidence intervals are determined.
TH1 *	fTotalHistogram
	Histogram for total number of events.
Double_t	fWeight
	Weight for all events (default = 1)
Protected Attributes inherited from TNamed
TString	fName
TString	fTitle
Protected Attributes inherited from TAttLine
Color_t	fLineColor
	Line color.
Style_t	fLineStyle
	Line style.
Width_t	fLineWidth
	Line width.
Protected Attributes inherited from TAttFill
Color_t	fFillColor
	Fill area color.
Style_t	fFillStyle
	Fill area style.
Protected Attributes inherited from TAttMarker
Color_t	fMarkerColor
	Marker color.
Size_t	fMarkerSize
	Marker size.
Style_t	fMarkerStyle
	Marker style.

#include <TEfficiency.h>

Inheritance diagram for TEfficiency:

Member Enumeration Documentation

EStatOption

enum TEfficiency::EStatOption

Enumeration type for different statistic options for calculating confidence intervals kF* ... frequentist methods; kB* ... bayesian methods.

Enumerator
kFCP	Clopper-Pearson interval (recommended by PDG)
kFNormal	Normal approximation.
kFWilson	Wilson interval.
kFAC	Agresti-Coull interval.
kFFC	Feldman-Cousins interval.
kBJeffrey	Jeffrey interval (Prior ~ Beta(0.5,0.5)
kBUniform	Prior ~ Uniform = Beta(1,1)
kBBayesian	User specified Prior ~ Beta(fBeta_alpha,fBeta_beta)
kMidP	Mid-P Lancaster interval.

Definition at line 32 of file TEfficiency.h.

EStatusBits

enum TEfficiency::EStatusBits

protected

Enumerator
kIsBayesian	Bayesian statistics are used.
kPosteriorMode	Use posterior mean for best estimate (Bayesian statistics)
kShortestInterval	Use shortest interval.
kUseBinPrior	Use a different prior for each bin.
kUseWeights	Use weights.

Definition at line 61 of file TEfficiency.h.

Constructor & Destructor Documentation

TEfficiency() [1/9]

TEfficiency::TEfficiency

(

)

Default constructor.

Should not be used explicitly

Definition at line 643 of file TEfficiency.cxx.

TEfficiency() [2/9]

TEfficiency::TEfficiency	(	const TH1 &	passed,
		const TH1 &	total
	)

Constructor using two existing histograms as input.

Input: passed - contains the events fulfilling some criteria total - contains all investigated events

Notes: - both histograms have to fulfill the conditions of CheckConsistency

• dimension of the resulting efficiency object depends on the dimension of the given histograms
• Clones of both histograms are stored internally
• The function SetName(total.GetName() + "_clone") is called to set the names of the new object and the internal histograms..
• The created TEfficiency object is NOT appended to a directory. It will not be written to disk during the next TFile::Write() command in order to prevent duplication of data. If you want to save this TEfficiency object anyway, you can either append it to a directory by calling SetDirectory(TDirectory*) or write it explicitly to disk by calling Write().

Definition at line 682 of file TEfficiency.cxx.

TEfficiency() [3/9]

TEfficiency::TEfficiency	(	const char *	name,
		const char *	title,
		Int_t	nbins,
		const Double_t *	xbins
	)

Create 1-dimensional TEfficiency object with variable bin size.

Constructor creates two new and empty histograms with a given binning

Input:

• name: the common part of the name for both histograms (no blanks) fTotalHistogram has name: name + "_total" fPassedHistogram has name: name + "_passed"
• title: the common part of the title for both histogram fTotalHistogram has title: title + " (total)" fPassedHistogram has title: title + " (passed)" It is possible to label the axis by passing a title with the following format: "title;xlabel;ylabel".
• nbins: number of bins on the x-axis
• xbins: array of length (nbins + 1) with low-edges for each bin xbins[nbinsx] ... lower edge for overflow bin

Definition at line 748 of file TEfficiency.cxx.

TEfficiency() [4/9]

TEfficiency::TEfficiency	(	const char *	name,
		const char *	title,
		Int_t	nbinsx,
		Double_t	xlow,
		Double_t	xup
	)

Create 1-dimensional TEfficiency object with fixed bins size.

Constructor creates two new and empty histograms with a fixed binning.

Input:

• name: the common part of the name for both histograms(no blanks) fTotalHistogram has name: name + "_total" fPassedHistogram has name: name + "_passed"
• title: the common part of the title for both histogram fTotalHistogram has title: title + " (total)" fPassedHistogram has title: title + " (passed)" It is possible to label the axis by passing a title with the following format: "title;xlabel;ylabel".
• nbinsx: number of bins on the x-axis
• xlow: lower edge of first bin
• xup: upper edge of last bin

Definition at line 789 of file TEfficiency.cxx.

TEfficiency() [5/9]

TEfficiency::TEfficiency	(	const char *	name,
		const char *	title,
		Int_t	nbinsx,
		Double_t	xlow,
		Double_t	xup,
		Int_t	nbinsy,
		Double_t	ylow,
		Double_t	yup
	)

Create 2-dimensional TEfficiency object with fixed bin size.

Constructor creates two new and empty histograms with a fixed binning.

Input:

• name: the common part of the name for both histograms(no blanks) fTotalHistogram has name: name + "_total" fPassedHistogram has name: name + "_passed"
• title: the common part of the title for both histogram fTotalHistogram has title: title + " (total)" fPassedHistogram has title: title + " (passed)" It is possible to label the axis by passing a title with the following format: "title;xlabel;ylabel;zlabel".
• nbinsx: number of bins on the x-axis
• xlow: lower edge of first x-bin
• xup: upper edge of last x-bin
• nbinsy: number of bins on the y-axis
• ylow: lower edge of first y-bin
• yup: upper edge of last y-bin

Definition at line 831 of file TEfficiency.cxx.

TEfficiency() [6/9]

TEfficiency::TEfficiency	(	const char *	name,
		const char *	title,
		Int_t	nbinsx,
		const Double_t *	xbins,
		Int_t	nbinsy,
		const Double_t *	ybins
	)

Create 2-dimensional TEfficiency object with variable bin size.

Constructor creates two new and empty histograms with a given binning.

Input:

• name: the common part of the name for both histograms(no blanks) fTotalHistogram has name: name + "_total" fPassedHistogram has name: name + "_passed"
• title: the common part of the title for both histogram fTotalHistogram has title: title + " (total)" fPassedHistogram has title: title + " (passed)" It is possible to label the axis by passing a title with the following format: "title;xlabel;ylabel;zlabel".
• nbinsx: number of bins on the x-axis
• xbins: array of length (nbins + 1) with low-edges for each bin xbins[nbinsx] ... lower edge for overflow x-bin
• nbinsy: number of bins on the y-axis
• ybins: array of length (nbins + 1) with low-edges for each bin ybins[nbinsy] ... lower edge for overflow y-bin

Definition at line 874 of file TEfficiency.cxx.

TEfficiency() [7/9]

TEfficiency::TEfficiency	(	const char *	name,
		const char *	title,
		Int_t	nbinsx,
		Double_t	xlow,
		Double_t	xup,
		Int_t	nbinsy,
		Double_t	ylow,
		Double_t	yup,
		Int_t	nbinsz,
		Double_t	zlow,
		Double_t	zup
	)

Create 3-dimensional TEfficiency object with fixed bin size.

Constructor creates two new and empty histograms with a fixed binning.

