#ifndef ROOT_TEfficiency_cxx #define ROOT_TEfficiency_cxx //standard header #include <vector> #include <string> #include <cmath> #include <stdlib.h> //ROOT headers #include "Math/DistFuncMathCore.h" #include "TBinomialEfficiencyFitter.h" #include "TDirectory.h" #include "TF1.h" #include "TGraphAsymmErrors.h" #include "TH1.h" #include "TH2.h" #include "TH3.h" #include "TList.h" #include "TMath.h" #include "TROOT.h" #include "TStyle.h" #include "TVirtualPad.h" #include "TError.h" #include "Math/BrentMinimizer1D.h" #include "Math/WrappedFunction.h" //custom headers #include "TEfficiency.h" // file with extra class for FC method #include "TEfficiencyHelper.h" //default values const Double_t kDefBetaAlpha = 1; const Double_t kDefBetaBeta = 1; const Double_t kDefConfLevel = 0.682689492137; // 1 sigma const TEfficiency::EStatOption kDefStatOpt = TEfficiency::kFCP; const Double_t kDefWeight = 1; ClassImp(TEfficiency) //______________________________________________________________________________ /***************************************************************************** Begin_Html <center><h2>TEfficiency - a class to handle efficiency histograms</h2></center> <ol style="list-style-type: upper-roman;"> <li><a href="#over">Overview</a></li> <li><a href="#create">Creating a TEfficiency object</a></li> <li><a href="#fill">Fill in events</a></li> <li><a href="#stat">Statistic options</a></li> <ol><li><a href="#compare">Coverage probabilities for different methods</a></li></ol> <li><a href="#cm">Merging and combining TEfficiency objects</a></li> <ol> <li><a href="#merge">When should I use merging?</a></li> <li><a href="#comb">When should I use combining?</a></li> </ol> <li><a href="#other">Further operations</a> <ol> <li><a href="#histo">Information about the internal histograms</a></li> <li><a href="#fit">Fitting</a></li> <li><a href="#draw">Draw a TEfficiency object</a></li> </ol> <li><a href="#class">TEfficiency class</a></li> </ol> <h3><a name="over">I. Overview</a></h3> 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. <br /> 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 (<a href="http://root.cern.ch/root/html/TEfficiency.html#TEfficiency:fTotalHistogram">fTotalHistogram</a> and <a href="http://root.cern.ch/root/html/TEfficiency.html#TEfficiency:fPassedHistogram">fPassedHistogram</a>). Then the efficiency as well as its upper and lower error an be calculated for each bin individually.<br /> As the efficiency can be regarded as a parameter of a binomial distribution, the number of pass ed 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 (<a href="http://root.cern.ch/root/html/TEfficiency.html#TEfficiency:SetWeight">SetWeight</a>). 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. <h3><a name="create">II. Creating a TEfficiency object</a></h3> 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 <a href="http://root.cern.ch/root/html/TFile.html#TFile:Write">TFile::Write</a> command. <div class="code"><pre> <b>Example:</b> create a twodimensional TEfficiency object with - name = "eff" - title = "my efficiency" - axistitles: x, y and LaTeX formated epsilon as label for Z axis - 10 bins with constant bin width (= 1) along X axis starting at 0 (lower edge from first bin) upto 10 (upper edge of last bin) - 20 bins with constant bin width (= 0.5) along Y axis starting at -5 (lower edge from first bin) upto 5 (upper edge of last bin) TEfficiency* pEff = new TEfficiency("eff","my efficiency;x;y;#epsilon",10,0,10,20,-5,5); </pre></div><div class="clear" /> If you already have two histograms filled with the number of passed and total events, you will use the constructor <a href="http://root.cern.ch/root/html/TEfficiency.html#TEfficiency:TEfficiency%1">TEfficiency(const TH1& passed,const TH1& total)</a> to construct the TEfficiency object. The histograms "passed" and "total" have to fullfill the conditions mentioned in <a href="http://root.cern.ch/root/html/TEfficiency.html#TEfficiency:CheckConsistency">CheckConsistency</a>, otherwise the construction will fail. As the histograms already exist, the new TEfficiency is by default <b>not</b> 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 <a href="http://root.cern.ch/root/html/TObject.html#TObject:Write">Write</a> or attach it to a directory using <a href="http://root.cern.ch/root/html/TEfficiency.html#TEfficiency:SetDirectory">SetDirectory</a>. This also applies for TEfficiency objects created by the copy constructor <a href="http://root.cern.ch/root/html/TEfficiency.html#TEfficiency:TEfficiency%8">TEfficiency(const TEfficiency& rEff)</a>. <div class="code"> <pre> <b>Example 1:</b> 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(); } <b>Example 2:</b> 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(); } </pre> </div><div class="clear" /> In the case that you already have two filled histograms and you only want to plot them as a graph, you should rather use <a href="http://root.cern.ch/root/html/TGraphAsymmErrors.html#TGraphAsymmErrors:TGraphAsymmErrors%8">TGraphAsymmErrors::TGraphAsymmErrors(const TH1* pass,const TH1* total,Option_t* opt)</a> to create a graph object. <h3><a name="fill">III. Filling with events</a></h3> You can fill the TEfficiency object by calling the <a href="http://root.cern.ch/root/html/TEfficiency.html#TEfficiency:Fill">Fill(Bool_t bPassed,Double_t x,Double_t y,Double_t z)</a> method. The boolean flag "bPassed" indicates whether the current event is a good (both histograms are filled) or not (only <a href="http://root.cern.ch/root/html/TEfficiency.html#TEfficiency:fTotalHistogram">fTotalHistogram</a> is filled). The variables x,y and z determine the bin which is filled. For lower dimensions the z- or even the y-value may be omitted. End_Html Begin_Macro(source) { //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"); //only for this documentation return c1; } End_Macro Begin_Html You can also set the number of passed or total events for a bin directly by using the <a href="http://root.cern.ch/root/html/TEfficiency.html#TEfficiency:SetPassedEvents">SetPassedEvents</a> or <a href="http://root.cern.ch/root/html/TEfficiency.html#TEfficiency:SetTotalEvents">SetTotalEvents</a> method. <h3><a name="stat">IV. Statistic options</a></h3> 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.<br /> <b>Frequentist methods</b><br /> 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. End_Html Begin_Latex #LT k #GT = #epsilon #times N #Rightarrow #hat{#varepsilon} = #frac{k}{N} End_Latex Begin_Html <b>Bayesian methods</b><br /> 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. End_Html Begin_Latex(separator='=',align='rl') 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_Latex Begin_Html By default the expectation value of this posterior distribution is used as estimator for the efficiency: End_Html Begin_Latex #hat{#varepsilon} = #frac{k + #alpha}{N + #alpha + #beta} End_Latex Begin_Html Optionally the mode can also be used as value for the estimated efficiency. This can be done by calling SetBit(kPosteriorMode) or <a href="http://root.cern.ch/root/html/TEfficiency.html#TEfficiency:SetPosteriorMode">SetPosteriorMode</a>. In this case the estimated efficiency is: End_Html Begin_Latex #hat{#varepsilon} = #frac{k + #alpha -1}{N + #alpha + #beta - 2} End_Latex Begin_Html 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 specifiy which confidence interval is used for calculating the uncertainties of the efficiency. The following properties define the error calculation: <ul> <li><b>fConfLevel:</b> desired confidence level: 0 < fConfLevel < 1 (<a href="http://root.cern.ch/root/html/TEfficiency.html#TEfficiency:GetConfidenceLevel">GetConfidenceLevel</a> / <a href="http://root.cern.ch/root/html/TEfficiency.html#TEfficiency:SetConfidenceLevel">SetConfidenceLevel</a>)</li> <li><b>fStatisticOption:</b> defines which method is used to calculate the boundaries of the confidence interval (<a href="http://root.cern.ch/root/html/TEfficiency.html#TEfficiency:SetStatisticOption">SetStatisticOption</a>)</li> <li><b>fBeta_alpha, fBeta_beta:</b> parameters for the prior distribution which is only used in the bayesian case (<a href="http://root.cern.ch/root/html/TEfficiency.html#TEfficiency:GetBetaAlpha">GetBetaAlpha</a> / <a href="http://root.cern.ch/root/html/TEfficiency.html#TEfficiency:GetBetaBeta">GetBetaBeta</a> / <a href="http://root.cern.ch/root/html/TEfficiency.html#TEfficiency:SetBetaAlpha">SetBetaAlpha</a> / <a href="http://root.cern.ch/root/html/TEfficiency.html#TEfficiency:SetBetaBeta">SetBetaBeta</a>)</li> <li><b>kIsBayesian:</b> flag whether bayesian statistics are used or not (<a href="http://root.cern.ch/root/html/TEfficiency.html#TEfficiency:UsesBayesianStat">UsesBayesianStat</a>)</li> <li><b>kShortestInterval:</b> flag whether shortest interval (instead of central one) are used in case of Bayesian statistics (<a href="http://root.cern.ch/root/html/TEfficiency.html#TEfficiency:UsesShortestInterval">UsesShortestInterval</a>). Normally shortest interval should be used in combination with the mode (see <a href="http://root.cern.ch/root/html/TEfficiency.html#TEfficiency:UsesPosteriorMode">UsesPosteriorMode</a>)</li> <li><b>fWeight:</b> global weight for this TEfficiency object which is used during combining or merging with other TEfficiency objects(<a href="http://root.cern.ch/root/html/TEfficiency.html#TEfficiency:GetWeight">GetWeight</a> / <a href="http://root.cern.ch/root/html/TEfficiency.html#TEfficiency:SetWeight">SetWeight</a>)</li> </ul> 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 the mentioned functions.<br /><br /> <table align="center" border="1" cellpadding="5" rules="rows" vspace="10"> <caption align="bottom">implemented confidence intervals and their options</caption> <tr> <th>name</th><th>statistic option</th><th>function</th><th>kIsBayesian</th><th>parameters</th> </tr> <tr> <td>Clopper-Pearson</td><td>kFCP</td> <td> <a href="http://root.cern.ch/root/html/TEfficiency.html#TEfficiency:ClopperPearson">ClopperPearson</a> </td> <td>false</td> <td> <ul><li>total events</li><li>passed events</li><li>confidence level</li></ul> </td> </tr> <tr> <td>normal approximation</td><td>kFNormal</td> <td> <a href="http://root.cern.ch/root/html/TEfficiency.html#TEfficiency:Normal">Normal</a> </td> <td>false</td> <td> <ul><li>total events</li><li>passed events</li><li>confidence level</li></ul> </td> </tr> <tr> <td>Wilson</td><td>kFWilson</td> <td> <a href="http://root.cern.ch/root/html/TEfficiency.html#TEfficiency:Wilson">Wilson</a> </td> <td>false</td> <td> <ul><li>total events</li><li>passed events</li><li>confidence level</li></ul> </td> </tr> <tr> <td>Agresti-Coull</td><td>kFAC</td> <td> <a href="http://root.cern.ch/root/html/TEfficiency.html#TEfficiency:AgrestiCoull">AgrestiCoull</a> </td> <td>false</td> <td> <ul><li>total events</li><li>passed events</li><li>confidence level</li></ul> </td> </tr> <tr> <td>Feldman-Cousins</td><td>kFFC</td> <td> <a href="http://root.cern.ch/root/html/TEfficiency.html#TEfficiency:FeldmanCousins">FeldmanCousins</a> </td> <td>false</td> <td> <ul><li>total events</li><li>passed events</li><li>confidence level</li></ul> </td> </tr> <tr> <td>Jeffrey</td><td>kBJeffrey</td> <td> <a href="http://root.cern.ch/root/html/TEfficiency.html#TEfficiency:Bayesian">Bayesian</a> </td> <td>true</td> <td> <ul><li>total