source: trunk/Mars/mtools/MRolke.cc@ 12740

// @(#)root/physics:$Name: not supported by cvs2svn $:$Id: MRolke.cc,v 1.2 2006-10-17 16:32:52 meyer Exp $
// Author: Jan Conrad    9/2/2004

/*************************************************************************
 * Copyright (C) 1995-2004, Rene Brun and Fons Rademakers.               *
 * All rights reserved.                                                  *
 *                                                                       *
 * For the licensing terms see $ROOTSYS/LICENSE.                         *
 * For the list of contributors see $ROOTSYS/README/CREDITS.             *
 *************************************************************************/

//////////////////////////////////////////////////////////////////////////////
//
//  MRolke
//
//  This class computes confidence intervals for the rate of a Poisson
//  in the presence of background and efficiency with a fully frequentist
//  treatment of the uncertainties in the efficiency and background estimate
//  using the profile likelihood method.
//
//  The signal is always assumed to be Poisson.
//
//  The method is very similar to the one used in MINUIT (MINOS).
//
//  Two options are offered to deal with cases where the maximum likelihood
//  estimate (MLE) is not in the physical region. Version "bounded likelihood" 
//  is the one used by MINOS if bounds for the physical region are chosen. Versi//  on "unbounded likelihood (the default) allows the MLE to be in the 
//  unphysical region. It has however better coverage. 
//  For more details consult the reference (see below). 
//
//
//   It allows the following Models:
//
//       1: Background - Poisson, Efficiency - Binomial  (cl,x,y,z,tau,m)
//       2: Background - Poisson, Efficiency - Gaussian  (cl,xd,y,em,tau,sde)
//       3: Background - Gaussian, Efficiency - Gaussian (cl,x,bm,em,sd)
//       4: Background - Poisson, Efficiency - known     (cl,x,y,tau,e)
//       5: Background - Gaussian, Efficiency - known    (cl,x,y,z,sdb,e)
//       6: Background - known, Efficiency - Binomial    (cl,x,z,m,b)
//       7: Background - known, Efficiency - Gaussian    (cl,x,em,sde,b)
//
//  Parameter definition:
//
//  cl  =  Confidence level
//
//  x = number of observed events
//
//  y = number of background events
//
//  z = number of simulated signal events
//
//  em = measurement of the efficiency.
//
//  bm = background estimate
//
//  tau = ratio between signal and background region (in case background is
//  observed) ratio between observed and simulated livetime in case
//  background is determined from MC.
//
//  sd(x) = sigma of the Gaussian
//
//  e = true efficiency (in case known)
//
//  b = expected background (in case known)
//
//  m = number of MC runs
//
//  mid = ID number of the model ...
//
//  For a description of the method and its properties:
//
//  W.Rolke, A. Lopez, J. Conrad and Fred James
//  "Limits and Confidence Intervals in presence of nuisance parameters"
//   http://lanl.arxiv.org/abs/physics/0403059
//   Nucl.Instrum.Meth.A551:493-503,2005
//
//  Should I use MRolke, TFeldmanCousins, TLimit?
//  ============================================
//  1. I guess MRolke makes TFeldmanCousins obsolete?
//
//  Certainly not. TFeldmanCousins is the fully frequentist construction and 
//  should be used in case of no (or negligible uncertainties). It is however 
//  not capable of treating uncertainties in nuisance parameters.
//  MRolke is desined for this case and it is shown in the reference above
//  that it has good coverage properties for most cases, ie it might be
//  used where FeldmannCousins can't.
//
//  2. What are the advantages of MRolke over TLimit?
//
//  MRolke is fully frequentist. TLimit treats nuisance parameters Bayesian. 
//  For a coverage study of a Bayesian method refer to 
//  physics/0408039 (Tegenfeldt & J.C). However, this note studies 
//  the coverage of Feldman&Cousins with Bayesian treatment of nuisance 
//  parameters. To make a long story short: using the Bayesian method you 
//  might introduce a small amount of over-coverage (though I haven't shown it 
//  for TLimit). On the other hand, coverage of course is a not so interesting 
//  when you consider yourself a Bayesian.
//
// Author: Jan Conrad (CERN)
//
// see example in tutorial Rolke.C
//
// Copyright CERN 2004                Jan.Conrad@cern.ch
//
///////////////////////////////////////////////////////////////////////////