Input:

• name: the common part of the name for both histograms(no blanks) fTotalHistogram has name: name + "_total" fPassedHistogram has name: name + "_passed"
• title: the common part of the title for both histogram fTotalHistogram has title: title + " (total)" fPassedHistogram has title: title + " (passed)" It is possible to label the axis by passing a title with the following format: "title;xlabel;ylabel;zlabel".
• nbinsx: number of bins on the x-axis
• xlow: lower edge of first x-bin
• xup: upper edge of last x-bin
• nbinsy: number of bins on the y-axis
• ylow: lower edge of first y-bin
• yup: upper edge of last y-bin
• nbinsz: number of bins on the z-axis
• zlow: lower edge of first z-bin
• zup: upper edge of last z-bin

Definition at line 920 of file TEfficiency.cxx.

TEfficiency() [8/9]

TEfficiency::TEfficiency	(	const char *	name,
		const char *	title,
		Int_t	nbinsx,
		const Double_t *	xbins,
		Int_t	nbinsy,
		const Double_t *	ybins,
		Int_t	nbinsz,
		const Double_t *	zbins
	)

Create 3-dimensional TEfficiency object with variable bin size.

Constructor creates two new and empty histograms with a given binning.

Input:

• name: the common part of the name for both histograms(no blanks) fTotalHistogram has name: name + "_total" fPassedHistogram has name: name + "_passed"
• title: the common part of the title for both histogram fTotalHistogram has title: title + " (total)" fPassedHistogram has title: title + " (passed)" It is possible to label the axis by passing a title with the following format: "title;xlabel;ylabel;zlabel".
• nbinsx: number of bins on the x-axis
• xbins: array of length (nbins + 1) with low-edges for each bin xbins[nbinsx] ... lower edge for overflow x-bin
• nbinsy: number of bins on the y-axis
• ybins: array of length (nbins + 1) with low-edges for each bin xbins[nbinsx] ... lower edge for overflow y-bin
• nbinsz: number of bins on the z-axis
• zbins: array of length (nbins + 1) with low-edges for each bin xbins[nbinsx] ... lower edge for overflow z-bin

Definition at line 967 of file TEfficiency.cxx.

TEfficiency() [9/9]

TEfficiency::TEfficiency

(

const TEfficiency &

rEff

)

Copy constructor.

The list of associated objects (e.g. fitted functions) is not copied.

Note:

• SetName(rEff.GetName() + "_copy") is called to set the names of the object and the histograms.
• The titles are set by calling SetTitle("[copy] " + rEff.GetTitle()).
• The copied TEfficiency object is NOT appended to a directory. It will not be written to disk during the next TFile::Write() command in order to prevent duplication of data. If you want to save this TEfficiency object anyway, you can either append it to a directory by calling SetDirectory(TDirectory*) or write it explicitly to disk by calling Write().

Definition at line 1006 of file TEfficiency.cxx.

~TEfficiency()

TEfficiency::~TEfficiency ( )

override

default destructor

Definition at line 1051 of file TEfficiency.cxx.

Member Function Documentation

Add()

void TEfficiency::Add ( const TEfficiency &  rEff )

inline

Definition at line 94 of file TEfficiency.h.

AgrestiCoull()

Double_t TEfficiency::AgrestiCoull ( Double_t  total, Double_t  passed, Double_t  level, Bool_t  bUpper  )

static

Calculates the boundaries for the frequentist Agresti-Coull interval.

Parameters

total	number of total events
passed	0 <= number of passed events <= total
level	confidence level
bUpper	true - upper boundary is returned false - lower boundary is returned

\begin{eqnarray*} \alpha &=& 1 - \frac{level}{2} \\ \kappa &=& \Phi^{-1}(1 - \alpha,1)\ ... normal\ quantile\ function\\ mode &=& \frac{passed + \frac{\kappa^{2}}{2}}{total + \kappa^{2}}\\ \Delta &=& \kappa * \sqrt{\frac{mode * (1 - mode)}{total + \kappa^{2}}}\\ return &=& max(0,mode - \Delta)\ or\ min(1,mode + \Delta) \end{eqnarray*}

Definition at line 1101 of file TEfficiency.cxx.

Bayesian()

Double_t TEfficiency::Bayesian ( Double_t  total, Double_t  passed, Double_t  level, Double_t  alpha, Double_t  beta, Bool_t  bUpper, Bool_t  bShortest = false  )

static

Calculates the boundaries for a Bayesian confidence interval (shortest or central interval depending on the option)

Parameters

[in]	total	number of total events
[in]	passed	0 <= number of passed events <= total
[in]	level	confidence level
[in]	alpha	shape parameter > 0 for the prior distribution (fBeta_alpha)
[in]	beta	shape parameter > 0 for the prior distribution (fBeta_beta)
[in]	bUpper	true - upper boundary is returned false - lower boundary is returned
[in]	bShortest	??

Note: In the case central confidence interval is calculated. when passed = 0 (or passed = total) the lower (or upper) interval values will be larger than 0 (or smaller than 1).

Calculation:

The posterior probability in bayesian statistics is given by:

\[ P(\varepsilon |k,N) \propto L(\varepsilon|k,N) \times Prior(\varepsilon) \]

As an efficiency can be interpreted as probability of a positive outcome of a Bernoullli trial the likelihood function is given by the binomial distribution:

\[ L(\varepsilon|k,N) = Binomial(N,k) \varepsilon ^{k} (1 - \varepsilon)^{N-k} \]

At the moment only beta distributions are supported as prior probabilities of the efficiency ( \( B(\alpha,\beta)\) is the beta function):

\[ Prior(\varepsilon) = \frac{1}{B(\alpha,\beta)} \varepsilon ^{\alpha - 1} (1 - \varepsilon)^{\beta - 1} \]

The posterior probability is therefore again given by a beta distribution:

\[ P(\varepsilon |k,N) \propto \varepsilon ^{k + \alpha - 1} (1 - \varepsilon)^{N - k + \beta - 1} \]

In case of central intervals the lower boundary for the equal-tailed confidence interval is given by the inverse cumulative (= quantile) function for the quantile \( \frac{1 - level}{2} \). The upper boundary for the equal-tailed confidence interval is given by the inverse cumulative (= quantile) function for the quantile \( \frac{1 + level}{2} \). Hence it is the solution \( \varepsilon \) of the following equation:

\[ I_{\varepsilon}(k + \alpha,N - k + \beta) = \frac{1}{norm} \int_{0}^{\varepsilon} dt t^{k + \alpha - 1} (1 - t)^{N - k + \beta - 1} = \frac{1 \pm level}{2} \]

In the case of shortest interval the minimum interval around the mode is found by minimizing the length of all intervals width the given probability content. See TEfficiency::BetaShortestInterval

Definition at line 1276 of file TEfficiency.cxx.

BetaCentralInterval()

Double_t TEfficiency::BetaCentralInterval ( Double_t  level, Double_t  a, Double_t  b, Bool_t  bUpper  )

static

Calculates the boundaries for a central confidence interval for a Beta distribution.

Parameters

[in]	level	confidence level
[in]	a	parameter > 0 for the beta distribution (for a posterior is passed + prior_alpha
[in]	b	parameter > 0 for the beta distribution (for a posterior is (total-passed) + prior_beta
[in]	bUpper	true - upper boundary is returned false - lower boundary is returned

Definition at line 1300 of file TEfficiency.cxx.