events</li><li>passed events</li><li>confidence level</li><li>fBeta_alpha = 0.5</li><li>fBeta_beta = 0.5</li></ul> </td> </tr> <td>Uniform prior</td><td>kBUniform</td> <td> <a href="http://root.cern.ch/root/html/TEfficiency.html#TEfficiency:Bayesian">Bayesian</a> </td> <td>true</td> <td> <ul><li>total events</li><li>passed events</li><li>confidence level</li><li>fBeta_alpha = 1</li><li>fBeta_beta = 1</li></ul> </td> </tr> <td>custom prior</td><td>kBBayesian</td> <td> <a href="http://root.cern.ch/root/html/TEfficiency.html#TEfficiency:Bayesian">Bayesian</a> </td> <td>true</td> <td> <ul><li>total events</li><li>passed events</li><li>confidence level</li><li>fBeta_alpha</li><li>fBeta_beta</li></ul> </td> </tr> </table> <br /> The following example demonstrates the effect of different statistic options and confidence levels. End_Html Begin_Macro(source) { //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"); //only for this documentation c1->cd(0); return c1; } End_Macro Begin_Html The prior probability of the efficiency in bayesian statistics can be given in terms of a beta distribution. The beta distribution has to positive shape parameters. The resulting priors for different combinations of these shape parameters are shown in the plot below. End_Html Begin_Macro(source) { //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"); //only for this documentation return c1; } End_Macro Begin_Html <h4><a name="compare">IV.1 Coverage probabilities for different methods</a></h4> 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. <p><img src="http://root.cern.ch/drupal/sites/default/files/images/normal95.gif" alt="normal approximation" width="600" height="400" /></p> <p><img src="http://root.cern.ch/drupal/sites/default/files/images/wilson95.gif" alt="wilson" width="600" height="400" /></p> <p><img src="http://root.cern.ch/drupal/sites/default/files/images/ac95.gif" alt="agresti coull" width="600" height="400" /></p> <p><img src="http://root.cern.ch/drupal/sites/default/files/images/cp95.gif" alt="clopper pearson" width="600" height="400" /></p> <p><img src="http://root.cern.ch/drupal/sites/default/files/images/uni95.gif" alt="uniform prior" width="600" height="400" /></p> <p><img src="http://root.cern.ch/drupal/sites/default/files/images/jeffrey95.gif" alt="jeffrey prior" width="600" height="400" /></p> The average (over all possible true efficiencies) coverage probability for different number of total events is shown in the next picture. <p><img src="http://root.cern.ch/drupal/sites/default/files/images/av_cov.png" alt="average coverage" width="600" height="400" /></p> <h3><a name="cm">V. Merging and combining TEfficiency objects</a></h3> In many applications the efficiency should be calculated for an inhomogenous 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 finsihing your analysis you can then construct the overall efficiency with its uncertainty.<br /> 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 efford 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.<br /><br /> <b>Example</b><br /> In high 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 End_Html Begin_Latex(separator='-') 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_Latex Begin_Html The reason for different weights can therefore be:<ul> <li>different processes</li> <li>other integrated luminosity</li> <li>varying trigger efficiency</li> <li>different sample sizes</li> <li>...</li> <li>or even combination of them</li> </ul> Depending on the actual meaning of different weights in your case, you should either merge or combine them to get the overall efficiency. <h4><a name="merge">V.1 When should I use merging?</a></h4> 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: <ul> <li> <b>eff1</b>.<a href="http://root.cern.ch/root/html/TEfficiency.html#TEfficiency:Add">Add</a>(eff2)</li> <li> <b>eff1</b> <a href="http://root.cern.ch/root/html/TEfficiency.html#TEfficiency:operator+=">+=</a> eff2</li> <li> <b>eff</b> = eff1 + eff2</li> </ul> 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. End_Html Begin_Latex #frac{1}{w_{new}} = #frac{1}{w_{1}} + #frac{1}{w_{2}}End_Latex Begin_Html <b>Example:</b><br /> 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. End_Html Begin_Latex 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}}} End_Latex Begin_Html <h4><a name="comb">V.2 When should I use combining?</a></h4> 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 <a href="http://root.cern.ch/root/html/TEfficiency.html#TEfficiency:Combine">Combine </a>only supports combination of 1-dimensional efficiencies in a bayesian approach.<br /> 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. End_Html Begin_Latex 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_Latex Begin_Html <b>Example:</b><br /> 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. End_Html Begin_Latex 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}} End_Latex Begin_Html <h3><a name="other">VI. Further operations</a></h3> <h4><a name="histo">VI.1 Information about the internal histograms</a></h4> The methods <a href="http://root.cern.ch/root/html/TEfficiency.html#TEfficiency:GetPassedHistogram">GetPassedHistogram</a> and <a href="http://root.cern.ch/root/html/TEfficiency.html#TEfficiency:GetTotalHistogram">GetTotalHistogram</a> 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 <a href="http://root.cern.ch/root/html/TEfficiency.html#TEfficiency:GetCopyPassedHisto">GetCopyPassedHisto</a> or <a href="http://root.cern.ch/root/html/TEfficiency.html#TEfficiency:GetCopyTotalHisto">GetCopyTotalHisto</a>. 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. <div class="code"> <pre> <b>Example:</b> //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 directoy and write the file //clone->SetDirectory(gDirectory); //pFile->Wrtie(); //delete histogram object delete clone; clone = 0; </pre> </div><div class="clear" /> It is also possible to set the internal total or passed histogram by using the methods <a href="http://root.cern.ch/root/html/TEfficiency.html#TEfficiency:SetPassedHistogram">SetPassedHistogram</a> or <a href="http://root.cern.ch/root/html/TEfficiency.html#TEfficiency:SetTotalHistogram">SetTotalHistogram</a>. In order to ensure the validity of the TEfficiency object, the consistency of the new histogram and the stored histogram is checked. It sometimes might be impossible to change the histograms in a consistent way. Therefore one can force the replacement by passing the option "f". 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. <h4><a name="fit">VI.2 Fitting</a></h4> The efficiency can be fitted using the <a href="http://root.cern.ch/root/html/TEfficiency.html#TEfficiency:Fit">Fit</a> function which uses internally the <a href="http://root.cern.ch/root/html/TBinomialEfficiencyFitter.html#TBinomialEfficiencyFitter:Fit">TBinomialEfficiencyFitter::Fit</a> 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 <a href="http://root.cern.ch/root/html/TEfficiency.html#TEfficiency:Draw">Draw</a> command. The list of associated function can be modified by using the pointer returned by <a href="http://root.cern.ch/root/html/TEfficiency.html#TEfficiency:GetListOfFunctions">GetListOfFunctions</a>. End_Html Begin_Macro(source) { //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"); //only for this documentation return c1; } End_Macro Begin_Html <h4><a name="draw">VI.3 Draw a TEfficiency object</a></h4> A TEfficiency object can be drawn by calling the usual <a href="http://root.cern.ch/root/html/TEfficiency.html#TEfficiency:Draw">Draw</a> 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 <br /> <a href="http://root.cern.ch/root/html/TGraph.html#TGraph:Draw">TGraphAsymmErrors::Draw</a> method. For 2-dimensional TEfficiency objects you can pass the same options as for a <a href="http://root.cern.ch/root/html/TH1.html#TH1:Draw">TH2::Draw</a> object. <h3 style="margin-bottom:-3em;"><a name="class">VII. TEfficiency class</a></h3> End_Html ********************************************************************************/ //______________________________________________________________________________ //______________________________________________________________________________ TEfficiency::TEfficiency(): fBeta_alpha(kDefBetaAlpha), fBeta_beta(kDefBetaBeta), fBoundary(0), fConfLevel(kDefConfLevel), fDirectory(0), fFunctions(0), fPaintGraph(0), fPaintHisto(0), fPassedHistogram(0), fTotalHistogram(0), fWeight(kDefWeight) { //default constructor // //should not be used explicitly SetStatisticOption(kDefStatOpt); // create 2 dummy histograms fPassedHistogram = new TH1F("h_passed","passed",10,0,10); fTotalHistogram = new TH1F("h_total","total",10,0,10); } //______________________________________________________________________________ TEfficiency::TEfficiency(const TH1& passed,const TH1& total): fBeta_alpha(kDefBetaAlpha), fBeta_beta(kDefBetaBeta), fConfLevel(kDefConfLevel), fDirectory(0), fFunctions(0), fPaintGraph(0), fPaintHisto(0), fWeight(kDefWeight) { //constructor using two existing histograms as input // //Input: passed - contains the events fullfilling some criteria // total - contains all investigated events // //Notes: - both histograms have to fullfill the conditions of CheckConsistency (with option 'w') // - dimension of the resulating 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(). //check consistency of histograms if(CheckConsistency(passed,total,"w")) { Bool_t bStatus = TH1::AddDirectoryStatus(); TH1::AddDirectory(kFALSE); fTotalHistogram = (TH1*)total.Clone(); fPassedHistogram = (TH1*)passed.Clone(); TH1::AddDirectory(bStatus); TString newName = total.GetName(); newName += TString("_clone"); SetName(newName); // are the histograms filled with weights? if(!CheckEntries(passed,total)) { Info("TEfficiency","given histograms are filled with weights"); SetUseWeightedEvents(); } } else { Error("TEfficiency(const TH1&,const TH1&)","histograms are not consistent -> results are useless"); Warning("TEfficiency(const TH1&,const TH1&)","using two empty TH1D('h1','h1',10,0,10)"); Bool_t bStatus = TH1::AddDirectoryStatus(); TH1::AddDirectory(kFALSE); fTotalHistogram = new TH1D("h1_total","h1 (total)",10,0,10); fPassedHistogram = new TH1D("h1_passed","h1 (passed)",10,0,10); TH1::AddDirectory(bStatus); } SetBit(kPosteriorMode,false); SetBit(kShortestInterval,false); SetStatisticOption(kDefStatOpt); SetDirectory(0); } //______________________________________________________________________________ TEfficiency::TEfficiency(const char* name,const char* title,Int_t nbins, const Double_t* xbins): fBeta_alpha(kDefBetaAlpha), fBeta_beta(kDefBetaBeta), fConfLevel(kDefConfLevel), fDirectory(0), fFunctions(0), fPaintGraph(0), fPaintHisto(0), fWeight(kDefWeight) { //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 Bool_t bStatus = TH1::AddDirectoryStatus(); TH1::AddDirectory(kFALSE); fTotalHistogram = new TH1D("total","total",nbins,xbins); fPassedHistogram = new TH1D("passed","passed",nbins,xbins); TH1::AddDirectory(bStatus); Build(name,title); } //______________________________________________________________________________ TEfficiency::TEfficiency(const char* name,const char* title,Int_t nbinsx, Double_t xlow,Double_t xup): fBeta_alpha(kDefBetaAlpha), fBeta_beta(kDefBetaBeta), fConfLevel(kDefConfLevel), fDirectory(0), fFunctions(0), fPaintGraph(0), fPaintHisto(0), fWeight(kDefWeight) { //create 