MultivariateGaussianTest.C File Reference

Detailed Description

Comparison of MCMC and PLC in a multi-variate gaussian problem

This tutorial produces an N-dimensional multivariate Gaussian with a non-trivial covariance matrix. By default N=4 (called "dim").

A subset of these are considered parameters of interest. This problem is tractable analytically.

We use this mainly as a test of Markov Chain Monte Carlo and we compare the result to the profile likelihood ratio.

We use the proposal helper to create a customized proposal function for this problem.

For N=4 and 2 parameters of interest it takes about 10-20 seconds and the acceptance rate is 37%

 
[#1] INFO:Minimization -- RooAbsMinimizerFcn::setOptimizeConst: activating const optimization
Minuit2Minimizer: Minimize with max-calls 2000 convergence for edm < 1 strategy 1
Minuit2Minimizer : Valid minimum - status = 0
FVAL  = 706.560063684865781
Edm   = 9.79361278577427602e-06
Nfcn  = 68
mu_x0   = 0.180728    +/-  0.17298  (limited)
mu_x1   = 0.207351    +/-  0.172978 (limited)
mu_x2   = -0.0159412  +/-  0.172984 (limited)
mu_x3   = 0.12343  +/-  0.172982 (limited)
[#1] INFO:Minimization -- RooAbsMinimizerFcn::setOptimizeConst: deactivating const optimization
Metropolis-Hastings progress: ....................................................................................................
[#1] INFO:Eval -- Proposal acceptance rate: 37.1%
[#1] INFO:Eval -- Number of steps in chain: 3710
[#1] INFO:InputArguments -- The deprecated RooFit::CloneData(1) option passed to createNLL() is ignored.
[#0] PROGRESS:Minimization -- ProfileLikelihoodCalcultor::DoGLobalFit - find MLE 
[#1] INFO:Minimization -- RooAbsMinimizerFcn::setOptimizeConst: activating const optimization
[#0] PROGRESS:Minimization -- ProfileLikelihoodCalcultor::DoMinimizeNLL - using Minuit2 / Migrad with strategy 1
[#1] INFO:Minimization -- 
  RooFitResult: minimized FCN value: 706.56, estimated distance to minimum: 3.96149e-12
                covariance matrix quality: Full, accurate covariance matrix
                Status : MINIMIZE=0 
 
    Floating Parameter    FinalValue +/-  Error   
  --------------------  --------------------------
                 mu_x0    1.8118e-01 +/-  1.73e-01
                 mu_x1    2.0792e-01 +/-  1.73e-01
                 mu_x2   -1.6067e-02 +/-  1.73e-01
                 mu_x3    1.2371e-01 +/-  1.73e-01
 
[#1] INFO:Minimization -- RooProfileLL::evaluate(nll_mvg_mvgData_Profile[mu_x0,mu_x1]) Creating instance of MINUIT
[#1] INFO:Minimization -- RooProfileLL::evaluate(nll_mvg_mvgData_Profile[mu_x0,mu_x1]) determining minimum likelihood for current configurations w.r.t all observable
[#1] INFO:Minimization -- RooProfileLL::evaluate(nll_mvg_mvgData_Profile[mu_x0,mu_x1]) minimum found at (mu_x0=0.181184, mu_x1=0.207918)
..[#1] INFO:Minimization -- LikelihoodInterval - Finding the contour of mu_x0 ( 0 ) and mu_x1 ( 1 ) 
MCMC interval on p0: [-0.28, 0.6]
MCMC interval on p1: [-0.2, 0.6]
Real time 0:00:01, CP time 1.460

 
#include "RooGlobalFunc.h"
#include <cstdlib>
#include "TMatrixDSym.h"
#include "RooMultiVarGaussian.h"
#include "RooArgList.h"
#include "RooRealVar.h"
#include "TH2F.h"
#include "TCanvas.h"
#include "RooAbsReal.h"
#include "RooFitResult.h"
#include "TStopwatch.h"
#include "RooStats/MCMCCalculator.h"
#include "RooStats/MetropolisHastings.h"
#include "RooStats/MarkovChain.h"
#include "RooStats/ConfInterval.h"
#include "RooStats/MCMCInterval.h"
#include "RooStats/MCMCIntervalPlot.h"
#include "RooStats/ModelConfig.h"
#include "RooStats/ProposalHelper.h"
#include "RooStats/ProposalFunction.h"
#include "RooStats/PdfProposal.h"
#include "RooStats/ProfileLikelihoodCalculator.h"
#include "RooStats/LikelihoodIntervalPlot.h"
#include "RooStats/LikelihoodInterval.h"
 
using namespace std;
using namespace RooFit;
using namespace RooStats;
 
void MultivariateGaussianTest(Int_t dim = 4, Int_t nPOI = 2)
{
   // let's time this challenging example
   TStopwatch t;
   t.Start();
 