BetaMean()

Double_t TEfficiency::BetaMean ( Double_t  a, Double_t  b  )

static

Compute the mean (average) of the beta distribution.

Parameters

[in]	a	parameter > 0 for the beta distribution (for a posterior is passed + prior_alpha
[in]	b	parameter > 0 for the beta distribution (for a posterior is (total-passed) + prior_beta

Definition at line 1410 of file TEfficiency.cxx.

BetaMode()

Double_t TEfficiency::BetaMode ( Double_t  a, Double_t  b  )

static

Compute the mode of the beta distribution.

Parameters

[in]	a	parameter > 0 for the beta distribution (for a posterior is passed + prior_alpha
[in]	b	parameter > 0 for the beta distribution (for a posterior is (total-passed) + prior_beta

note the mode is defined for a Beta(a,b) only if (a,b)>1 (a = passed+alpha; b = total-passed+beta) return then the following in case (a,b) < 1:

• if (a==b) return 0.5 (it is really undefined)
• if (a < b) return 0;
• if (a > b) return 1;

Definition at line 1433 of file TEfficiency.cxx.

BetaShortestInterval()

Bool_t TEfficiency::BetaShortestInterval ( Double_t  level, Double_t  a, Double_t  b, Double_t &  lower, Double_t &  upper  )

static

Calculates the boundaries for a shortest confidence interval for a Beta distribution.

Parameters

[in]	level	confidence level
[in]	a	parameter > 0 for the beta distribution (for a posterior is passed + prior_alpha
[in]	b	parameter > 0 for the beta distribution (for a posterior is (total-passed) + prior_beta
[out]	upper	upper boundary is returned
[out]	lower	lower boundary is returned

The lower/upper boundary are then obtained by finding the shortest interval of the beta distribution contained the desired probability level. The length of all possible intervals is minimized in order to find the shortest one

Definition at line 1356 of file TEfficiency.cxx.

Browse()

void TEfficiency::Browse ( TBrowser *  b )

inlineoverridevirtual

Browse object. May be overridden for another default action.

Reimplemented from TObject.

Definition at line 95 of file TEfficiency.h.

Build()

void TEfficiency::Build ( const char *  name, const char *  title  )

protected

Building standard data structure of a TEfficiency object.

Notes:

• calls: SetName(name), SetTitle(title)
• set the statistic option to the default (kFCP)
• appends this object to the current directory SetDirectory(gDirectory)

Definition at line 1457 of file TEfficiency.cxx.

CheckBinning()

Bool_t TEfficiency::CheckBinning ( const TH1 &  pass, const TH1 &  total  )

static

Checks binning for each axis.

It is assumed that the passed histograms have the same dimension.

Definition at line 1479 of file TEfficiency.cxx.

CheckConsistency()

Bool_t TEfficiency::CheckConsistency ( const TH1 &  pass, const TH1 &  total, Option_t *  opt = ""  )

static

Checks the consistence of the given histograms.

The histograms are considered as consistent if:

• both have the same dimension
• both have the same binning
• pass.GetBinContent(i) <= total.GetBinContent(i) for each bin i

Definition at line 1528 of file TEfficiency.cxx.

CheckEntries()

Bool_t TEfficiency::CheckEntries ( const TH1 &  pass, const TH1 &  total, Option_t *  opt = ""  )

static

Checks whether bin contents are compatible with binomial statistics.

The following inequality has to be valid for each bin i: total.GetBinContent(i) >= pass.GetBinContent(i)

Note:

• It is assumed that both histograms have the same dimension and binning.

Definition at line 1560 of file TEfficiency.cxx.

CheckWeights()

Bool_t TEfficiency::CheckWeights ( const TH1 &  pass, const TH1 &  total  )

static

Check if both histogram are weighted.

If they are weighted a true is returned

Definition at line 1590 of file TEfficiency.cxx.

Class()

static TClass * TEfficiency::Class ( )

static

Returns

TClass describing this class

Class_Name()

static const char * TEfficiency::Class_Name ( )

static

Returns

Name of this class

Class_Version()

static constexpr Version_t TEfficiency::Class_Version ( )

inlinestaticconstexpr

Returns

Version of this class

Definition at line 188 of file TEfficiency.h.

ClopperPearson()

Double_t TEfficiency::ClopperPearson ( Double_t  total, Double_t  passed, Double_t  level, Bool_t  bUpper  )

static

Calculates the boundaries for the frequentist Clopper-Pearson interval.

This interval is recommended by the PDG.

Parameters

[in]	total	number of total events
[in]	passed	0 <= number of passed events <= total
[in]	level	confidence level
[in]	bUpper	true - upper boundary is returned ;false - lower boundary is returned

Calculation:

The lower boundary of the Clopper-Pearson interval is the "exact" inversion of the test:

\begin{eqnarray*} P(x \geq passed; total) &=& \frac{1 - level}{2}\\ P(x \geq passed; total) &=& 1 - P(x \leq passed - 1; total)\\ &=& 1 - \frac{1}{norm} * \int_{0}^{1 - \varepsilon} t^{total - passed} (1 - t)^{passed - 1} dt\\ &=& 1 - \frac{1}{norm} * \int_{\varepsilon}^{1} t^{passed - 1} (1 - t)^{total - passed} dt\\ &=& \frac{1}{norm} * \int_{0}^{\varepsilon} t^{passed - 1} (1 - t)^{total - passed} dt\\ &=& I_{\varepsilon}(passed,total - passed + 1) \end{eqnarray*}

The lower boundary is therefore given by the \( \frac{1 - level}{2}\) quantile of the beta distribution.

The upper boundary of the Clopper-Pearson interval is the "exact" inversion of the test:

\begin{eqnarray*} P(x \leq passed; total) &=& \frac{1 - level}{2}\\ P(x \leq passed; total) &=& \frac{1}{norm} * \int_{0}^{1 - \varepsilon} t^{total - passed - 1} (1 - t)^{passed} dt\\ &=& \frac{1}{norm} * \int_{\varepsilon}^{1} t^{passed} (1 - t)^{total - passed - 1} dt\\ &=& 1 - \frac{1}{norm} * \int_{0}^{\varepsilon} t^{passed} (1 - t)^{total - passed - 1} dt\\ \Rightarrow 1 - \frac{1 - level}{2} &=& \frac{1}{norm} * \int_{0}^{\varepsilon} t^{passed} (1 - t)^{total - passed -1} dt\\ \frac{1 + level}{2} &=& I_{\varepsilon}(passed + 1,total - passed) \end{eqnarray*}

The upper boundary is therefore given by the \(\frac{1 + level}{2}\) quantile of the beta distribution.

Note: The connection between the binomial distribution and the regularized incomplete beta function \( I_{\varepsilon}(\alpha,\beta)\) has been used.

Definition at line 1848 of file TEfficiency.cxx.

Combine() [1/2]

Double_t TEfficiency::Combine ( Double_t &  up, Double_t &  low, Int_t  n, const Int_t *  pass, const Int_t *  total, Double_t  alpha, Double_t  beta, Double_t  level = 0.683, const Double_t *  w = nullptr, Option_t *  opt = ""  )

static

Calculates the combined efficiency and its uncertainties

This method does a bayesian combination of the given samples.