1-dimensional TEfficiency object with fixed bins isze // //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 Bool_t bStatus = TH1::AddDirectoryStatus(); TH1::AddDirectory(kFALSE); fTotalHistogram = new TH1D("total","total",nbinsx,xlow,xup); fPassedHistogram = new TH1D("passed","passed",nbinsx,xlow,xup); TH1::AddDirectory(bStatus); Build(name,title); } //______________________________________________________________________________ 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): fBeta_alpha(kDefBetaAlpha), fBeta_beta(kDefBetaBeta), fConfLevel(kDefConfLevel), fDirectory(0), fFunctions(0), fPaintGraph(0), fPaintHisto(0), fWeight(kDefWeight) { //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 Bool_t bStatus = TH1::AddDirectoryStatus(); TH1::AddDirectory(kFALSE); fTotalHistogram = new TH2D("total","total",nbinsx,xlow,xup,nbinsy,ylow,yup); fPassedHistogram = new TH2D("passed","passed",nbinsx,xlow,xup,nbinsy,ylow,yup); TH1::AddDirectory(bStatus); Build(name,title); } //______________________________________________________________________________ TEfficiency::TEfficiency(const char* name,const char* title,Int_t nbinsx, const Double_t* xbins,Int_t nbinsy, const Double_t* ybins): fBeta_alpha(kDefBetaAlpha), fBeta_beta(kDefBetaBeta), fConfLevel(kDefConfLevel), fDirectory(0), fFunctions(0), fPaintGraph(0), fPaintHisto(0), fWeight(kDefWeight) { //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 Bool_t bStatus = TH1::AddDirectoryStatus(); TH1::AddDirectory(kFALSE); fTotalHistogram = new TH2D("total","total",nbinsx,xbins,nbinsy,ybins); fPassedHistogram = new TH2D("passed","passed",nbinsx,xbins,nbinsy,ybins); TH1::AddDirectory(bStatus); Build(name,title); } //______________________________________________________________________________ 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): fBeta_alpha(kDefBetaAlpha), fBeta_beta(kDefBetaBeta), fConfLevel(kDefConfLevel), fDirectory(0), fFunctions(0), fPaintGraph(0), fPaintHisto(0), fWeight(kDefWeight) { //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 Bool_t bStatus = TH1::AddDirectoryStatus(); TH1::AddDirectory(kFALSE); fTotalHistogram = new TH3D("total","total",nbinsx,xlow,xup,nbinsy,ylow,yup,nbinsz,zlow,zup); fPassedHistogram = new TH3D("passed","passed",nbinsx,xlow,xup,nbinsy,ylow,yup,nbinsz,zlow,zup); TH1::AddDirectory(bStatus); Build(name,title); } //______________________________________________________________________________ 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): fBeta_alpha(kDefBetaAlpha), fBeta_beta(kDefBetaBeta), fConfLevel(kDefConfLevel), fDirectory(0), fFunctions(0), fPaintGraph(0), fPaintHisto(0), fWeight(kDefWeight) { //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 Bool_t bStatus = TH1::AddDirectoryStatus(); TH1::AddDirectory(kFALSE); fTotalHistogram = new TH3D("total","total",nbinsx,xbins,nbinsy,ybins,nbinsz,zbins); fPassedHistogram = new TH3D("passed","passed",nbinsx,xbins,nbinsy,ybins,nbinsz,zbins); TH1::AddDirectory(bStatus); Build(name,title); } //______________________________________________________________________________ TEfficiency::TEfficiency(const TEfficiency& rEff): TNamed(), TAttLine(), TAttFill(), TAttMarker(), fBeta_alpha(rEff.fBeta_alpha), fBeta_beta(rEff.fBeta_beta), fBeta_bin_params(rEff.fBeta_bin_params), fConfLevel(rEff.fConfLevel), fDirectory(0), fFunctions(0), fPaintGraph(0), fPaintHisto(0), fWeight(rEff.fWeight) { //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(). // copy TObject bits ((TObject&)rEff).Copy(*this); Bool_t bStatus = TH1::AddDirectoryStatus(); TH1::AddDirectory(kFALSE); fTotalHistogram = (TH1*)((rEff.fTotalHistogram)->Clone()); fPassedHistogram = (TH1*)((rEff.fPassedHistogram)->Clone()); TH1::AddDirectory(bStatus); TString name = rEff.GetName(); name += "_copy"; SetName(name); TString title = "[copy] "; title += rEff.GetTitle(); SetTitle(title); SetStatisticOption(rEff.GetStatisticOption()); SetDirectory(0); //copy style rEff.TAttLine::Copy(*this); rEff.TAttFill::Copy(*this); rEff.TAttMarker::Copy(*this); } //______________________________________________________________________________ TEfficiency::~TEfficiency() { //default destructor //delete all function in fFunctions // use same logic as in TH1 destructor // (see TH1::~TH1 code in TH1.cxx) if(fFunctions) { fFunctions->SetBit(kInvalidObject); TObject* obj = 0; while ((obj = fFunctions->First())) { while(fFunctions->Remove(obj)) { } if (!obj->TestBit(kNotDeleted)) { break; } delete obj; obj = 0; } delete fFunctions; fFunctions = 0; } if(fDirectory) fDirectory->Remove(this); delete fTotalHistogram; delete fPassedHistogram; delete fPaintGraph; delete fPaintHisto; } //______________________________________________________________________________ Double_t TEfficiency::AgrestiCoull(Int_t total,Int_t passed,Double_t level,Bool_t bUpper) { //calculates the boundaries for the frequentist Agresti-Coull interval // //Input: - 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 // //calculation: //Begin_Latex(separator='=',align='rl') // #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_Latex Double_t alpha = (1.0 - level)/2; Double_t kappa = ROOT::Math::normal_quantile(1 - alpha,1); Double_t mode = (passed + 0.5 * kappa * kappa) / (total + kappa * kappa); Double_t delta = kappa * std::sqrt(mode * (1 - mode) / (total + kappa * kappa)); if(bUpper) return ((mode + delta) > 1) ? 1.0 : (mode + delta); else return ((mode - delta) < 0) ? 0.0 : (mode - delta); } //______________________________________________________________________________ Double_t TEfficiency::FeldmanCousins(Int_t total,Int_t passed,Double_t level,Bool_t bUpper) { //calculates the boundaries for the frequentist Feldman-Cousins interval // //Input: - 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 // // Double_t lower = 0; Double_t upper = 1; if (!FeldmanCousinsInterval(total,passed,level, lower, upper)) { ::Error("FeldmanCousins","Error running FC method - return 0 or 1"); } return (bUpper) ? upper : lower; } Bool_t TEfficiency::FeldmanCousinsInterval(Int_t total,Int_t passed,Double_t level,Double_t & lower, Double_t & upper) { //calculates the interval boundaries using the frequentist methods of Feldman-Cousins // //Input: - total : number of total events // - passed: 0 <= number of passed events <= total // - level : confidence level //Output: // - lower : lower boundary returned on exit // - upper : lower boundary returned on exit // //Return 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: //Begin_Latex(separator='=',align='rl') // LR = #frac{Binomial(k | N, #epsilon)}{Binomial(k | N, #hat{#epsilon} ) } //End_Latex //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 // FeldmanCousinsBinomialInterval fc; double alpha = 1.-level; fc.Init(alpha); fc.Calculate(passed, total); lower = fc.Lower(); upper = fc.Upper(); return true; } //______________________________________________________________________________ Double_t TEfficiency::Bayesian(Int_t total,Int_t passed,Double_t level,Double_t alpha,Double_t beta,Bool_t bUpper, Bool_t bShortest) { //calculates the boundaries for a Bayesian confidence interval (shortest or central interval depending on the option) // //Input: - total : number of total events // - passed: 0 <= number of passed events <= total // - level : confidence level // - alpha : shape parameter > 0 for the prior distribution (fBeta_alpha) // - beta : shape parameter > 0 for the prior distribution (fBeta_beta) // - bUpper: true - upper boundary is returned // false - lower boundary is returned // //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: //Begin_Latex P(#varepsilon |k,N) #propto L(#varepsilon|k,N) #times Prior(#varepsilon)End_Latex //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: //Begin_Latex L(#varepsilon|k,N) = Binomial(N,k) #varepsilon ^{k} (1 - #varepsilon)^{N-k}End_Latex //At the moment only beta distributions are supported as prior probabilities //of the efficiency (Begin_Latex #scale[0.8]{B(#alpha,#beta)}End_Latex is the beta function): //Begin_Latex Prior(#varepsilon) = #frac{1}{B(#alpha,#beta)} #varepsilon ^{#alpha - 1} (1 - #varepsilon)^{#beta - 1}End_Latex //The posterior probability is therefore again given by a beta distribution: //Begin_Latex P(#varepsilon |k,N) #propto #varepsilon ^{k + #alpha - 1} (1 - #varepsilon)^{N - k + #beta - 1} End_Latex //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 Begin_Latex #frac{1 - level}{2} End_Latex. //The upper boundary for the equal-tailed confidence interval is given by the //inverse cumulative (= quantile) function for the quantile Begin_Latex #frac{1 + level}{2} End_Latex. //Hence it is the solution Begin_Latex #varepsilon End_Latex of the following equation: //Begin_Latex 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} End_Latex // In the case of shortest interval the minimum interval aorund the mode is found by minimizing the length of all intervals whith the // given probability content. See TEfficiency::BetaShortestInterval Double_t a = double(passed)+alpha; Double_t b = double(total-passed)+beta; if (bShortest) { double lower = 0; double upper = 1; BetaShortestInterval(level,a,b,lower,upper); return (bUpper) ? upper : lower; } else return BetaCentralInterval(level, a, b, bUpper); } //______________________________________________________________________________ Double_t TEfficiency::BetaCentralInterval(Double_t level,Double_t a,Double_t b,Bool_t bUpper) { //calculates the boundaries for a central confidence interval for a Beta distribution // //Input: - level : confidence level // - a : parameter > 0 for the beta distribution (for a posterior is passed + prior_alpha // - b : parameter > 0 for the beta distribution (for a posterior is (total-passed) + prior_beta // - bUpper: true - upper boundary is returned // false - lower boundary is returned // if(bUpper) { if((a > 0) && (b > 0)) return ROOT::Math::beta_quantile((1+level)/2,a,b); else { gROOT->Error("TEfficiency::BayesianCentral","Invalid input parameters - return 1"); return 1; } } else { if((a > 0) && (b > 0)) return ROOT::Math::beta_quantile((1-level)/2,a,b); else { gROOT->Error("TEfficiency::BayesianCentral","Invalid input parameters - return 0"); return 0; } } } struct Beta_interval_length { Beta_interval_length(Double_t level,Double_t alpha,Double_t beta ) : fCL(level), fAlpha(alpha), fBeta(beta) {} Double_t LowerMax() { // max allowed value of lower given the interval size return ROOT::Math::beta_quantile_c(fCL, fAlpha,fBeta); } Double_t operator() (double lower) const { // return length of interval Double_t plow = ROOT::Math::beta_cdf(lower, fAlpha, fBeta); Double_t pup = plow + fCL; double upper = ROOT::Math::beta_quantile(pup, fAlpha,fBeta); return upper-lower; } Double_t fCL; // interval size (confidence level) Double_t fAlpha; // beta distribution alpha parameter Double_t fBeta; // beta distribution beta parameter }; //______________________________________________________________________________ Bool_t TEfficiency::BetaShortestInterval(Double_t level,Double_t a,Double_t b, Double_t & lower, Double_t & upper) { //calculates the boundaries for a shortest confidence interval for a Beta distribution // //Input: - level : confidence level // - a : parameter > 0 for the beta distribution (for a posterior is passed + prior_alpha // - b : parameter > 0 for