/ Elementi di C++ Introduzione a ROOT , Laboratorio Informatico ROOT warm up , Laboratorio Informatico Introduzione a RooFit Primo esercizio con RooFit , Laboratorio Informatico RooFit (Workspace, Factory, Composite Model) , Laboratorio Informatico Introduzione a RooStats , Laboratorio Informatico RooStats, TMVA Analisi Statistica dei Dati per HEP (Laboratorio)

/57 [1] roostats_ex1.C Si modifichi il modello: Specificare i componenti del modello per i tool statistici di roostat: osservabile e parametro di interesse. Utilizzare il numero di eventi di segnale come unico parametro di interesse. Fissare costanti tutti gli altri parametri del modello. Importare la configurazione nel workspace e salvare su file. Si modifichi luso del modello: - leggere il modelConfig dal workspace esempio: ModelConfig* mc = (ModelConfig*) w.obj("ModelConfig"); - calcolare un Confidence Interval utilizzando il ProfileLikelihoodCalculator - Disegnare il profilo della likelihood e sovrapporre lintervallo - calcolare la discovery significance utilizzando il profilelikelihoodcalculator come test di ipotesi - scrivere sulla console i limiti dellintervallo e la significatività [2] roostats_ex1.C Aggiungere lintervallo calcolato con Feldman-Cousins suggerimento : guardare il codice in : $ROOTSYS/tuturials/roostats/IntervalExamples.CIntervalExamples.C RECAP: Esercizio RooStats

/57 void makeModel(RooWorkspace& w ) { // Construct model here w.factory("Exponential:bkg(x[110,120], expr('(-1./tau)',tau[10,-1000,1000]) )"); w.factory("Gaussian:sig(x, mean[115,105,120], sigma[1.0,0,10])"); w.factory("SUM:model( Ns[100,0,120]*sig, Nb[1000,0,10000]*bkg)"); // Getting variables and functions out of a workspace RooAbsPdf * model = w.pdf("model"); RooRealVar * x = w.var("x"); // the observable RooRealVar * Ns = w.var("Ns"); // the parameter x->setBins(30); // set constant parameters w.var("mean")->setConstant(true); w.var("sigma")->setConstant(true); w.var("tau")->setConstant(true); w.var("Nb")->setConstant(true); //specify components of model for statistical tools ModelConfig modelConfig("ModelConfig"); modelConfig.SetWorkspace(w); //set components using the name of ws objects modelConfig.SetPdf( "model" ); modelConfig.SetObservables("x"); modelConfig.SetParametersOfInterest("Ns"); //import modelCong into workspace too w.import(modelConfig); // generate the data (nsig = 50, nbkg=1000) w.var("Ns")->setVal(50); w.var("Nb")->setVal(1000); // use fixed random numbers for reproducibility RooRandom::randomGenerator()->SetSeed(111); int N = 1100 ; // will generate accordint to total S+B events RooDataSet * data = model->generate( *x ); data->Print(); // add data to the workspace data->SetName("data"); w.import(*data); // write the workspace in the file const char * fileName = "model.root"; w.writeToFile(fileName,true); } non metto N

/57 void useModel(RooWorkspace& w ) { // Getting variables and functions out of a workspace RooAbsPdf * model = w.pdf("model"); RooRealVar * x = w.var("x"); // the observable RooRealVar * Ns = w.var("Ns"); // the parameter RooDataSet * data = w.data("data"); // the data // get the modelConfig out of the ModelConfig* mc = (ModelConfig*) w.obj("ModelConfig"); // create the class using data and model ProfileLikelihoodCalculator plc(*data, *mc); // set the confidence level plc.SetConfidenceLevel(0.683); // compute the interval LikelihoodInterval* interval = plc.GetInterval(); double lowerLimit = interval->LowerLimit(*Ns); double upperLimit = interval->UpperLimit(*Ns); // plot the interval LikelihoodIntervalPlot plot(interval); plot.Draw(); // Calculate the significance: SL2 estimator Ns->setVal(0); plc.SetNullParameters(*Ns); HypoTestResult* hypotest = plc.GetHypoTest(); double SL2 = hypotest->Significance(); cout << "Limits: [" << lowerLimit << " - " << upperLimit << "]" << endl; cout << "Significance (SL2): " << SL2 << endl; } Chiama il ProfileLikelihoodCalculator Passando gli argomenti (dati, modelconfig)

/57 Limits: [ ] Significance (SL2): % Ns 68% C.L. Ricordare che Nb era fissato costante, così come gli altri parametri del modello (tau, mass, sigma).

/57 void useModel(RooWorkspace& w ) { // Getting variables and functions out of a workspace RooAbsPdf * model = w.pdf("model"); RooRealVar * x = w.var("x"); // the observable RooRealVar * Ns = w.var("Ns"); // the parameter RooDataSet * data = w.data("data"); // the data // get the modelConfig out of the ModelConfig* mc = (ModelConfig*) w.obj("ModelConfig"); FeldmanCousins fc(*data, *modelConfig); fc.SetConfidenceLevel( 0.95 ); fc.SetNBins(40); // number of points to test per parameter fc.UseAdaptiveSampling(true); // make it go faster // The PDF could be extended and this could be removed // fc.FluctuateNumDataEntries(false); // Proof // ProofConfig pc(*wspace, 4, "workers=4", kFALSE); // proof-lite //ProofConfig pc(w, 8, "localhost"); // proof cluster at "localhost" // ToyMCSampler* toymcsampler = (ToyMCSampler*) fc.GetTestStatSampler(); // toymcsampler->SetProofConfig(&pc); // enable proof PointSetInterval* interval = (PointSetInterval*) fc.GetInterval(); std::cout << Feldman-Cousins Limits: ["<< interval->LowerLimit(*Ns) << ", " << interval->UpperLimit(*Ns) << "]" << endl; } Chiama il FeldmanCousins calculator Passando sempre gli argomenti (dati, modelconfig) ESERCIZIO 2 Intervallo Feldman-Cousins