\param[in] up  contains the upper limit of the confidence interval afterwards
\param[in] low  contains the lower limit of the confidence interval afterwards
\param[in] n    number of samples which are combined
\param[in] pass array of length n containing the number of passed events
\param[in] total array of length n containing the corresponding numbers of total events
\param[in] alpha  shape parameters for the beta distribution as prior
\param[in] beta   shape parameters for the beta distribution as prior
\param[in] level  desired confidence level
\param[in] w weights for each sample; if not given, all samples get the weight 1
          The weights do not need to be normalized, since they are internally renormalized
          to the number of effective entries.
\param[in] opt
  -  mode : The mode is returned instead of the mean of the posterior as best value
            When using the mode the shortest interval is also computed instead of the central one
  -  shortest: compute shortest interval (done by default if mode option is set)
  -  central: compute central interval (done by default if mode option is NOT set)

Calculation:

The combined posterior distributions is calculated from the Bayes theorem assuming a common prior Beta distribution.
    It is easy to proof that the combined posterior is then:

\begin{eqnarray*} P_{comb}(\epsilon |{w_{i}}; {k_{i}}; {N_{i}}) &=& B(\epsilon, \sum_{i}{ w_{i} k_{i}} + \alpha, \sum_{i}{ w_{i}(n_{i}-k_{i})}+\beta)\\ w_{i} &=& weight\ for\ each\ sample\ renormalized\ to\ the\ effective\ entries\\ w^{'}_{i} &=& w_{i} \frac{ \sum_{i} {w_{i} } } { \sum_{i} {w_{i}^{2} } } \end{eqnarray*}

The estimated efficiency is the mode (or the mean) of the obtained posterior distribution

The boundaries of the confidence interval for a confidence level (1 - a) are given by the a/2 and 1-a/2 quantiles of the resulting cumulative distribution.

Example (uniform prior distribution):

{
     TCanvas* c1 = new TCanvas("c1","",600,800);
     c1->Divide(1,2);
     c1->SetFillStyle(1001);
     c1->SetFillColor(kWhite);
 
     TF1* p1 = new TF1("p1","TMath::BetaDist(x,19,9)",0,1);
     TF1* p2 = new TF1("p2","TMath::BetaDist(x,4,8)",0,1);
     TF1* comb = new TF1("comb2","TMath::BetaDist(x,[0],[1])",0,1);
     double nrm = 1./(0.6*0.6+0.4*0.4); // weight normalization
     double a = 0.6*18.0 + 0.4*3.0 + 1.0;  // new alpha parameter of combined beta dist.
     double b = 0.6*10+0.4*7+1.0;  // new beta parameter of combined beta dist.
     comb->SetParameters(nrm*a ,nrm *b );
     TF1* const1 = new TF1("const1","0.05",0,1);
     TF1* const2 = new TF1("const2","0.95",0,1);
 
     p1->SetLineColor(kRed);
     p1->SetTitle("combined posteriors;#epsilon;P(#epsilon|k,N)");
     p2->SetLineColor(kBlue);
     comb->SetLineColor(kGreen+2);
 
     TLegend* leg1 = new TLegend(0.12,0.65,0.5,0.85);
     leg1->AddEntry(p1,"k1 = 18, N1 = 26","l");
     leg1->AddEntry(p2,"k2 = 3, N2 = 10","l");
     leg1->AddEntry(comb,"combined: p1 = 0.6, p2=0.4","l");
 
     c1->cd(1);
     comb->Draw();
     p1->Draw("same");
     p2->Draw("same");
     leg1->Draw("same");
     c1->cd(2);
     const1->SetLineWidth(1);
     const2->SetLineWidth(1);
     TGraph* gr = (TGraph*)comb->DrawIntegral();
     gr->SetTitle("cumulative function of combined posterior with boundaries for cl = 95%;#epsilon;CDF");
     const1->Draw("same");
     const2->Draw("same");
 
     c1->cd(0);
     return c1;
}

Definition at line 1944 of file TEfficiency.cxx.

Combine() [2/2]

TGraphAsymmErrors * TEfficiency::Combine ( TCollection *  pList, Option_t *  option = "", Int_t  n = 0, const Double_t *  w = nullptr  )

static

Combines a list of 1-dimensional TEfficiency objects.

A TGraphAsymmErrors object is returned which contains the estimated efficiency and its uncertainty for each bin. If the combination fails, a zero pointer is returned.

At the moment the combining is only implemented for bayesian statistics.

Parameters

[in]	pList	list containing TEfficiency objects which should be combined only one-dimensional efficiencies are taken into account
[in]	option	s : strict combining; only TEfficiency objects with the same beta prior and the flag kIsBayesian == true are combined If not specified the prior parameter of the first TEfficiency object is used v : verbose mode; print information about combining cl=x : set confidence level (0 < cl < 1). If not specified, the confidence level of the first TEfficiency object is used. mode Use mode of combined posterior as estimated value for the efficiency shortest: compute shortest interval (done by default if mode option is set) central: compute central interval (done by default if mode option is NOT set)
[in]	n	number of weights (has to be the number of one-dimensional TEfficiency objects in pList) If no weights are passed, the internal weights GetWeight() of the given TEfficiency objects are used.
[in]	w	array of length n with weights for each TEfficiency object in pList (w[0] correspond to pList->First ... w[n-1] -> pList->Last) The weights do not have to be normalised.

For each bin the calculation is done by the Combine(double&, double& ...) method.

Definition at line 2034 of file TEfficiency.cxx.

CreateGraph()

TGraphAsymmErrors * TEfficiency::CreateGraph

(

Option_t *

opt = ""

)

const

Create the graph used be painted (for dim=1 TEfficiency) The return object is managed by the caller.

Definition at line 1619 of file TEfficiency.cxx.

CreateHistogram()

TH2 * TEfficiency::CreateHistogram

(

Option_t *

opt = ""

)

const

Create the histogram used to be painted (for dim=2 TEfficiency) The return object is managed by the caller.

Definition at line 1724 of file TEfficiency.cxx.

DeclFileName()

static const char * TEfficiency::DeclFileName ( )

inlinestatic

Returns

Name of the file containing the class declaration

Definition at line 188 of file TEfficiency.h.

DistancetoPrimitive()

Int_t TEfficiency::DistancetoPrimitive ( Int_t  px, Int_t  py  )

overridevirtual

Compute distance from point px,py to a graph.

Compute the closest distance of approach from point px,py to this line. The distance is computed in pixels units.

Forward the call to the painted graph

Reimplemented from TObject.

Definition at line 2219 of file TEfficiency.cxx.

Draw()

void TEfficiency::Draw ( Option_t *  opt = "" )

overridevirtual

Draws the current TEfficiency object.

Parameters

[in]

opt

1-dimensional case: same options as TGraphAsymmErrors::Draw() but as default "AP" is used 2-dimensional case: same options as TH2::Draw() 3-dimensional case: not yet supported

Specific TEfficiency drawing options:

• E0 - plot bins where the total number of passed events is zero (the error interval will be [0,1] )

Reimplemented from TObject.

Definition at line 2240 of file TEfficiency.cxx.

ExecuteEvent()

void TEfficiency::ExecuteEvent ( Int_t  event, Int_t  px, Int_t  py  )

overridevirtual

Execute action corresponding to one event.

This member function is called when the drawn class is clicked with the locator If Left button clicked on one of the line end points, this point follows the cursor until button is released.

if Middle button clicked, the line is moved parallel to itself until the button is released. Forward the call to the underlying graph

Reimplemented from TObject.