the beta distribution (for a posterior is (total-passed) + prior_beta // - bUpper: true - upper boundary is returned // false - lower boundary is returned // // //The lower/upper boundary are then obtained by finding the shortest interval of the beta distribbution // contained the desired probability level. // The length of all possible intervals is minimized in order to find the shortest one if (a <= 0 || b <= 0) { lower = 0; upper = 1; gROOT->Error("TEfficiency::BayesianShortest","Invalid input parameters - return [0,1]"); return kFALSE; } // treat here special cases when mode == 0 or 1 double mode = BetaMode(a,b); if (mode == 0.0) { lower = 0; upper = ROOT::Math::beta_quantile(level, a, b); return kTRUE; } if (mode == 1.0) { lower = ROOT::Math::beta_quantile_c(level, a, b); upper = 1.0; return kTRUE; } // special case when the shortest interval is undefined return the central interval // can happen for a posterior when passed=total=0 // if ( a==b && a<=1.0) { lower = BetaCentralInterval(level,a,b,kFALSE); upper = BetaCentralInterval(level,a,b,kTRUE); return kTRUE; } // for the other case perform a minimization // make a function of the length of the posterior interval as a function of lower bound Beta_interval_length intervalLength(level,a,b); // minimize the interval length ROOT::Math::WrappedFunction<const Beta_interval_length &> func(intervalLength); ROOT::Math::BrentMinimizer1D minim; minim.SetFunction(func, 0, intervalLength.LowerMax() ); minim.SetNpx(2); // no need to bracket with many iterations. Just do few times to estimate some better points bool ret = minim.Minimize(100, 1.E-10,1.E-10); if (!ret) { gROOT->Error("TEfficiency::BayesianShortes","Error finding the shortest interval"); return kFALSE; } lower = minim.XMinimum(); upper = lower + minim.FValMinimum(); return kTRUE; } //______________________________________________________________________________ Double_t TEfficiency::BetaMean(Double_t a,Double_t b) { // compute the mean (average) of the beta distribution // //Input: a : parameter > 0 for the beta distribution (for a posterior is passed + prior_alpha // b : parameter > 0 for the beta distribution (for a posterior is (total-passed) + prior_beta // if (a <= 0 || b <= 0 ) { gROOT->Error("TEfficiency::BayesianMean","Invalid input parameters - return 0"); return 0; } Double_t mean = a / (a + b); return mean; } //______________________________________________________________________________ Double_t TEfficiency::BetaMode(Double_t a,Double_t b) { // compute the mode of the beta distribution // //Input: a : parameter > 0 for the beta distribution (for a posterior is passed + prior_alpha // 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; if (a <= 0 || b <= 0 ) { gROOT->Error("TEfficiency::BayesianMode","Invalid input parameters - return 0"); return 0; } if ( a <= 1 || b <= 1) { if ( a < b) return 0; if ( a > b) return 1; if (a == b) return 0.5; // cannot do otherwise } // since a and b are > 1 here denominator cannot be 0 or < 0 Double_t mode = (a - 1.0) / (a + b -2.0); return mode; } //______________________________________________________________________________ void TEfficiency::Build(const char* name,const char* title) { //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) SetName(name); SetTitle(title); SetStatisticOption(kDefStatOpt); SetDirectory(gDirectory); SetBit(kPosteriorMode,false); SetBit(kShortestInterval,false); SetBit(kUseWeights,false); //set normalisation factors to 0, otherwise the += may not work properly fPassedHistogram->SetNormFactor(0); fTotalHistogram->SetNormFactor(0); } //______________________________________________________________________________ Bool_t TEfficiency::CheckBinning(const TH1& pass,const TH1& total) { //checks binning for each axis // //It is assumed that the passed histograms have the same dimension. const TAxis* ax1 = 0; const TAxis* ax2 = 0; //check binning along axis for(Int_t j = 0; j < pass.GetDimension(); ++j) { switch(j) { case 0: ax1 = pass.GetXaxis(); ax2 = total.GetXaxis(); break; case 1: ax1 = pass.GetYaxis(); ax2 = total.GetYaxis(); break; case 2: ax1 = pass.GetZaxis(); ax2 = total.GetZaxis(); break; } if(ax1->GetNbins() != ax2->GetNbins()) { gROOT->Info("TEfficiency::CheckBinning","Histograms are not consistent: they have different number of bins"); return false; } for(Int_t i = 1; i <= ax1->GetNbins() + 1; ++i) if(!TMath::AreEqualRel(ax1->GetBinLowEdge(i), ax2->GetBinLowEdge(i), 1.E-15)) { gROOT->Info("TEfficiency::CheckBinning","Histograms are not consistent: they have different bin edges"); return false; } if(!TMath::AreEqualRel(ax1->GetXmax(), ax2->GetXmax(), 1.E-15)) { gROOT->Info("TEfficiency::CheckBinning","Histograms are not consistent: they have different axis max value"); return false; } } return true; } //______________________________________________________________________________ Bool_t TEfficiency::CheckConsistency(const TH1& pass,const TH1& total,Option_t* opt) { //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 // //Option: - w: The check for unit weights is skipped and therefore histograms // filled with weights are accepted. if(pass.GetDimension() != total.GetDimension()) { gROOT->Error("TEfficiency::CheckConsistency","passed TEfficiency objects have different dimensions"); return false; } if(!CheckBinning(pass,total)) { gROOT->Error("TEfficiency::CheckConsistency","passed TEfficiency objects have different binning"); return false; } if(!CheckEntries(pass,total,opt)) { gROOT->Error("TEfficiency::CheckConsistency","passed TEfficiency objects do not have consistent bin contents"); return false; } return true; } //______________________________________________________________________________ Bool_t TEfficiency::CheckEntries(const TH1& pass,const TH1& total,Option_t* opt) { //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) // //and the histogram have to be filled with unit weights. // //Option: - w: Do not check for unit weights -> accept histograms filled with // weights // //Note: - It is assumed that both histograms have the same dimension and // binning. TString option = opt; option.ToLower(); //check for unit weights if(!option.Contains("w")) { Double_t statpass[TH1::kNstat]; Double_t stattotal[TH1::kNstat]; pass.GetStats(statpass); total.GetStats(stattotal); //require: sum of weights == sum of weights^2 if((TMath::Abs(statpass[0]-statpass[1]) > 1e-5) || (TMath::Abs(stattotal[0]-stattotal[1]) > 1e-5)) { gROOT->Info("TEfficiency::CheckEntries","Histograms are filled with weights"); return false; } } //check: pass <= total Int_t nbinsx, nbinsy, nbinsz, nbins; nbinsx = pass.GetNbinsX(); nbinsy = pass.GetNbinsY(); nbinsz = pass.GetNbinsZ(); switch(pass.GetDimension()) { case 1: nbins = nbinsx + 2; break; case 2: nbins = (nbinsx + 2) * (nbinsy + 2); break; case 3: nbins = (nbinsx + 2) * (nbinsy + 2) * (nbinsz + 2); break; default: nbins = 0; } for(Int_t i = 0; i < nbins; ++i) { if(pass.GetBinContent(i) > total.GetBinContent(i)) { gROOT->Info("TEfficiency::CheckEntries","Histograms are not consistent: passed bin content > total bin content"); return false; } } return true; } //______________________________________________________________________________ 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 if (GetDimension() != 1) { Error("CreatePaintingGraph","Call this function only for dimension == 1"); return 0; } Int_t npoints = fTotalHistogram->GetNbinsX(); TGraphAsymmErrors * graph = new TGraphAsymmErrors(npoints); graph->SetName("eff_graph"); FillGraph(graph,opt); return graph; } //______________________________________________________________________________ void TEfficiency::FillGraph(TGraphAsymmErrors * graph, Option_t * opt) const { // Fill the graph to be painted with information from TEfficiency // Internal metyhod called by TEfficiency::Paint or TEfficiency::CreateGraph TString option = opt; option.ToLower(); Bool_t plot0Bins = false; if (option.Contains("e0") ) plot0Bins = true; Double_t x,y,xlow,xup,ylow,yup; //point i corresponds to bin i+1 in histogram // point j is point graph index // LM: cannot use TGraph::SetPoint because it deletes the underlying // histogram each time (see TGraph::SetPoint) // so use it only when extra points are added to the graph Int_t j = 0; double * px = graph->GetX(); double * py = graph->GetY(); double * exl = graph->GetEXlow(); double * exh = graph->GetEXhigh(); double * eyl = graph->GetEYlow(); double * eyh = graph->GetEYhigh(); Int_t npoints = fTotalHistogram->GetNbinsX(); for (Int_t i = 0; i < npoints; ++i) { if (!plot0Bins && fTotalHistogram->GetBinContent(i+1) == 0 ) continue; x = fTotalHistogram->GetBinCenter(i+1); y = GetEfficiency(i+1); xlow = fTotalHistogram->GetBinCenter(i+1) - fTotalHistogram->GetBinLowEdge(i+1); xup = fTotalHistogram->GetBinWidth(i+1) - xlow; ylow = GetEfficiencyErrorLow(i+1); yup = GetEfficiencyErrorUp(i+1); // in the case the graph already existed and extra points have been added if (j >= graph->GetN() ) { graph->SetPoint(j,x,y); graph->SetPointError(j,xlow,xup,ylow,yup); } else { px[j] = x; py[j] = y; exl[j] = xlow; exh[j] = xup; eyl[j] = ylow; eyh[j] = yup; } j++; } // tell the graph the effective number of points graph->Set(j); //refresh title before painting if changed TString oldTitle = graph->GetTitle(); TString newTitle = GetTitle(); if (oldTitle != newTitle ) { graph->SetTitle(newTitle); } // set the axis labels TString xlabel = fTotalHistogram->GetXaxis()->GetTitle(); TString ylabel = fTotalHistogram->GetYaxis()->GetTitle(); if (xlabel) graph->GetXaxis()->SetTitle(xlabel); if (ylabel) graph->GetYaxis()->SetTitle(ylabel); //copying style information TAttLine::Copy(*graph); TAttFill::Copy(*graph); TAttMarker::Copy(*graph); // this method forces the graph to compute correctly the axis // according to the given points graph->GetHistogram(); } //______________________________________________________________________________ TH2 * TEfficiency::CreateHistogram(Option_t *) const { // Create the histogram used to be painted (for dim=2 TEfficiency) // The return object is managed by the caller if (GetDimension() != 2) { Error("CreatePaintingistogram","Call this function only for dimension == 2"); return 0; } Int_t nbinsx = fTotalHistogram->GetNbinsX(); Int_t nbinsy = fTotalHistogram->GetNbinsY(); TAxis * xaxis = fTotalHistogram->GetXaxis(); TAxis * yaxis = fTotalHistogram->GetYaxis(); TH2 * hist = 0; if (xaxis->IsVariableBinSize() && yaxis->IsVariableBinSize() ) hist = new TH2F("eff_histo",GetTitle(),nbinsx,xaxis->GetXbins()->GetArray(), nbinsy,yaxis->GetXbins()->GetArray()); else if (xaxis->IsVariableBinSize() && ! yaxis->IsVariableBinSize() ) hist = new TH2F("eff_histo",GetTitle(),nbinsx,xaxis->GetXbins()->GetArray(), nbinsy,yaxis->GetXmin(), yaxis->GetXmax()); else if (!xaxis->IsVariableBinSize() && yaxis->IsVariableBinSize() ) hist = new TH2F("eff_histo",GetTitle(),nbinsx,xaxis->GetXmin(), xaxis->GetXmax(), nbinsy,yaxis->GetXbins()->GetArray()); else hist = new TH2F("eff_histo",GetTitle(),nbinsx,xaxis->GetXmin(), xaxis->GetXmax(), nbinsy,yaxis->GetXmin(), yaxis->GetXmax()); hist->SetDirectory(0); FillHistogram(hist); return hist; } //______________________________________________________________________________ void TEfficiency::FillHistogram(TH2 * hist ) const { // Fill the 2d histogram to be painted with information from TEfficiency 2D // Internal metyhod