#include "MRolke.h"
#include "TMath.h"
#include "Riostream.h"

ClassImp(MRolke)

//__________________________________________________________________________
MRolke::MRolke(Double_t CL, Option_t * /*option*/)
{
   //constructor
   fUpperLimit  = 0.0;
   fLowerLimit  = 0.0;
   fCL          = CL;
   fSwitch      = 0; // 0: unbounded likelihood
                    // 1: bounded likelihood
}

//___________________________________________________________________________
MRolke::~MRolke()
{
}


//___________________________________________________________________________
Double_t MRolke::CalculateInterval(Int_t x, Int_t y, Int_t z, Double_t bm, Double_t em,Double_t e, Int_t mid, Double_t sde, Double_t sdb, Double_t tau, Double_t b, Int_t m)
{
   //calculate interval
   Int_t done = 0;
   Double_t limit[2];

   limit[1] = Interval(x,y,z,bm,em,e,mid, sde,sdb,tau,b,m);

   if (limit[1] > 0) {
      done = 1;
   }

   if (fSwitch == 0) {

      Int_t trial_x = x;

      while (done == 0) {
         trial_x++;
         limit[1] = Interval(trial_x,y,z,bm,em,e,mid, sde,sdb,tau,b,m);
         if (limit[1] > 0) done = 1;
      } 
   }

   return limit[1];
}




//_____________________________________________________________________
Double_t MRolke::Interval(Int_t x, Int_t y, Int_t z, Double_t bm, Double_t em,Double_t e, Int_t mid, Double_t sde, Double_t sdb, Double_t tau, Double_t b, Int_t m)
{
   // Calculates the Confidence Interval

   //Double_t dchi2 =  Chi2Percentile(1,1-fCL);
   Double_t dchi2 = TMath::ChisquareQuantile(fCL, 1);

   Double_t tempxy[2],limits[2] = {0,0};
   Double_t slope,fmid,low,flow,high,fhigh,test,ftest,mu0,maximum,target,l,f0;
   Double_t med = 0;
   Double_t maxiter=1000, acc = 0.00001;
   Int_t i;
   Int_t bp = 0;

   if ((mid != 3) && (mid != 5)) bm = (Double_t)y;

   if ((mid == 3) || (mid == 5)) {
      if (bm == 0) bm = 0.00001;
   } 

   if ((mid <= 2) || (mid == 4)) bp = 1;
  

   if (bp == 1 && x == 0 && bm > 0 ){

      for(Int_t i = 0; i < 2; i++) {
         x++;
         tempxy[i] = Interval(x,y,z,bm,em,e,mid,sde,sdb,tau,b,m);
      }
      slope = tempxy[1] - tempxy[0];
      limits[1] = tempxy[0] - slope;
      limits[0] = 0.0;
      if (limits[1] < 0) limits[1] = 0.0;
      goto done;
   }

   if (bp != 1 && x == 0){

      for(Int_t i = 0; i < 2; i++) {
         x++;
         tempxy[i] = Interval(x,y,z,bm,em,e,mid,sde,sdb,tau,b,m);
      }
      slope = tempxy[1] - tempxy[0];
      limits[1] = tempxy[0] - slope;
      limits[0] = 0.0;
      if (limits[1] < 0) limits[1] = 0.0;
      goto done;
   }

   if (bp != 1  && bm == 0){
      for(Int_t i = 0; i < 2; i++) {
         bm++;
         limits[1] = Interval(x,y,z,bm,em,e,mid,sde,sdb,tau,b,m);
         tempxy[i] = limits[1];
      }
      slope = tempxy[1] - tempxy[0];
      limits[1] = tempxy[0] - slope;
      if (limits[1] < 0) limits[1] = 0;
      goto done;
   }


   if (x == 0 && bm == 0){
      x++;
      bm++;