Definition at line 2274 of file TEfficiency.cxx.

FeldmanCousins()

Double_t TEfficiency::FeldmanCousins ( Double_t  total, Double_t  passed, Double_t  level, Bool_t  bUpper  )

static

Calculates the boundaries for the frequentist Feldman-Cousins interval.

Parameters

total	number of total events
passed	0 <= number of passed events <= total
level	confidence level
bUpper	true - upper boundary is returned false - lower boundary is returned

Definition at line 1124 of file TEfficiency.cxx.

FeldmanCousinsInterval()

Bool_t TEfficiency::FeldmanCousinsInterval ( Double_t  total, Double_t  passed, Double_t  level, Double_t &  lower, Double_t &  upper  )

static

Calculates the interval boundaries using the frequentist methods of Feldman-Cousins.

Parameters

[in]	total	number of total events
[in]	passed	0 <= number of passed events <= total
[in]	level	confidence level
[out]	lower	lower boundary returned on exit
[out]	upper	lower boundary returned on exit

Returns

a flag with the status of the calculation

Calculation:

The Feldman-Cousins is a frequentist method where the interval is estimated using a Neyman construction where the ordering is based on the likelihood ratio:

\[ LR = \frac{Binomial(k | N, \epsilon)}{Binomial(k | N, \hat{\epsilon} ) } \]

See G. J. Feldman and R. D. Cousins, Phys. Rev. D57 (1998) 3873 and R. D. Cousins, K. E. Hymes, J. Tucker, Nuclear Instruments and Methods in Physics Research A 612 (2010) 388

Implemented using classes developed by Jordan Tucker and Luca Lista See File hist/hist/src/TEfficiencyHelper.h

Definition at line 1157 of file TEfficiency.cxx.

Fill()

void TEfficiency::Fill	(	Bool_t	bPassed,
		Double_t	x,
		Double_t	y = 0,
		Double_t	z = 0
	)

This function is used for filling the two histograms.

Parameters

[in]	bPassed	flag whether the current event passed the selection true: both histograms are filled false: only the total histogram is filled
[in]	x	x-value
[in]	y	y-value (use default=0 for 1-D efficiencies)
[in]	z	z-value (use default=0 for 2-D or 1-D efficiencies)

Definition at line 2290 of file TEfficiency.cxx.

FillGraph()

void TEfficiency::FillGraph ( TGraphAsymmErrors *  graph, Option_t *  opt  ) const

protected

Fill the graph to be painted with information from TEfficiency Internal method called by TEfficiency::Paint or TEfficiency::CreateGraph.

Definition at line 1640 of file TEfficiency.cxx.

FillHistogram()

void TEfficiency::FillHistogram ( TH2 *  h2 ) const

protected

Fill the 2d histogram to be painted with information from TEfficiency 2D Internal method called by TEfficiency::Paint or TEfficiency::CreatePaintingGraph.

Definition at line 1762 of file TEfficiency.cxx.

FillWeighted()

void TEfficiency::FillWeighted	(	Bool_t	bPassed,
		Double_t	weight,
		Double_t	x,
		Double_t	y = 0,
		Double_t	z = 0
	)

This function is used for filling the two histograms with a weight.

Parameters

[in]	bPassed	flag whether the current event passed the selection true: both histograms are filled false: only the total histogram is filled
[in]	weight	weight for the event
[in]	x	x-value
[in]	y	y-value (use default=0 for 1-D efficiencies)
[in]	z	z-value (use default=0 for 2-D or 1-D efficiencies)

Note: - this function will call SetUseWeightedEvents if it was not called by the user before

Definition at line 2324 of file TEfficiency.cxx.

FindFixBin()

Int_t TEfficiency::FindFixBin	(	Double_t	x,
		Double_t	y = 0,
		Double_t	z = 0
	)		const

Returns the global bin number containing the given values.

Note:

• values which belong to dimensions higher than the current dimension of the TEfficiency object are ignored (i.e. for 1-dimensional efficiencies only the x-value is considered)

Definition at line 2360 of file TEfficiency.cxx.

Fit()

TFitResultPtr TEfficiency::Fit	(	TF1 *	f1,
		Option_t *	opt = ""
	)

Fits the efficiency using the TBinomialEfficiencyFitter class.

The resulting fit function is added to the list of associated functions.

Options:

• "+": previous fitted functions in the list are kept, by default all functions in the list are deleted
• for more fitting options see TBinomialEfficiencyFitter::Fit

Definition at line 2384 of file TEfficiency.cxx.

GetBetaAlpha()

Double_t TEfficiency::GetBetaAlpha ( Int_t  bin = -1 ) const

inline

Definition at line 106 of file TEfficiency.h.

GetBetaBeta()

Double_t TEfficiency::GetBetaBeta ( Int_t  bin = -1 ) const

inline

Definition at line 107 of file TEfficiency.h.

GetConfidenceLevel()

Double_t TEfficiency::GetConfidenceLevel ( ) const

inline

Definition at line 108 of file TEfficiency.h.

GetCopyPassedHisto()

TH1 * TEfficiency::GetCopyPassedHisto

(

)

const

Returns a cloned version of fPassedHistogram.

Notes:

• The histogram is filled with unit weights. You might want to scale it with the global weight GetWeight().
• The returned object is owned by the user who has to care about the deletion of the new TH1 object.
• This histogram is by default NOT attached to the current directory to avoid duplication of data. If you want to store it automatically during the next TFile::Write() command, you have to attach it to the corresponding directory.

TFile* pFile = new TFile("passed.root","update");
TEfficiency* pEff = (TEfficiency*)gDirectory->Get("my_eff");
TH1* copy = pEff->GetCopyPassedHisto();
copy->SetDirectory(gDirectory);
pFile->Write();

Definition at line 2444 of file TEfficiency.cxx.

GetCopyTotalHisto()

TH1 * TEfficiency::GetCopyTotalHisto

(

)

const

Returns a cloned version of fTotalHistogram.

Notes:

• The histogram is filled with unit weights. You might want to scale it with the global weight GetWeight().
• The returned object is owned by the user who has to care about the deletion of the new TH1 object.
• This histogram is by default NOT attached to the current directory to avoid duplication of data. If you want to store it automatically during the next TFile::Write() command, you have to attach it to the corresponding directory.

TFile* pFile = new TFile("total.root","update");
TEfficiency* pEff = (TEfficiency*)gDirectory->Get("my_eff");
TH1* copy = pEff->GetCopyTotalHisto();
copy->SetDirectory(gDirectory);
pFile->Write();

Definition at line 2474 of file TEfficiency.cxx.

GetDimension()

Int_t TEfficiency::GetDimension

(

)

const

returns the dimension of the current TEfficiency object

Definition at line 2486 of file TEfficiency.cxx.

GetDirectory()

TDirectory * TEfficiency::GetDirectory ( ) const

inline

Definition at line 112 of file TEfficiency.h.

GetEfficiency()

Double_t TEfficiency::GetEfficiency

(

Int_t

bin

)

const

Returns the efficiency in the given global bin.