called by TEfficiency::Paint or TEfficiency::CreatePaintingGraph //refresh title before each painting hist->SetTitle(GetTitle()); // set the axis labels TString xlabel = fTotalHistogram->GetXaxis()->GetTitle(); TString ylabel = fTotalHistogram->GetYaxis()->GetTitle(); if (xlabel) hist->GetXaxis()->SetTitle(xlabel); if (ylabel) hist->GetYaxis()->SetTitle(ylabel); Int_t bin; Int_t nbinsx = hist->GetNbinsX(); Int_t nbinsy = hist->GetNbinsY(); for(Int_t i = 0; i < nbinsx + 2; ++i) { for(Int_t j = 0; j < nbinsy + 2; ++j) { bin = GetGlobalBin(i,j); hist->SetBinContent(bin,GetEfficiency(bin)); } } //copying style information TAttLine::Copy(*hist); TAttFill::Copy(*hist); TAttMarker::Copy(*hist); hist->SetStats(0); return; } //______________________________________________________________________________ Double_t TEfficiency::ClopperPearson(Int_t total,Int_t passed,Double_t level,Bool_t bUpper) { //calculates the boundaries for the frequentist Clopper-Pearson interval // //This interval is recommended by the PDG. // //Input: - 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 // //calculation: // //The lower boundary of the Clopper-Pearson interval is the "exact" inversion //of the test: //Begin_Latex(separator='=',align='rl') //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_Latex //The lower boundary is therfore given by the Begin_Latex #frac{1 - level}{2}End_Latex quantile //of the beta distribution. // //The upper boundary of the Clopper-Pearson interval is the "exact" inversion //of the test: //Begin_Latex(separator='=',align='rl') //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_Latex //The upper boundary is therfore given by the Begin_Latex #frac{1 + level}{2}End_Latex quantile //of the beta distribution. // //Note: The connection between the binomial distribution and the regularized // incomplete beta function Begin_Latex I_{#varepsilon}(#alpha,#beta)End_Latex has been used. Double_t alpha = (1.0 - level) / 2; if(bUpper) return ((passed == total) ? 1.0 : ROOT::Math::beta_quantile(1 - alpha,passed + 1,total-passed)); else return ((passed == 0) ? 0.0 : ROOT::Math::beta_quantile(alpha,passed,total-passed+1.0)); } //______________________________________________________________________________ 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,const Double_t* w,Option_t* opt) { //calculates the combined efficiency and its uncertainties // //This method does a bayesian combination of the given samples. // //Input: //- up : contains the upper limit of the confidence interval afterwards //- low : contains the lower limit of the confidence interval afterwards //- n : number of samples which are combined //- pass : array of length n containing the number of passed events //- total : array of length n containing the corresponding numbers of total // events //- alpha : shape parameters for the beta distribution as prior //- beta : shape parameters for the beta distribution as prior //- level : desired confidence level //- 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. //- options: // // + 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) // //Begin_Html //Calculation: //<ol> //<li>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:</li> //Begin_Latex(separator='=',align='rl') //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_Latex //Begin_Html //<li>The estimated efficiency is the mode (or the mean) of the obtained posterior distribution </li> //End_Html //Begin_Html //<li>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.</li> //</ol> //End_Html //Example (uniform prior distribution): //Begin_Macro(source) //{ // 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; //} //End_Macro TString option(opt); option.ToLower(); //LM: new formula for combination // works only if alpha beta are the same always // the weights are normalized to w(i) -> N_eff w(i)/ Sum w(i) // i.e. w(i) -> Sum (w(i) / Sum (w(i)^2) * w(i) // norm = Sum (w(i) / Sum (w(i)^2) double ntot = 0; double ktot = 0; double sumw = 0; double sumw2 = 0; for (int i = 0; i < n ; ++i) { if(pass[i] > total[i]) { ::Error("TEfficiency::Combine","total events = %i < passed events %i",total[i],pass[i]); ::Info("TEfficiency::Combine","stop combining"); return -1; } ntot += w[i] * total[i]; ktot += w[i] * pass[i]; sumw += w[i]; sumw2 += w[i]*w[i]; //mean += w[i] * (pass[i] + alpha[i])/(total[i] + alpha[i] + beta[i]); } double norm = sumw/sumw2; ntot *= norm; ktot *= norm; if(ktot > ntot) { ::Error("TEfficiency::Combine","total = %f < passed %f",ntot,ktot); ::Info("TEfficiency::Combine","stop combining"); return -1; } double a = ktot + alpha; double b = ntot - ktot + beta; double mean = a/(a+b); double mode = BetaMode(a,b); Bool_t shortestInterval = option.Contains("sh") || ( option.Contains("mode") && !option.Contains("cent") ); if (shortestInterval) BetaShortestInterval(level, a, b, low, up); else { low = BetaCentralInterval(level, a, b, false); up = BetaCentralInterval(level, a, b, true); } if (option.Contains("mode")) return mode; return mean; } //______________________________________________________________________________ TGraphAsymmErrors* TEfficiency::Combine(TCollection* pList,Option_t* option, Int_t n,const Double_t* w) { //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. // //Input: //- pList : list containing TEfficiency objects which should be combined // only one-dimensional efficiencies are taken into account //- options // + 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) // //- 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. //- 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. TString opt = option; opt.ToLower(); //parameter of prior distribution, confidence level and normalisation factor Double_t alpha = -1; Double_t beta = -1; Double_t level = 0; //flags for combining Bool_t bStrict = false; Bool_t bOutput = false; Bool_t bWeights = false; //list of all information needed to weight and combine efficiencies std::vector<TH1*> vTotal; vTotal.reserve(n); std::vector<TH1*> vPassed; vPassed.reserve(n); std::vector<Double_t> vWeights; vWeights.reserve(n); // std::vector<Double_t> vAlpha; // std::vector<Double_t> vBeta; if(opt.Contains("s")) { opt.ReplaceAll("s",""); bStrict = true; } if(opt.Contains("v")) { opt.ReplaceAll("v",""); bOutput = true; } if(opt.Contains("cl=")) { Ssiz_t pos = opt.Index("cl=") + 3; level = atof( opt(pos,opt.Length() ).Data() ); if((level <= 0) || (level >= 1)) level = 0; opt.ReplaceAll("cl=",""); } //are weights explicitly given if(n && w) { bWeights = true; for(Int_t k = 0; k < n; ++k) { if(w[k] > 0) vWeights.push_back(w[k]); else { gROOT->Error("TEfficiency::Combine","invalid custom weight found w = %.2lf",w[k]); gROOT->Info("TEfficiency::Combine","stop combining"); return 0; } } } TIter next(pList); TObject* obj = 0; TEfficiency* pEff = 0; while((obj = next())) { pEff = dynamic_cast<TEfficiency*>(obj); //is object a TEfficiency object? if(pEff) { if(pEff->GetDimension() > 1) continue; if(!level) level = pEff->GetConfidenceLevel(); if(alpha<1) alpha = pEff->GetBetaAlpha(); if(beta<1) beta = pEff->GetBetaBeta(); //if strict combining, check priors, confidence level and statistic if(bStrict) { if(alpha != pEff->GetBetaAlpha()) continue; if(beta != pEff->GetBetaBeta()) continue; if(!pEff->UsesBayesianStat()) continue; } vTotal.push_back(pEff->fTotalHistogram); vPassed.push_back(pEff->fPassedHistogram); //no weights given -> use weights of TEfficiency objects if(!bWeights) vWeights.push_back(pEff->fWeight); //strict combining -> using global prior // if(bStrict) { // vAlpha.push_back(alpha); // vBeta.push_back(beta); // } // else { // vAlpha.push_back(pEff->GetBetaAlpha()); // vBeta.push_back(pEff->GetBetaBeta()); // } } } //no TEfficiency objects found if(vTotal.empty()) { gROOT->Error("TEfficiency::Combine","no TEfficiency objects in given list"); gROOT->Info("TEfficiency::Combine","stop combining"); return 0; } //invalid number of custom weights if(bWeights && (n != (Int_t)vTotal.size())) { gROOT->Error("TEfficiency::Combine","number of weights n=%i differs from number of TEfficiency objects k=%i which should be combined",n,(Int_t)vTotal.size()); gROOT->Info("TEfficiency::Combine","stop combining"); return 0; } Int_t nbins_max = vTotal.at(0)->GetNbinsX(); //check binning of all histograms for(UInt_t i=0; i<vTotal.size(); ++i) { if (!TEfficiency::CheckBinning(*vTotal.at(0),*vTotal.at(i)) ) gROOT->Warning("TEfficiency::Combine","histograms have not the same binning -> results may be useless"); if(vTotal.at(i)->GetNbinsX() < nbins_max) nbins_max = vTotal.at(i)->GetNbinsX(); } //display information about combining if(bOutput) { gROOT->Info("TEfficiency::Combine","combining %i TEfficiency objects",(Int_t)vTotal.size()); if(bWeights) gROOT->Info("TEfficiency::Combine","using custom weights"); if(bStrict) { gROOT->Info("TEfficiency::Combine","using the following prior probability for the efficiency: P(e) ~ Beta(e,%.3lf,%.3lf)",alpha,beta); } else gROOT->Info("TEfficiency::Combine","using individual priors of each TEfficiency object"); gROOT->Info("TEfficiency::Combine","confidence level = %.2lf",level); } //create TGraphAsymmErrors with efficiency std::vector<Double_t> x(nbins_max); std::vector<Double_t> xlow(nbins_max); std::vector<Double_t> xhigh(nbins_max); std::vector<Double_t> eff(nbins_max); std::vector<Double_t> efflow(nbins_max); std::vector<Double_t> effhigh(nbins_max); //parameters for combining: //number of objects Int_t num = vTotal.size(); std::vector<Int_t> pass(num); std::vector<Int_t> total(num); //loop over all bins Double_t low = 0; Double_t up = 0; for(Int_t i=1; i <= nbins_max; ++i) { //the binning of the x-axis is taken from the first total histogram x[i-1] = vTotal.at(0)->GetBinCenter(i); xlow[i-1] = x[i-1] - vTotal.at(0)->GetBinLowEdge(i); xhigh[i-1] = vTotal.at(0)->GetBinWidth(i) - xlow[i-1]; for(Int_t j = 0; j < num; ++j) { pass[j] = (Int_t)(vPassed.at(j)->GetBinContent(i) + 0.5); total[j] = (Int_t)(vTotal.at(j)->GetBinContent(i) + 0.5); } //fill efficiency and errors eff[i-1] = Combine(up,low,num,&pass[0],&total[0],alpha,beta,level,&vWeights[0],opt.Data()); //did an error occured ? if(eff[i-1] == -1) { gROOT->Error("TEfficiency::Combine","error occured during combining"); gROOT->Info("TEfficiency::Combine","stop combining"); return 0; } efflow[i-1]= eff[i-1] - low; effhigh[i-1]= up - eff[i-1]; }//loop over all bins TGraphAsymmErrors* gr = new TGraphAsymmErrors(nbins_max,&x[0],&eff[0],&xlow[0],&xhigh[0],&efflow[0],&effhigh[0]); return gr; } //______________________________________________________________________________ Int_t TEfficiency::DistancetoPrimitive(Int_t px, Int_t py) { // 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 if (fPaintGraph) return fPaintGraph->DistancetoPrimitive(px,py); if (fPaintHisto) return fPaintHisto->DistancetoPrimitive(px,py); return 0; } //______________________________________________________________________________ void TEfficiency::Draw(Option_t* opt) { //draws the current TEfficiency object // //options: //- 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] ) //check options TString option = opt; option.ToLower(); // use by default "AP" if (option.IsNull() ) option = "ap"; if(gPad && !option.Contains("same")) gPad->Clear(); else { // add always "a" if not present if (!option.Contains("a") ) option += "a"; } // add always p to