   RooArgList xVec;
   RooArgList muVec;
   RooArgSet poi;
 
   // make the observable and means
   Int_t i, j;
   RooRealVar *x;
   RooRealVar *mu_x;
   for (i = 0; i < dim; i++) {
      char *name = Form("x%d", i);
      x = new RooRealVar(name, name, 0, -3, 3);
      xVec.add(*x);
 
      char *mu_name = Form("mu_x%d", i);
      mu_x = new RooRealVar(mu_name, mu_name, 0, -2, 2);
      muVec.add(*mu_x);
   }
 
   // put them into the list of parameters of interest
   for (i = 0; i < nPOI; i++) {
      poi.add(*muVec.at(i));
   }
 
   // make a covariance matrix that is all 1's
   TMatrixDSym cov(dim);
   for (i = 0; i < dim; i++) {
      for (j = 0; j < dim; j++) {
         if (i == j)
            cov(i, j) = 3.;
         else
            cov(i, j) = 1.0;
      }
   }
 
   // now make the multivariate Gaussian
   RooMultiVarGaussian mvg("mvg", "mvg", xVec, muVec, cov);
 
   // --------------------
   // make a toy dataset
   std::unique_ptr<RooDataSet> data{mvg.generate(xVec, 100)};
 
   // now create the model config for this problem
   RooWorkspace *w = new RooWorkspace("MVG");
   ModelConfig modelConfig(w);
   modelConfig.SetPdf(mvg);
   modelConfig.SetParametersOfInterest(poi);
 
   // -------------------------------------------------------
   // Setup calculators
 
   // MCMC
   // we want to setup an efficient proposal function
   // using the covariance matrix from a fit to the data
   std::unique_ptr<RooFitResult> fit{mvg.fitTo(*data, Save(true))};
   ProposalHelper ph;
   ph.SetVariables((RooArgSet &)fit->floatParsFinal());
   ph.SetCovMatrix(fit->covarianceMatrix());
   ph.SetUpdateProposalParameters(true);
   ph.SetCacheSize(100);
   ProposalFunction *pdfProp = ph.GetProposalFunction();
 
   // now create the calculator
   MCMCCalculator mc(*data, modelConfig);
   mc.SetConfidenceLevel(0.95);
   mc.SetNumBurnInSteps(100);
   mc.SetNumIters(10000);
   mc.SetNumBins(50);
   mc.SetProposalFunction(*pdfProp);
 
   MCMCInterval *mcInt = mc.GetInterval();
   RooArgList *poiList = mcInt->GetAxes();
 
   // now setup the profile likelihood calculator
   ProfileLikelihoodCalculator plc(*data, modelConfig);
   plc.SetConfidenceLevel(0.95);
   LikelihoodInterval *plInt = (LikelihoodInterval *)plc.GetInterval();
 
   // make some plots
   MCMCIntervalPlot mcPlot(*mcInt);
 
   TCanvas *c1 = new TCanvas();
   mcPlot.SetLineColor(kGreen);
   mcPlot.SetLineWidth(2);
   mcPlot.Draw();
 
   LikelihoodIntervalPlot plPlot(plInt);
   plPlot.Draw("same");
 
   if (poiList->getSize() == 1) {
      RooRealVar *p = (RooRealVar *)poiList->at(0);
      Double_t ll = mcInt->LowerLimit(*p);
      Double_t ul = mcInt->UpperLimit(*p);
      cout << "MCMC interval: [" << ll << ", " << ul << "]" << endl;
   }
 
   if (poiList->getSize() == 2) {
      RooRealVar *p0 = (RooRealVar *)poiList->at(0);
      RooRealVar *p1 = (RooRealVar *)poiList->at(1);
      TCanvas *scatter = new TCanvas();
      Double_t ll = mcInt->LowerLimit(*p0);
      Double_t ul = mcInt->UpperLimit(*p0);
      cout << "MCMC interval on p0: [" << ll << ", " << ul << "]" << endl;
      ll = mcInt->LowerLimit(*p1);
      ul = mcInt->UpperLimit(*p1);
      cout << "MCMC interval on p1: [" << ll << ", " << ul << "]" << endl;
 
      // MCMC interval on p0: [-0.2, 0.6]
      // MCMC interval on p1: [-0.2, 0.6]
 
      mcPlot.DrawChainScatter(*p0, *p1);
      scatter->Update();
   }
 
   t.Print();
}

Authors

Kevin Belasco, Kyle Cranmer

Definition in file MultivariateGaussianTest.C.