Note:

• The estimated efficiency depends on the chosen statistic option: for frequentist ones: \( \hat{\varepsilon} = \frac{passed}{total} \) for bayesian ones the expectation value of the resulting posterior distribution is returned: \( \hat{\varepsilon} = \frac{passed + \alpha}{total + \alpha + \beta} \) If the bit kPosteriorMode is set (or the method TEfficiency::UsePosteriorMode() has been called ) the mode (most probable value) of the posterior is returned: \( \hat{\varepsilon} = \frac{passed + \alpha -1}{total + \alpha + \beta -2} \)
  • If the denominator is equal to 0, an efficiency of 0 is returned.
  • When \( passed + \alpha < 1 \) or \( total - passed + \beta < 1 \) the above formula for the mode is not valid. In these cases values the estimated efficiency is 0 or 1.

Definition at line 2508 of file TEfficiency.cxx.

GetEfficiencyErrorLow()

Double_t TEfficiency::GetEfficiencyErrorLow

(

Int_t

bin

)

const

Returns the lower error on the efficiency in the given global bin.

The result depends on the current confidence level fConfLevel and the chosen statistic option fStatisticOption. See SetStatisticOption(Int_t) for more details.

Note: If the histograms are filled with weights, only bayesian methods and the normal approximation are supported.

Definition at line 2559 of file TEfficiency.cxx.

GetEfficiencyErrorUp()

Double_t TEfficiency::GetEfficiencyErrorUp

(

Int_t

bin

)

const

Returns the upper error on the efficiency in the given global bin.

The result depends on the current confidence level fConfLevel and the chosen statistic option fStatisticOption. See SetStatisticOption(Int_t) for more details.

Note: If the histograms are filled with weights, only bayesian methods and the normal approximation are supported.

Definition at line 2639 of file TEfficiency.cxx.

GetGlobalBin()

Int_t TEfficiency::GetGlobalBin	(	Int_t	binx,
		Int_t	biny = 0,
		Int_t	binz = 0
	)		const

Returns the global bin number which can be used as argument for the following functions:

• GetEfficiency(bin), GetEfficiencyErrorLow(bin), GetEfficiencyErrorUp(bin)
• SetPassedEvents(bin), SetTotalEvents(bin)

see TH1::GetBin() for conventions on numbering bins

Definition at line 2717 of file TEfficiency.cxx.

GetListOfFunctions()

TList * TEfficiency::GetListOfFunctions

(

)

Definition at line 2724 of file TEfficiency.cxx.

GetPaintedGraph()

TGraphAsymmErrors * TEfficiency::GetPaintedGraph ( ) const

inline

Definition at line 117 of file TEfficiency.h.

GetPaintedHistogram()

TH2 * TEfficiency::GetPaintedHistogram ( ) const

inline

Definition at line 118 of file TEfficiency.h.

GetPassedHistogram()

const TH1 * TEfficiency::GetPassedHistogram ( ) const

inline

Definition at line 120 of file TEfficiency.h.

GetStatisticOption()

EStatOption TEfficiency::GetStatisticOption ( ) const

inline

Definition at line 121 of file TEfficiency.h.

GetTotalHistogram()

const TH1 * TEfficiency::GetTotalHistogram ( ) const

inline

Definition at line 122 of file TEfficiency.h.

GetWeight()

Double_t TEfficiency::GetWeight ( ) const

inline

Definition at line 123 of file TEfficiency.h.

IsA()

TClass * TEfficiency::IsA ( ) const

inlineoverridevirtual

Returns

TClass describing current object

Reimplemented from TObject.

Definition at line 188 of file TEfficiency.h.

Merge()

Long64_t TEfficiency::Merge

(

TCollection *

pList

)

Merges the TEfficiency objects in the given list to the given TEfficiency object using the operator+=(TEfficiency&)

The merged result is stored in the current object. The statistic options and the confidence level are taken from the current object.

This function should be used when all TEfficiency objects correspond to the same process.

The new weight is set according to: \( \frac{1}{w_{new}} = \sum_{i} \frac{1}{w_{i}} \)

Definition at line 2742 of file TEfficiency.cxx.

MidPInterval()

Double_t TEfficiency::MidPInterval ( Double_t  total, Double_t  passed, Double_t  level, Bool_t  bUpper  )

static

Calculates the boundaries using the mid-P binomial interval (Lancaster method) from B.

Cousing and J. Tucker. See http://arxiv.org/abs/0905.3831 for a description and references for the method

Modify equal_tailed to get the kind of interval you want. Can also be converted to interval on ratio of poisson means X/Y by the substitutions

X = passed
total = X + Y
lower_poisson = lower/(1 - lower)
upper_poisson = upper/(1 - upper)

Definition at line 1182 of file TEfficiency.cxx.

Normal()

Double_t TEfficiency::Normal ( Double_t  total, Double_t  passed, Double_t  level, Bool_t  bUpper  )

static

Returns the confidence limits for the efficiency supposing that the efficiency follows a normal distribution with the rms below.

Parameters

[in]	total	number of total events
[in]	passed	0 <= number of passed events <= total
[in]	level	confidence level
[in]	bUpper	true - upper boundary is returned false - lower boundary is returned

Calculation:

\begin{eqnarray*} \hat{\varepsilon} &=& \frac{passed}{total}\\ \sigma_{\varepsilon} &=& \sqrt{\frac{\hat{\varepsilon} (1 - \hat{\varepsilon})}{total}}\\ \varepsilon_{low} &=& \hat{\varepsilon} \pm \Phi^{-1}(\frac{level}{2},\sigma_{\varepsilon}) \end{eqnarray*}

Definition at line 2779 of file TEfficiency.cxx.

operator+=()

TEfficiency & TEfficiency::operator+=

(

const TEfficiency &

rhs

)

Adds the histograms of another TEfficiency object to current histograms.

The statistic options and the confidence level remain unchanged.

fTotalHistogram += rhs.fTotalHistogram; fPassedHistogram += rhs.fPassedHistogram;

calculates a new weight: current weight of this TEfficiency object = \( w_{1} \) weight of rhs = \( w_{2} \) \( w_{new} = \frac{w_{1} \times w_{2}}{w_{1} + w_{2}} \)

Definition at line 2806 of file TEfficiency.cxx.

operator=()

TEfficiency & TEfficiency::operator=

(

const TEfficiency &

rhs

)

Assignment operator.

The histograms, statistic option, confidence level, weight and paint styles of rhs are copied to the this TEfficiency object.

Note: - The list of associated functions is not copied. After this operation the list of associated functions is empty.

Definition at line 2848 of file TEfficiency.cxx.

Paint()

void TEfficiency::Paint ( Option_t *  opt )

overridevirtual

Paints this TEfficiency object.

For details on the possible option see Draw(Option_t*)

Note for 1D classes In 1D the TEfficiency uses a TGraphAsymmErrors for drawing The TGraph is created only the first time Paint is used. The user can manipulate the TGraph via the method TEfficiency::GetPaintedGraph() The TGraph creates behing an histogram for the axis. The histogram is created also only the first time. If the axis needs to be updated because in the meantime the class changed use this trick which will trigger a re-calculation of the axis of the graph TEfficiency::GetPaintedGraph()->Set(0)

Note that in order to access the painted graph via GetPaintedGraph() you need either to call Paint or better gPad->Update();

Reimplemented from TObject.

Definition at line 2906 of file TEfficiency.cxx.

SavePrimitive()

void TEfficiency::SavePrimitive ( std::ostream &  out, Option_t *  opt = ""  )

overridevirtual

Have histograms fixed bins along each axis?

Reimplemented from TObject.

Definition at line 2963 of file TEfficiency.cxx.

SetBetaAlpha()

void TEfficiency::SetBetaAlpha

(

Double_t

alpha

)

Sets the shape parameter α.