the option if (!option.Contains("p") ) option += "p"; AppendPad(option.Data()); } //______________________________________________________________________________ void TEfficiency::ExecuteEvent(Int_t event, Int_t px, Int_t py) { // 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 if (fPaintGraph) fPaintGraph->ExecuteEvent(event,px,py); else if (fPaintHisto) fPaintHisto->ExecuteEvent(event,px,py); } //______________________________________________________________________________ void TEfficiency::Fill(Bool_t bPassed,Double_t x,Double_t y,Double_t z) { //This function is used for filling the two histograms. // //Input: bPassed - flag whether the current event passed the selection // true: both histograms are filled // false: only the total histogram is filled // x - x value // y - y value (use default=0 for 1-D efficiencies) // z - z value (use default=0 for 2-D or 1-D efficiencies) switch(GetDimension()) { case 1: fTotalHistogram->Fill(x); if(bPassed) fPassedHistogram->Fill(x); break; case 2: ((TH2*)(fTotalHistogram))->Fill(x,y); if(bPassed) ((TH2*)(fPassedHistogram))->Fill(x,y); break; case 3: ((TH3*)(fTotalHistogram))->Fill(x,y,z); if(bPassed) ((TH3*)(fPassedHistogram))->Fill(x,y,z); break; } } //______________________________________________________________________________ void TEfficiency::FillWeighted(Bool_t bPassed,Double_t weight,Double_t x,Double_t y,Double_t z) { //This function is used for filling the two histograms with a weight. // //Input: bPassed - flag whether the current event passed the selection // true: both histograms are filled // false: only the total histogram is filled // weight - weight for the event // x - x value // y - y value (use default=0 for 1-D efficiencies) // 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 if(!TestBit(kUseWeights)) { Info("FillWeighted","call SetUseWeightedEvents() manually to ensure correct storage of sum of weights squared"); SetUseWeightedEvents(); } switch(GetDimension()) { case 1: fTotalHistogram->Fill(x,weight); if(bPassed) fPassedHistogram->Fill(x,weight); break; case 2: ((TH2*)(fTotalHistogram))->Fill(x,y,weight); if(bPassed) ((TH2*)(fPassedHistogram))->Fill(x,y,weight); break; case 3: ((TH3*)(fTotalHistogram))->Fill(x,y,z,weight); if(bPassed) ((TH3*)(fPassedHistogram))->Fill(x,y,z,weight); break; } } //______________________________________________________________________________ Int_t TEfficiency::FindFixBin(Double_t x,Double_t y,Double_t z) 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) Int_t nx = fTotalHistogram->GetXaxis()->FindFixBin(x); Int_t ny = 0; Int_t nz = 0; switch(GetDimension()) { case 3: nz = fTotalHistogram->GetZaxis()->FindFixBin(z); case 2: ny = fTotalHistogram->GetYaxis()->FindFixBin(y);break; } return GetGlobalBin(nx,ny,nz); } //______________________________________________________________________________ Int_t 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 TString option = opt; option.ToLower(); //replace existing functions in list with same name Bool_t bDeleteOld = true; if(option.Contains("+")) { option.ReplaceAll("+",""); bDeleteOld = false; } TBinomialEfficiencyFitter Fitter(fPassedHistogram,fTotalHistogram); Int_t result = Fitter.Fit(f1,option.Data()); //create copy which is appended to the list TF1* pFunc = new TF1(*f1); if(bDeleteOld) { TIter next(fFunctions); TObject* obj = 0; while((obj = next())) { if(obj->InheritsFrom(TF1::Class())) { fFunctions->Remove(obj); delete obj; } } } // create list if necessary if(!fFunctions) fFunctions = new TList(); fFunctions->Add(pFunc); return result; } //______________________________________________________________________________ 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. //Begin_html //<div class="code"><pre> // TFile* pFile = new TFile("passed.root","update"); // TEfficiency* pEff = (TEfficiency*)gDirectory->Get("my_eff"); // TH1* copy = pEff->GetCopyPassedHisto(); // copy->SetDirectory(gDirectory); // pFile->Write(); //</pre></div> //<div class="clear"></div> //End_Html Bool_t bStatus = TH1::AddDirectoryStatus(); TH1::AddDirectory(kFALSE); TH1* tmp = (TH1*)(fPassedHistogram->Clone()); TH1::AddDirectory(bStatus); return tmp; } //______________________________________________________________________________ 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. //Begin_Html //<div class="code"><pre> // TFile* pFile = new TFile("total.root","update"); // TEfficiency* pEff = (TEfficiency*)gDirectory->Get("my_eff"); // TH1* copy = pEff->GetCopyTotalHisto(); // copy->SetDirectory(gDirectory); // pFile->Write(); //</pre></div> //<div class="clear"></div> //End_Html Bool_t bStatus = TH1::AddDirectoryStatus(); TH1::AddDirectory(kFALSE); TH1* tmp = (TH1*)(fTotalHistogram->Clone()); TH1::AddDirectory(bStatus); return tmp; } //______________________________________________________________________________ Int_t TEfficiency::GetDimension() const { //returns the dimension of the current TEfficiency object return fTotalHistogram->GetDimension(); } //______________________________________________________________________________ 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: // Begin_Latex #hat{#varepsilon} = #frac{passed}{total} End_Latex // for bayesian ones the expectation value of the resulting posterior // distribution is returned: // Begin_Latex #hat{#varepsilon} = #frac{passed + #alpha}{total + #alpha + #beta} End_Latex // If the bit kPosteriorMode is set (or the method TEfficiency::UsePosteriorMode() has been called ) the // mode (most probable value) of the posterior is returned: // Begin_Latex #hat{#varepsilon} = #frac{passed + #alpha -1}{total + #alpha + #beta -2} End_Latex // // - If the denominator is equal to 0, an efficiency of 0 is returned. // - When Begin_Latex passed + #alpha < 1 End_Latex or Begin_Latex total - passed + #beta < 1 End_latex the above // formula for the mode is not valid. In these cases values the estimated efficiency is 0 or 1. Double_t total = fTotalHistogram->GetBinContent(bin); Double_t passed = fPassedHistogram->GetBinContent(bin); if(TestBit(kIsBayesian)) { // parameters for the beta prior distribution Double_t alpha = TestBit(kUseBinPrior) ? GetBetaAlpha(bin) : GetBetaAlpha(); Double_t beta = TestBit(kUseBinPrior) ? GetBetaBeta(bin) : GetBetaBeta(); Double_t aa,bb; if(TestBit(kUseWeights)) { Double_t tw = fTotalHistogram->GetBinContent(bin); Double_t tw2 = fTotalHistogram->GetSumw2()->At(bin); Double_t pw = fPassedHistogram->GetBinContent(bin); if (tw2 <= 0 ) return pw/tw; // tw/tw2 renormalize the weights double norm = tw/tw2; aa = pw * norm + alpha; bb = (tw - pw) * norm + beta; } else { aa = passed + alpha; bb = total - passed + beta; } if (!TestBit(kPosteriorMode) ) return BetaMean(aa,bb); else return BetaMode(aa,bb); } else return (total)? ((Double_t)passed)/total : 0; } //______________________________________________________________________________ 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. Int_t total = (Int_t)fTotalHistogram->GetBinContent(bin); Int_t passed = (Int_t)fPassedHistogram->GetBinContent(bin); Double_t eff = GetEfficiency(bin); // check whether weights have been used if(TestBit(kUseWeights)) { Double_t tw = fTotalHistogram->GetBinContent(bin); Double_t tw2 = fTotalHistogram->GetSumw2()->At(bin); Double_t pw = fPassedHistogram->GetBinContent(bin); Double_t pw2 = fPassedHistogram->GetSumw2()->At(bin); if(TestBit(kIsBayesian)) { Double_t alpha = TestBit(kUseBinPrior) ? GetBetaAlpha(bin) : GetBetaAlpha(); Double_t beta = TestBit(kUseBinPrior) ? GetBetaBeta(bin) : GetBetaBeta(); if (tw2 <= 0) return 0; // tw/tw2 renormalize the weights Double_t norm = tw/tw2; Double_t aa = pw * norm + alpha; Double_t bb = (tw - pw) * norm + beta; Double_t low = 0; Double_t upper = 1; if(TestBit(kShortestInterval)) { TEfficiency::BetaShortestInterval(fConfLevel,aa,bb,low,upper); } else { low = TEfficiency::BetaCentralInterval(fConfLevel,aa,bb,false); } return eff - low; } else { if(fStatisticOption != kFNormal) { Warning("GetEfficiencyErrorLow","frequentist confidence intervals for weights are only supported by the normal approximation"); Info("GetEfficiencyErrorLow","setting statistic option to kFNormal"); const_cast<TEfficiency*>(this)->SetStatisticOption(kFNormal); } Double_t variance = ( pw2 * (1. - 2 * eff) + tw2 * eff *eff ) / ( tw * tw) ; Double_t sigma = sqrt(variance); Double_t prob = 0.5 * (1.- fConfLevel); Double_t delta = ROOT::Math::normal_quantile_c(prob, sigma); // avoid to return errors which makes eff-err < 0 return (eff - delta < 0) ? eff : delta; } } else { if(TestBit(kIsBayesian)) { // parameters for the beta prior distribution Double_t alpha = TestBit(kUseBinPrior) ? GetBetaAlpha(bin) : GetBetaAlpha(); Double_t beta = TestBit(kUseBinPrior) ? GetBetaBeta(bin) : GetBetaBeta(); return (eff - Bayesian(total,passed,fConfLevel,alpha,beta,false,TestBit(kShortestInterval))); } else return (eff - fBoundary(total,passed,fConfLevel,false)); } } //______________________________________________________________________________ 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. Int_t total = (Int_t)fTotalHistogram->GetBinContent(bin); Int_t passed = (Int_t)fPassedHistogram->GetBinContent(bin); Double_t eff = GetEfficiency(bin); // check whether weights have been used if(TestBit(kUseWeights)) { Double_t tw = fTotalHistogram->GetBinContent(bin); Double_t tw2 = fTotalHistogram->GetSumw2()->At(bin); Double_t pw = fPassedHistogram->GetBinContent(bin); Double_t pw2 = fPassedHistogram->GetSumw2()->At(bin); if(TestBit(kIsBayesian)) { Double_t alpha = TestBit(kUseBinPrior) ? GetBetaAlpha(bin) : GetBetaAlpha(); Double_t beta = TestBit(kUseBinPrior) ? GetBetaBeta(bin) : GetBetaBeta(); if (tw2 <= 0) return 0; // tw/tw2 renormalize the weights Double_t norm = tw/tw2; Double_t aa = pw * norm + alpha; Double_t bb = (tw - pw) * norm + beta; Double_t low = 0; Double_t upper = 1; if(TestBit(kShortestInterval)) { TEfficiency::BetaShortestInterval(fConfLevel,aa,bb,low,upper); } else { upper = TEfficiency::BetaCentralInterval(fConfLevel,aa,bb,true); } return upper - eff; } else { if(fStatisticOption != kFNormal) { Warning("GetEfficiencyErrorUp","frequentist confidence intervals for weights are only supported by the normal approximation"); Info("GetEfficiencyErrorUp","setting statistic option to kFNormal"); const_cast<TEfficiency*>(this)->SetStatisticOption(kFNormal); } Double_t variance = ( pw2 * (1. - 2 * eff) + tw2 * eff *eff ) / ( tw * tw) ; Double_t sigma = sqrt(variance); Double_t prob = 0.5 * (1.- fConfLevel); Double_t delta = ROOT::Math::normal_quantile_c(prob, sigma); return (eff + delta > 1) ? 1.