/57 Feldman Cousins Interval is [16.5, 85.5 ] ProfileLikeLihood Interval is [ , ] Significance (SL2): % C.L. Ns Ricordare che Nb era fissato costante, così come gli altri parametri del modello (tau, mass, sigma). 95% C.L. === Using the following for ModelConfig === Observables: RooArgSet:: = (x) Parameters of Interest: RooArgSet:: = (Ns) PDF: RooAddPdf::model[ Ns * sig + Nb * bkg ] FeldmanCousins: ntoys per point: adaptive FeldmanCousins: nEvents per toy will fluctuate about expectation FeldmanCousins: Model has no nuisance parameters FeldmanCousins: # points to test = 40

/57 void makeModel(RooWorkspace& w ) { // Construct model here w.factory("Exponential:bkg(x[110,120], expr('(-1./tau)',tau[10,-1000,1000]) )"); w.factory("Gaussian:sig(x, mean[115,105,120], sigma[1.0,0,10])"); w.factory("SUM:model( Ns[100,0,120]*sig, Nb[1000,0,10000]*bkg)"); // Getting variables and functions out of a workspace RooAbsPdf * model = w.pdf("model"); RooRealVar * x = w.var("x"); // the observable RooRealVar * Ns = w.var("Ns"); // the parameter x->setBins(30); // set constant parameters w.var("mean")->setConstant(true); w.var("sigma")->setConstant(true); // define set of nuisance parameters w.defineSet("nuisParams", "tau,Nb"); //specify components of model for statistical tools ModelConfig modelConfig("ModelConfig"); modelConfig.SetWorkspace(w); //set components using the name of ws objects modelConfig.SetPdf( "model" ); modelConfig.SetObservables("x"); modelConfig.SetParametersOfInterest("Ns"); modelConfig.SetNuisanceParameters(*w.set("nuisParams")); //import modelCong into workspace too w.import(modelConfig); VARIAZIONE Introduciamo Nuisance Parameters

/57 Feldman Cousins Interval is [16.5, ] ProfileLikeLihood Interval is [ ] Significance (SL2): % C.L. Ns 95% C.L.

/57 VARIAZIONE Intervallo Bayesian // example use of BayesianCalculator // now we also need to specify a prior in the ModelConfig wspace->factory("Uniform::prior(mu)"); modelConfig->SetPriorPdf(*wspace->pdf("prior")); // example usage of BayesianCalculator BayesianCalculator bc(*data, *modelConfig); bc.SetConfidenceLevel( confidenceLevel); SimpleInterval* bcInt = bc.GetInterval(); Chiama il BayesianCalculator Passando sempre gli argomenti (dati, modelconfig)

/57 We create the class and configure it. Since the numerical integration can be tricky, it is better in this case to set a reasonable range from the nuisance parameters which will be integrated to obtain the marginalised posterior function. A sensible choice is to use for example an interval using 10-sigma around the best fit values for the nuisance parameters, RooRealVar *nuisPar = w->var("..."); nuisPar->setRange(nuisPar->getVal() - 10*nuisPar->getError(), nuisPar->getVal() + 10* nuisPar->getError() ) ; It is also recommended to use this class on a model which has a binned dataset or not having too many events (<~ 100), otherwise it will take a long time to obtain the result. We create the class can set the type of interval (central, upper/lower limit or shortest) and the type of integration method BayesianCalculator bayesianCalc(*data,*mc); bayesianCalc.SetConfidenceLevel(0.683); // 68% interval // set the type of interval (not really needed for central which is the default) bayesianCalc.SetLeftSideTailFraction(0.5); // for central interval //bayesianCalc.SetLeftSideTailFraction(0.); // for upper limit //bayesianCalc.SetShortestInterval(); // for shortest interval // set the integration type (not really needed for the default ADAPTIVE) // possible alternative values are "VEGAS", "MISER", or "PLAIN" (MC integration from libMathMore) // "TOYMC" (toy MC integration, work when nuisances exist and they have a constraints pdf) TString integrationType = ""; // this is needed if using TOYMC if (integrationType.Contains("TOYMC") ) { RooAbsPdf * nuisPdf = RooStats::MakeNuisancePdf(*mc, "nuisance_pdf"); if (nuisPdf) bayesianCalc.ForceNuisancePdf(*nuisPdf); } bayesianCalc.SetIntegrationType(integrationType);

/57 // compute interval by scanning the posterior function // it is done by default when computing shortest intervals bayesianCalc.SetScanOfPosterior(100); RooRealVar* firstPOI = (RooRealVar*) mc->GetParametersOfInterest()->first(); SimpleInterval* interval = bayesianCalc.GetInterval(); if (!interval) { cout << "Error computing Bayesian interval - exit " << endl; return; } double lowerLimit = interval->LowerLimit(); double upperLimit = interval->UpperLimit(); cout GetName()<<" is : ["<< lowerLimit << ", "<< upperLimit <<"] "<<endl; // draw plot of posterior function RooPlot * plot = bayesianCalc.GetPosteriorPlot(); if (plot) plot->Draw();

/57 TMVA