The prior probability of the efficiency is given by the beta distribution:

\[ f(\varepsilon;\alpha;\beta) = \frac{1}{B(\alpha,\beta)} \varepsilon^{\alpha-1} (1 - \varepsilon)^{\beta-1} \]

Note: - both shape parameters have to be positive (i.e. > 0)

Definition at line 3137 of file TEfficiency.cxx.

SetBetaBeta()

void TEfficiency::SetBetaBeta

(

Double_t

beta

)

Sets the shape parameter β.

The prior probability of the efficiency is given by the beta distribution:

\[ f(\varepsilon;\alpha,\beta) = \frac{1}{B(\alpha,\beta)} \varepsilon^{\alpha-1} (1 - \varepsilon)^{\beta-1} \]

Note: - both shape parameters have to be positive (i.e. > 0)

Definition at line 3155 of file TEfficiency.cxx.

SetBetaBinParameters()

void TEfficiency::SetBetaBinParameters	(	Int_t	bin,
		Double_t	alpha,
		Double_t	beta
	)

Sets different shape parameter α and β for the prior distribution for each bin.

By default the global parameter are used if they are not set for the specific bin The prior probability of the efficiency is given by the beta distribution:

\[ f(\varepsilon;\alpha;\beta) = \frac{1}{B(\alpha,\beta)} \varepsilon^{\alpha-1} (1 - \varepsilon)^{\beta-1} \]

Note:

• both shape parameters have to be positive (i.e. > 0)
• bin gives the global bin number (cf. GetGlobalBin)

Definition at line 3176 of file TEfficiency.cxx.

SetBins() [1/6]

Bool_t TEfficiency::SetBins	(	Int_t	nx,
		const Double_t *	xBins
	)

Set the bins for the underlined passed and total histograms If the class have been already filled the previous contents will be lost.

Definition at line 3217 of file TEfficiency.cxx.

SetBins() [2/6]

Bool_t TEfficiency::SetBins	(	Int_t	nx,
		const Double_t *	xBins,
		Int_t	ny,
		const Double_t *	yBins
	)

Set the bins for the underlined passed and total histograms If the class have been already filled the previous contents will be lost.

Definition at line 3257 of file TEfficiency.cxx.

SetBins() [3/6]

Bool_t TEfficiency::SetBins	(	Int_t	nx,
		const Double_t *	xBins,
		Int_t	ny,
		const Double_t *	yBins,
		Int_t	nz,
		const Double_t *	zBins
	)

Set the bins for the underlined passed and total histograms If the class have been already filled the previous contents will be lost.

Definition at line 3298 of file TEfficiency.cxx.

SetBins() [4/6]

Bool_t TEfficiency::SetBins	(	Int_t	nx,
		Double_t	xmin,
		Double_t	xmax
	)

Set the bins for the underlined passed and total histograms If the class have been already filled the previous contents will be lost.

Definition at line 3197 of file TEfficiency.cxx.

SetBins() [5/6]

Bool_t TEfficiency::SetBins	(	Int_t	nx,
		Double_t	xmin,
		Double_t	xmax,
		Int_t	ny,
		Double_t	ymin,
		Double_t	ymax
	)

Set the bins for the underlined passed and total histograms If the class have been already filled the previous contents will be lost.

Definition at line 3237 of file TEfficiency.cxx.

SetBins() [6/6]

Bool_t TEfficiency::SetBins	(	Int_t	nx,
		Double_t	xmin,
		Double_t	xmax,
		Int_t	ny,
		Double_t	ymin,
		Double_t	ymax,
		Int_t	nz,
		Double_t	zmin,
		Double_t	zmax
	)

Set the bins for the underlined passed and total histograms If the class have been already filled the previous contents will be lost.

Definition at line 3277 of file TEfficiency.cxx.

SetCentralInterval()

void TEfficiency::SetCentralInterval ( Bool_t  on = true )

inline

Definition at line 140 of file TEfficiency.h.

SetConfidenceLevel()

void TEfficiency::SetConfidenceLevel

(

Double_t

level

)

Sets the confidence level (0 < level < 1) The default value is 1-sigma :~ 0.683.

Definition at line 3319 of file TEfficiency.cxx.

SetDirectory()

void TEfficiency::SetDirectory

(

TDirectory *

dir

)

Sets the directory holding this TEfficiency object.

A reference to this TEfficiency object is removed from the current directory (if it exists) and a new reference to this TEfficiency object is added to the given directory.

Notes: - If the given directory is 0, the TEfficiency object does not belong to any directory and will not be written to file during the next TFile::Write() command.

Definition at line 3338 of file TEfficiency.cxx.

SetName()

void TEfficiency::SetName ( const char *  name )

overridevirtual

Sets the name.

Note: The names of the internal histograms are set to "name + _total" and "name + _passed" respectively.

Reimplemented from TNamed.

Definition at line 3355 of file TEfficiency.cxx.

SetPassedEvents()

Bool_t TEfficiency::SetPassedEvents	(	Int_t	bin,
		Int_t	events
	)

Sets the number of passed events in the given global bin.

returns "true" if the number of passed events has been updated otherwise "false" ist returned

Note: - requires: 0 <= events <= fTotalHistogram->GetBinContent(bin)

Definition at line 3374 of file TEfficiency.cxx.

SetPassedHistogram()

Bool_t TEfficiency::SetPassedHistogram	(	const TH1 &	rPassed,
		Option_t *	opt
	)

Sets the histogram containing the passed events.

The given histogram is cloned and stored internally as histogram containing the passed events. The given histogram has to be consistent with the current fTotalHistogram (see CheckConsistency(const TH1&,const TH1&)). The method returns whether the fPassedHistogram has been replaced (true) or not (false).

Note: The list of associated functions fFunctions is cleared.

Option:

• "f": force the replacement without checking the consistency This can lead to inconsistent histograms and useless results or unexpected behaviour. But sometimes it might be the only way to change the histograms. If you use this option, you should ensure that the fTotalHistogram is replaced by a consistent one (with respect to rPassed) as well.

Definition at line 3405 of file TEfficiency.cxx.

SetPosteriorAverage()

void TEfficiency::SetPosteriorAverage ( Bool_t  on = true )

inline

Definition at line 138 of file TEfficiency.h.

SetPosteriorMode()

void TEfficiency::SetPosteriorMode ( Bool_t  on = true )

inline

Definition at line 137 of file TEfficiency.h.

SetShortestInterval()

void TEfficiency::SetShortestInterval ( Bool_t  on = true )

inline

Definition at line 139 of file TEfficiency.h.

SetStatisticOption()

void TEfficiency::SetStatisticOption

(

EStatOption

option

)

Sets the statistic option which affects the calculation of the confidence interval.

Options:

• kFCP (=0)(default): using the Clopper-Pearson interval (recommended by PDG) sets kIsBayesian = false see also ClopperPearson
• kFNormal (=1) : using the normal approximation sets kIsBayesian = false see also Normal
• kFWilson (=2) : using the Wilson interval sets kIsBayesian = false see also Wilson
• kFAC (=3) : using the Agresti-Coull interval sets kIsBayesian = false see also AgrestiCoull
• kFFC (=4) : using the Feldman-Cousins frequentist method sets kIsBayesian = false see also FeldmanCousins
• kBJeffrey (=5) : using the Jeffrey interval sets kIsBayesian = true, fBeta_alpha = 0.5 and fBeta_beta = 0.5 see also Bayesian
• kBUniform (=6) : using a uniform prior sets kIsBayesian = true, fBeta_alpha = 1 and fBeta_beta = 1 see also Bayesian
• kBBayesian (=7) : using a custom prior defined by fBeta_alpha and fBeta_beta sets kIsBayesian = true see also Bayesian
• kMidP (=8) : using the Lancaster Mid-P method sets kIsBayesian = false

Definition at line 3470 of file TEfficiency.cxx.