-eff : delta; } } else { if(TestBit(kIsBayesian)) { // parameters for the beta prior distribution Double_t alpha = TestBit(kUseBinPrior) ? GetBetaAlpha(bin) : GetBetaAlpha(); Double_t beta = TestBit(kUseBinPrior) ? GetBetaBeta(bin) : GetBetaBeta(); return (Bayesian(total,passed,fConfLevel,alpha,beta,true,TestBit(kShortestInterval)) - eff); } else return fBoundary(total,passed,fConfLevel,true) - eff; } } //______________________________________________________________________________ Int_t TEfficiency::GetGlobalBin(Int_t binx,Int_t biny,Int_t binz) 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 return fTotalHistogram->GetBin(binx,biny,binz); } //______________________________________________________________________________ TList* TEfficiency::GetListOfFunctions() { return (fFunctions) ? fFunctions : fFunctions = new TList(); } //______________________________________________________________________________ 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: //Begin_Latex #frac{1}{w_{new}} = #sum_{i} \frac{1}{w_{i}}End_Latex if(!pList->IsEmpty()) { TIter next(pList); TObject* obj = 0; TEfficiency* pEff = 0; while((obj = next())) { pEff = dynamic_cast<TEfficiency*>(obj); if(pEff) { *this += *pEff; } } } return (Long64_t)fTotalHistogram->GetEntries(); } //______________________________________________________________________________ Double_t TEfficiency::Normal(Int_t total,Int_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 // //Input: - 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 // //calculation: //Begin_Latex(separator='=',align='rl') // #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_Latex Double_t alpha = (1.0 - level)/2; if (total == 0) return (bUpper) ? 1 : 0; Double_t average = ((Double_t)passed) / total; Double_t sigma = std::sqrt(average * (1 - average) / total); Double_t delta = ROOT::Math::normal_quantile(1 - alpha,sigma); if(bUpper) return ((average + delta) > 1) ? 1.0 : (average + delta); else return ((average - delta) < 0) ? 0.0 : (average - delta); } //______________________________________________________________________________ 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 = Begin_Latex w_{1} End_Latex //weight of rhs = Begin_Latex w_{2} End_Latex //Begin_Latex w_{new} = \frac{w_{1} \times w_{2}}{w_{1} + w_{2}}End_Latex if (fTotalHistogram == 0 && fPassedHistogram == 0) { // efficiency is empty just copy it over *this = rhs; return *this; } else if (fTotalHistogram == 0 || fPassedHistogram == 0) { Fatal("operator+=","Adding to a non consistent TEfficiency object which has not a total or a passed histogram "); return *this; } if (rhs.fTotalHistogram == 0 && rhs.fTotalHistogram == 0 ) { Warning("operator+=","no operation: adding an empty object"); return *this; } else if (rhs.fTotalHistogram == 0 || rhs.fTotalHistogram == 0 ) { Fatal("operator+=","Adding a non consistent TEfficiency object which has not a total or a passed histogram "); return *this; } fTotalHistogram->ResetBit(TH1::kIsAverage); fPassedHistogram->ResetBit(TH1::kIsAverage); fTotalHistogram->Add(rhs.fTotalHistogram); fPassedHistogram->Add(rhs.fPassedHistogram); SetWeight((fWeight * rhs.GetWeight())/(fWeight + rhs.GetWeight())); return *this; } //______________________________________________________________________________ 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. if(this != &rhs) { //statistic options SetStatisticOption(rhs.GetStatisticOption()); SetConfidenceLevel(rhs.GetConfidenceLevel()); SetBetaAlpha(rhs.GetBetaAlpha()); SetBetaBeta(rhs.GetBetaBeta()); SetWeight(rhs.GetWeight()); //associated list of functions if(fFunctions) fFunctions->Delete(); //copy histograms delete fTotalHistogram; delete fPassedHistogram; Bool_t bStatus = TH1::AddDirectoryStatus(); TH1::AddDirectory(kFALSE); fTotalHistogram = (TH1*)(rhs.fTotalHistogram->Clone()); fPassedHistogram = (TH1*)(rhs.fPassedHistogram->Clone()); TH1::AddDirectory(bStatus); //delete temporary paint objects delete fPaintHisto; delete fPaintGraph; fPaintHisto = 0; fPaintGraph = 0; //copy style rhs.TAttLine::Copy(*this); rhs.TAttFill::Copy(*this); rhs.TAttMarker::Copy(*this); } return *this; } //______________________________________________________________________________ void TEfficiency::Paint(const Option_t* opt) { //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(); // if(!gPad) return; //use TGraphAsymmErrors for painting if(GetDimension() == 1) { if(!fPaintGraph) { fPaintGraph = CreateGraph(opt); } else // update existing graph already created FillGraph(fPaintGraph, opt); //paint graph fPaintGraph->Paint(opt); //paint all associated functions if(fFunctions) { //paint box with fit parameters gStyle->SetOptFit(1); TIter next(fFunctions); TObject* obj = 0; while((obj = next())) { if(obj->InheritsFrom(TF1::Class())) { fPaintGraph->PaintStats((TF1*)obj); ((TF1*)obj)->Paint("sameC"); } } } return; } //use TH2 for painting if(GetDimension() == 2) { if(!fPaintHisto) { fPaintHisto = CreateHistogram(); } else FillHistogram(fPaintHisto); //paint histogram fPaintHisto->Paint(opt); return; } Warning("Paint","Painting 3D efficiency is not implemented"); } //______________________________________________________________________________ void TEfficiency::SavePrimitive(std::ostream& out,Option_t* opt) { //have histograms fixed bins along each axis? Bool_t equi_bins = true; //indentation TString indent = " "; //names for arrays containing the bin edges //static counter needed if more objects are saved static Int_t naxis = 0; TString sxaxis="xAxis",syaxis="yAxis",szaxis="zAxis"; //note the missing break statements! switch(GetDimension()) { case 3: equi_bins = equi_bins && !fTotalHistogram->GetZaxis()->GetXbins()->fArray && !fTotalHistogram->GetZaxis()->GetXbins()->fN; case 2: equi_bins = equi_bins && !fTotalHistogram->GetYaxis()->GetXbins()->fArray && !fTotalHistogram->GetYaxis()->GetXbins()->fN; case 1: equi_bins = equi_bins && !fTotalHistogram->GetXaxis()->GetXbins()->fArray && !fTotalHistogram->GetXaxis()->GetXbins()->fN; } //create arrays containing the variable binning if(!equi_bins) { Int_t i; ++naxis; sxaxis += naxis; syaxis += naxis; szaxis += naxis; //x axis out << indent << "Double_t " << sxaxis << "[" << fTotalHistogram->GetXaxis()->GetXbins()->fN << "] = {"; for (i = 0; i < fTotalHistogram->GetXaxis()->GetXbins()->fN; ++i) { if (i != 0) out << ", "; out << fTotalHistogram->GetXaxis()->GetXbins()->fArray[i]; } out << "}; " << std::endl; //y axis if(GetDimension() > 1) { out << indent << "Double_t " << syaxis << "[" << fTotalHistogram->GetYaxis()->GetXbins()->fN << "] = {"; for (i = 0; i < fTotalHistogram->GetYaxis()->GetXbins()->fN; ++i) { if (i != 0) out << ", "; out << fTotalHistogram->GetYaxis()->GetXbins()->fArray[i]; } out << "}; " << std::endl; } //z axis if(GetDimension() > 2) { out << indent << "Double_t " << szaxis << "[" << fTotalHistogram->GetZaxis()->GetXbins()->fN << "] = {"; for (i = 0; i < fTotalHistogram->GetZaxis()->GetXbins()->fN; ++i) { if (i != 0) out << ", "; out << fTotalHistogram->GetZaxis()->GetXbins()->fArray[i]; } out << "}; " << std::endl; } }//creating variable binning //TEfficiency pointer has efficiency name + counter static Int_t eff_count = 0; ++eff_count; TString eff_name = GetName(); eff_name += eff_count; const char* name = eff_name.Data(); //construct TEfficiency object const char quote = '"'; out << indent << std::endl; out << indent << ClassName() << " * " << name << " = new " << ClassName() << "(" << quote << GetName() << quote << "," << quote << GetTitle() << quote <<","; //fixed bin size -> use n,min,max constructor if(equi_bins) { out << fTotalHistogram->GetXaxis()->GetNbins() << "," << fTotalHistogram->GetXaxis()->GetXmin() << "," << fTotalHistogram->GetXaxis()->GetXmax(); if(GetDimension() > 1) { out << "," << fTotalHistogram->GetYaxis()->GetNbins() << "," << fTotalHistogram->GetYaxis()->GetXmin() << "," << fTotalHistogram->GetYaxis()->GetXmax(); } if(GetDimension() > 2) { out << "," << fTotalHistogram->GetZaxis()->GetNbins() << "," << fTotalHistogram->GetZaxis()->GetXmin() << "," << fTotalHistogram->GetZaxis()->GetXmax(); } } //variable bin size -> use n,*bins constructor else { out << fTotalHistogram->GetXaxis()->GetNbins() << "," << sxaxis; if(GetDimension() > 1) out << "," << fTotalHistogram->GetYaxis()->GetNbins() << "," << syaxis; if(GetDimension() > 2) out << "," << fTotalHistogram->GetZaxis()->GetNbins() << "," << szaxis; } out << ");" << std::endl; out << indent << std::endl; //set statistic options out << indent << name << "->SetConfidenceLevel(" << fConfLevel << ");" << std::endl; out << indent << name << "->SetBetaAlpha(" << fBeta_alpha << ");" << std::endl; out << indent << name << "->SetBetaBeta(" << fBeta_beta << ");" << std::endl; out << indent << name << "->SetWeight(" << fWeight << ");" << std::endl; out << indent << name << "->SetStatisticOption(" << fStatisticOption << ");" << std::endl; out << indent << name << "->SetPosteriorMode(" << TestBit(kPosteriorMode) << ");" << std::endl; out << indent << name << "->SetShortestInterval(" << TestBit(kShortestInterval) << ");" << std::endl; if(TestBit(kUseWeights)) out << indent << name << "->SetUseWeightedEvents();" << std::endl; // save bin-by-bin prior parameters for(unsigned int i = 0; i < fBeta_bin_params.size(); ++i) { out << indent << name << "->SetBetaBinParameters(" << i << "," << fBeta_bin_params.at(i).first << "," << fBeta_bin_params.at(i).second << ");" << std::endl; } //set bin contents Int_t nbins = fTotalHistogram->GetNbinsX() + 2; if(GetDimension() > 1) nbins *= fTotalHistogram->GetNbinsY() + 2; if(GetDimension() > 2) nbins *= fTotalHistogram->GetNbinsZ() + 2; //important: set first total number than passed number for(Int_t i = 0; i < nbins; ++i) { out << indent << name <<"->SetTotalEvents(" << i << "," << fTotalHistogram->GetBinContent(i) << ");" << std::endl; out << indent << name <<"->SetPassedEvents(" << i << "," << fPassedHistogram->GetBinContent(i) << ");" << std::endl; } //save list of functions TIter next(fFunctions); TObject* obj = 0; while((obj = next())) { obj->SavePrimitive(out,"nodraw"); if(obj->InheritsFrom(TF1::Class())) { out << indent << name << "->GetListOfFunctions()->Add(" << obj->GetName() << ");" << std::endl; } } //set style SaveFillAttributes(out,name); SaveLineAttributes(out,name); SaveMarkerAttributes(out,name); //draw TEfficiency object TString option = opt; option.ToLower(); if (!option.Contains("nodraw")) out<< indent << name<< "->Draw(" << quote << opt << quote << ");" << std::endl; } //______________________________________________________________________________ void TEfficiency::SetBetaAlpha(Double_t alpha) { //sets the shape parameter Begin_Latex \alpha End_Latex // //The prior probability of the efficiency is given by the beta distribution: //Begin_Latex // f(\varepsilon;\alpha;\beta) = \frac{1}{B(\alpha,\beta)} \varepsilon^{\alpha-1} (1 - \varepsilon)^{\beta-1} //End_Latex // //Note: - both shape parameters have to be positive (i.e. > 0) if(alpha > 0) fBeta_alpha = alpha; else Warning("SetBetaAlpha(Double_t)","invalid shape parameter %.2lf",alpha); } //______________________________________________________________________________ void TEfficiency::SetBetaBeta(Double_t beta) { //sets the shape parameter Begin_Latex \beta End_Latex // //The prior probability of the efficiency is given by the beta distribution: //Begin_Latex // f(\varepsilon;\alpha,\beta) = \frac{1}{B(\alpha,\beta)} \varepsilon^{\alpha-1} (1 - \varepsilon)^{\beta-1} //End_Latex // //Note: - both shape parameters have to be positive (i.e. > 0) if(beta > 0) fBeta_beta = beta; else Warning("SetBetaBeta(Double_t)","invalid shape parameter %.2lf",beta); } //______________________________________________________________________________ void TEfficiency::SetBetaBinParameters(Int_t bin, Double_t alpha, Double_t beta) { //sets different shape parameter Begin_Latex \alpha and \beta End_Latex // 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: //Begin_Latex // f(\varepsilon;\alpha;\beta) = \frac{1}{B(\alpha,\beta)} \varepsilon^{\alpha-1} (1 - \varepsilon)^{\beta-1} //End_Latex // //Note: - both shape parameters have to be positive (i.e. > 0) // - bin gives the global bin number (cf. GetGlobalBin) if (!fPassedHistogram || !fTotalHistogram) return; TH1 * h1 = fTotalHistogram; // doing this I get h1->fN which is available only for a TH1D UInt_t n = h1->GetBin(h1->GetNbinsX()+1, h1->GetNbinsY()+1, h1->GetNbinsZ()+1 ) + 1; // in case vector is not created do with default alpha, beta params if (fBeta_bin_params.size() != n ) fBeta_bin_params = std::vector<std::pair<Double_t, Double_t> >(n, std::make_pair(fBeta_alpha, fBeta_beta) ); // vector contains also values for under/overflows fBeta_bin_params[bin] = std::make_pair(alpha,beta); SetBit(kUseBinPrior,true); } //______________________________________________________________________________ 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 if (GetDimension() != 1) { Error("SetBins","Using wrong SetBins function for a %d-d histogram",GetDimension()); return kFALSE; } if (fTotalHistogram->GetEntries() != 0 ) { Warning("SetBins","Histogram entries will be lost after SetBins"); fPassedHistogram->Reset(); fTotalHistogram->Reset(); } fPassedHistogram->SetBins(nx,xmin,xmax); fTotalHistogram->SetBins(nx,xmin,xmax); return kTRUE; } //______________________________________________________________________________ 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 if (GetDimension() != 1) { Error("SetBins","Using wrong SetBins function for a %d-d histogram",GetDimension()); return kFALSE; } if (fTotalHistogram->GetEntries() != 0 ) { Warning("SetBins","Histogram entries will be lost after SetBins"); fPassedHistogram->Reset(); fTotalHistogram->Reset(); } fPassedHistogram->SetBins(nx,xBins); fTotalHistogram->SetBins(nx,xBins); return kTRUE; } //______________________________________________________________________________ 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 if (GetDimension() != 2) { Error("SetBins","Using wrong SetBins function for a %d-d histogram",GetDimension()); return kFALSE; } if (fTotalHistogram->GetEntries() != 0 ) { Warning("SetBins","Histogram entries will be lost after SetBins"); fPassedHistogram->Reset(); fTotalHistogram->Reset(); } fPassedHistogram->SetBins(nx,xmin,xmax,ny,ymin,ymax); fTotalHistogram->SetBins(nx,xmin,xmax,ny,ymin,ymax); return kTRUE; } //______________________________________________________________________________ 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 if (GetDimension() != 2) { Error("SetBins","Using wrong SetBins function for a %d-d histogram",GetDimension()); return kFALSE; } if (fTotalHistogram->GetEntries() != 0 ) { Warning("SetBins","Histogram entries will be lost after SetBins"); fPassedHistogram->Reset(); fTotalHistogram->Reset(); } fPassedHistogram->SetBins(nx,xBins,ny,yBins); fTotalHistogram->SetBins(nx,xBins,ny,yBins); return kTRUE; } //______________________________________________________________________________ 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 if (GetDimension() != 3) { Error("SetBins","Using wrong SetBins function for a %d-d histogram",GetDimension()); return kFALSE; } if (fTotalHistogram->GetEntries() != 0 ) { Warning("SetBins","Histogram entries will be lost after SetBins"); fPassedHistogram->Reset(); fTotalHistogram->Reset(); } fPassedHistogram->SetBins(nx,xmin,xmax,ny,ymin,ymax,nz,zmin,zmax); fTotalHistogram->SetBins (nx,xmin,xmax,ny,ymin,ymax,nz,zmin,zmax); return kTRUE; } //______________________________________________________________________________ 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 if (GetDimension() != 3) { Error("SetBins","Using wrong SetBins function for a %d-d histogram",GetDimension()); return kFALSE; } if (fTotalHistogram->GetEntries() != 0 ) { Warning("SetBins","Histogram entries will be lost after SetBins"); fPassedHistogram->Reset(); fTotalHistogram->Reset(); } fPassedHistogram->SetBins(nx,xBins,ny,yBins,nz,zBins); fTotalHistogram->SetBins(nx,xBins,ny,yBins,nz,zBins); return kTRUE; } //______________________________________________________________________________ void TEfficiency::SetConfidenceLevel(Double_t level) { //sets the confidence level (0 < level < 1) // The default value is 1-sigma :~ 0.683 if((level > 0) && (level < 1)) fConfLevel = level; else Warning("SetConfidenceLevel(Double_t)","invalid confidence level %.2lf",level); } //______________________________________________________________________________ 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. if(fDirectory == dir) return; if(fDirectory) fDirectory->Remove(this); fDirectory = dir; if(fDirectory) fDirectory->Append(this); } //______________________________________________________________________________ void TEfficiency::SetName(const char* name) { //sets the name // //Note: The names of the internal histograms are set to "name + _total" and // "name + _passed" respectively. TNamed::SetName(name); //setting the names (appending the correct ending) TString name_total = name + TString("_total"); TString name_passed = name + TString("_passed"); fTotalHistogram->SetName(name_total); fPassedHistogram->SetName(name_passed); } //______________________________________________________________________________ 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) if(events <= fTotalHistogram->GetBinContent(bin)) { fPassedHistogram->SetBinContent(bin,events); return true; } else { Error("SetPassedEvents(Int_t,Int_t)","total number of events (%.1lf) in bin %i is less than given number of passed events %i",fTotalHistogram->GetBinContent(bin),bin,events); return false; } } //______________________________________________________________________________ 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. TString option = opt; option.ToLower(); Bool_t bReplace = option.Contains("f"); if(!bReplace) bReplace = CheckConsistency(rPassed,*fTotalHistogram,"w"); if(bReplace) { delete fPassedHistogram; Bool_t bStatus = TH1::AddDirectoryStatus(); TH1::AddDirectory(kFALSE); fPassedHistogram = (TH1*)(rPassed.Clone()); fPassedHistogram->SetNormFactor(0); TH1::AddDirectory(bStatus); if(fFunctions) fFunctions->Delete(); //check whether histogram is filled with weights Double_t statpass[TH1::kNstat]; rPassed.GetStats(statpass); //require: sum of weights == sum of weights^2 if(TMath::Abs(statpass[0]-statpass[1]) > 1e-5) SetUseWeightedEvents(); return true; } else return false; } //______________________________________________________________________________ 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 fStatisticOption = option; switch(option) { case kFCP: fBoundary = &ClopperPearson; SetBit(kIsBayesian,false); break; case kFNormal: fBoundary = &Normal; SetBit(kIsBayesian,false); break; case kFWilson: fBoundary = &Wilson; SetBit(kIsBayesian,false); break; case kFAC: fBoundary = &AgrestiCoull; SetBit(kIsBayesian,false); break; case kFFC: fBoundary = &FeldmanCousins; SetBit(kIsBayesian,false); break; case kBJeffrey: fBeta_alpha = 0.5; fBeta_beta = 0.5; SetBit(kIsBayesian,true); SetBit(kUseBinPrior,false); break; case kBUniform: fBeta_alpha = 1; fBeta_beta = 1; SetBit(kIsBayesian,true); SetBit(kUseBinPrior,false); break; case kBBayesian: SetBit(kIsBayesian,true); break; default: fStatisticOption = kFCP; fBoundary = &ClopperPearson; SetBit(kIsBayesian,false); } } //______________________________________________________________________________ void TEfficiency::SetTitle(const char* title) { //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 //Begin_Html //<div class="code"><pre> //pEff->SetTitle("My Efficiency;x label;eff"); //</pre></div> //<div class="clear"></div> //End_Html //setting the titles (looking for the first semicolon and insert the tokens there) TString title_passed = title; TString title_total = title; Ssiz_t pos = title_passed.First(";"); if (pos != kNPOS) { title_passed.Insert(pos," (passed)"); title_total.Insert(pos," (total)"); } else { title_passed.Append(" (passed)"); title_total.Append(" (total)"); } fPassedHistogram->SetTitle(title_passed); fTotalHistogram->SetTitle(title_total); // strip (total) for the TEfficiency title // HIstogram SetTitle has already stripped the axis TString teffTitle = fTotalHistogram->GetTitle(); teffTitle.ReplaceAll(" (total)",""); TNamed::SetTitle(teffTitle); } //______________________________________________________________________________ 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 if(events >= fPassedHistogram->GetBinContent(bin)) { fTotalHistogram->SetBinContent(bin,events); return true; } else { Error("SetTotalEvents(Int_t,Int_t)","passed number of events (%.1lf) in bin %i is bigger than given number of total events %i",fPassedHistogram->GetBinContent(bin),bin,events); return false; } } //______________________________________________________________________________ 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. TString option = opt; option.ToLower(); Bool_t bReplace = option.Contains("f"); if(!bReplace) bReplace = CheckConsistency(*fPassedHistogram,rTotal,"w"); if(bReplace) { delete fTotalHistogram; Bool_t bStatus = TH1::AddDirectoryStatus(); TH1::AddDirectory(kFALSE); fTotalHistogram = (TH1*)(rTotal.Clone()); fTotalHistogram->SetNormFactor(0); TH1::AddDirectory(bStatus); if(fFunctions) fFunctions->Delete(); //check whether histogram is filled with weights Double_t stattotal[TH1::kNstat]; rTotal.GetStats(stattotal); //require: sum of weights == sum of weights^2 if(TMath::Abs(stattotal[0]-stattotal[1]) > 1e-5) SetUseWeightedEvents(); return true; } else return false; } //______________________________________________________________________________ void TEfficiency::SetUseWeightedEvents() { SetBit(kUseWeights,true); fTotalHistogram->Sumw2(); fPassedHistogram->Sumw2(); } //______________________________________________________________________________ void TEfficiency::SetWeight(Double_t weight) { //sets the global weight for this TEfficiency object // //Note: - weight has to be positive ( > 0) if(weight > 0) fWeight = weight; else Warning("SetWeight","invalid weight %.2lf",weight); } //______________________________________________________________________________ Double_t TEfficiency::Wilson(Int_t total,Int_t passed,Double_t level,Bool_t bUpper) { //calculates the boundaries for the frequentist Wilson interval // //Input: - 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 // //calculation: //Begin_Latex(separator='=',align='rl') // #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_Latex Double_t alpha = (1.0 - level)/2; if (total == 0) return (bUpper) ? 1 : 0; Double_t average = ((Double_t)passed) / total; Double_t kappa = ROOT::Math::normal_quantile(1 - alpha,1); Double_t mode = (passed + 0.5 * kappa * kappa) / (total + kappa * kappa); Double_t delta = kappa / (total + kappa*kappa) * std::sqrt(total * average * (1 - average) + kappa * kappa / 4); if(bUpper) return ((mode + delta) > 1) ? 1.0 : (mode + delta); else return ((mode - delta) < 0) ? 0.0 : (mode - delta); } //______________________________________________________________________________ const TEfficiency operator+(const TEfficiency& lhs,const TEfficiency& rhs) { // addition operator // // adds the corresponding histograms: // lhs.GetTotalHistogram() + rhs.GetTotalHistogram() // lhs.GetPassedHistogram() + rhs.GetPassedHistogram() // // the statistic option and the confidence level are taken from lhs TEfficiency tmp(lhs); tmp += rhs; return tmp; } #endif