SetTitle()

void TEfficiency::SetTitle ( const char *  title )

overridevirtual

Sets the title.

Notes:

• The titles of the internal histograms are set to "title + (total)" or "title + (passed)" respectively.
• It is possible to label the axis of the histograms as usual (see TH1::SetTitle).

Example: Setting the title to "My Efficiency" and label the axis pEff->SetTitle("My Efficiency;x label;eff");

Reimplemented from TNamed.

Definition at line 3534 of file TEfficiency.cxx.

SetTotalEvents()

Bool_t TEfficiency::SetTotalEvents	(	Int_t	bin,
		Int_t	events
	)

Sets the number of total events in the given global bin.

returns "true" if the number of total events has been updated otherwise "false" ist returned

Note: - requires: fPassedHistogram->GetBinContent(bin) <= events

Definition at line 3568 of file TEfficiency.cxx.

SetTotalHistogram()

Bool_t TEfficiency::SetTotalHistogram	(	const TH1 &	rTotal,
		Option_t *	opt
	)

Sets the histogram containing all events.

The given histogram is cloned and stored internally as histogram containing all events. The given histogram has to be consistent with the current fPassedHistogram (see CheckConsistency(const TH1&,const TH1&)). The method returns whether the fTotalHistogram has been replaced (true) or not (false).

Note: The list of associated functions fFunctions is cleared.

Option:

• "f": force the replacement without checking the consistency This can lead to inconsistent histograms and useless results or unexpected behaviour. But sometimes it might be the only way to change the histograms. If you use this option, you should ensure that the fPassedHistogram is replaced by a consistent one (with respect to rTotal) as well.

Definition at line 3599 of file TEfficiency.cxx.

SetUseWeightedEvents()

void TEfficiency::SetUseWeightedEvents

(

Bool_t

on = kTRUE

)

Definition at line 3633 of file TEfficiency.cxx.

SetWeight()

void TEfficiency::SetWeight

(

Double_t

weight

)

Sets the global weight for this TEfficiency object.

Note: - weight has to be positive ( > 0)

Definition at line 3651 of file TEfficiency.cxx.

Streamer()

void TEfficiency::Streamer ( TBuffer &  R__b )

overridevirtual

Stream an object of class TObject.

Reimplemented from TObject.

StreamerNVirtual()

void TEfficiency::StreamerNVirtual ( TBuffer &  ClassDef_StreamerNVirtual_b )

inline

Definition at line 188 of file TEfficiency.h.

UsesBayesianStat()

Bool_t TEfficiency::UsesBayesianStat ( ) const

inline

Definition at line 156 of file TEfficiency.h.

UsesCentralInterval()

Bool_t TEfficiency::UsesCentralInterval ( ) const

inline

Definition at line 160 of file TEfficiency.h.

UsesPosteriorAverage()

Bool_t TEfficiency::UsesPosteriorAverage ( ) const

inline

Definition at line 159 of file TEfficiency.h.

UsesPosteriorMode()

Bool_t TEfficiency::UsesPosteriorMode ( ) const

inline

Definition at line 157 of file TEfficiency.h.

UsesShortestInterval()

Bool_t TEfficiency::UsesShortestInterval ( ) const

inline

Definition at line 158 of file TEfficiency.h.

UsesWeights()

Bool_t TEfficiency::UsesWeights ( ) const

inline

Definition at line 161 of file TEfficiency.h.

Wilson()

Double_t TEfficiency::Wilson ( Double_t  total, Double_t  passed, Double_t  level, Bool_t  bUpper  )

static

Calculates the boundaries for the frequentist Wilson interval.

Parameters

[in]	total	number of total events
[in]	passed	0 <= number of passed events <= total
[in]	level	confidence level
[in]	bUpper	true - upper boundary is returned false - lower boundary is returned

Calculation:

\begin{eqnarray*} \alpha &=& 1 - \frac{level}{2}\\ \kappa &=& \Phi^{-1}(1 - \alpha,1) ...\ normal\ quantile\ function\\ mode &=& \frac{passed + \frac{\kappa^{2}}{2}}{total + \kappa^{2}}\\ \Delta &=& \frac{\kappa}{total + \kappa^{2}} * \sqrt{passed (1 - \frac{passed}{total}) + \frac{\kappa^{2}}{4}}\\ return &=& max(0,mode - \Delta)\ or\ min(1,mode + \Delta) \end{eqnarray*}

Definition at line 3681 of file TEfficiency.cxx.

Member Data Documentation

fBeta_alpha

Double_t TEfficiency::fBeta_alpha

protected

Global parameter for prior beta distribution (default = 1)

Definition at line 46 of file TEfficiency.h.

fBeta_beta

Double_t TEfficiency::fBeta_beta

protected

Global parameter for prior beta distribution (default = 1)

Definition at line 47 of file TEfficiency.h.

fBeta_bin_params

std::vector<std::pair<Double_t, Double_t> > TEfficiency::fBeta_bin_params

protected

Parameter for prior beta distribution different bin by bin (default vector is empty)

Definition at line 48 of file TEfficiency.h.

fBoundary

Double_t(* TEfficiency::fBoundary) (Double_t, Double_t, Double_t, Bool_t)

protected

! Pointer to a method calculating the boundaries of confidence intervals

Definition at line 50 of file TEfficiency.h.

fConfLevel

Double_t TEfficiency::fConfLevel

protected

Confidence level (default = 0.683, 1 sigma)

Definition at line 51 of file TEfficiency.h.

fDirectory

TDirectory* TEfficiency::fDirectory

protected

! Pointer to directory holding this TEfficiency object

Definition at line 52 of file TEfficiency.h.

fFunctions

TList* TEfficiency::fFunctions

protected

->Pointer to list of functions

Definition at line 53 of file TEfficiency.h.

fPaintGraph

TGraphAsymmErrors* TEfficiency::fPaintGraph

protected

! Temporary graph for painting

Definition at line 54 of file TEfficiency.h.

fPaintHisto

TH2* TEfficiency::fPaintHisto

protected

! Temporary histogram for painting

Definition at line 55 of file TEfficiency.h.

fPassedHistogram

TH1* TEfficiency::fPassedHistogram

protected

Histogram for events which passed certain criteria.

Definition at line 56 of file TEfficiency.h.

fStatisticOption

EStatOption TEfficiency::fStatisticOption

protected

Defines how the confidence intervals are determined.

Definition at line 57 of file TEfficiency.h.

fTotalHistogram

TH1* TEfficiency::fTotalHistogram

protected

Histogram for total number of events.

Definition at line 58 of file TEfficiency.h.

fWeight

Double_t TEfficiency::fWeight

protected

Weight for all events (default = 1)

Definition at line 59 of file TEfficiency.h.

Libraries for TEfficiency:

---

The documentation for this class was generated from the following files:

• hist/hist/inc/TEfficiency.h
• hist/hist/src/TEfficiency.cxx