      limits[1] = Interval(x,y,z,bm,em,e,mid,sde,sdb,tau,b,m);
      tempxy[0] = limits[1];
      x  = 1;
      bm = 2;
      limits[1] = Interval(x,y,z,bm,em,e,mid,sde,sdb,tau,b,m);
      tempxy[1] = limits[1];
      x  = 2;
      bm = 1;
      limits[1] = Interval(x,y,z,bm,em,e,mid,sde,sdb,tau,b,m);
      limits[1] = 3*tempxy[0] -tempxy[1] - limits[1];
      if (limits[1] < 0) limits[1] = 0;
      goto done;
   }

   mu0 = Likelihood(0,x,y,z,bm,em,e,mid,sde,sdb,tau,b,m,1);

   maximum = Likelihood(0,x,y,z,bm,em,e,mid,sde,sdb,tau,b,m,2);

   test = 0;

   f0 = Likelihood(test,x,y,z,bm,em,e,mid,sde,sdb,tau,b,m,3);

   if ( fSwitch == 1 ) {  // do this only for the unbounded likelihood case
      if ( mu0 < 0 ) maximum = f0;
   }

   target = maximum - dchi2;

   if (f0 > target) {
      limits[0] = 0;
   } else {
      if (mu0 < 0){
         limits[0] = 0;
         limits[1] = 0;
      }

      low   = 0;
      flow  = f0;
      high  = mu0;
      fhigh = maximum;

      for(Int_t i = 0; i < maxiter; i++) {
         l = (target-fhigh)/(flow-fhigh);
         if (l < 0.2) l = 0.2;
         if (l > 0.8) l = 0.8;

         med = l*low + (1-l)*high;
         if(med < 0.01){
            limits[1]=0.0;                           
            goto done;
         }

         fmid = Likelihood(med,x,y,z,bm,em,e,mid,sde,sdb,tau,b,m,3);

         if (fmid > target) {
            high  = med;
            fhigh = fmid;
         } else {
            low  = med;
            flow = fmid;
         }
         if ((high-low) < acc*high) break;
      }
      limits[0] = med;
   }


   if(mu0 > 0) {
      low  = mu0;
      flow = maximum;
   } else {
      low  = 0;
      flow = f0;
   }

   test = low +1 ;

   ftest = Likelihood(test,x,y,z,bm,em,e,mid,sde,sdb,tau,b,m,3);

   if (ftest < target) {
      high  = test;
      fhigh = ftest;
   } else {
      slope = (ftest - flow)/(test - low);
      high  = test + (target -ftest)/slope;
      fhigh = Likelihood(high,x,y,z,bm,em,e,mid,sde,sdb,tau,b,m,3);
   }

   for(i = 0; i < maxiter; i++) {
      l = (target-fhigh)/(flow-fhigh);
      if (l < 0.2) l = 0.2;
      if (l > 0.8) l = 0.8;
      med  = l * low + (1.-l)*high;
      fmid = Likelihood(med,x,y,z,bm,em,e,mid,sde,sdb,tau,b,m,3);

      if (fmid < target) {
         high  = med;
         fhigh = fmid;
      } else {
         low  = med;
         flow = fmid;
      }
      if (high-low < acc*high) break;
   }
   limits[1] = med;

done:

   if ( (mid == 4) || (mid==5) ) {
      limits[0] /= e;
      limits[1] /= e;
   }


   fUpperLimit = limits[1];
   fLowerLimit = TMath::Max(limits[0],0.0);

  
   return limits[1];
}


//___________________________________________________________________________
Double_t MRolke::Likelihood(Double_t mu, Int_t x, Int_t y, Int_t z, Double_t bm,Double_t em, Double_t e, Int_t mid, Double_t sde, Double_t sdb, Double_t tau, Double_t b, Int_t m, Int_t what)
{
   // Chooses between the different profile likelihood functions to use for the
   // different models.
   // Returns evaluation of the profile likelihood functions.

   switch (mid) {
      case 1: return EvalLikeMod1(mu,x,y,z,e,tau,b,m,what);
      case 2: return EvalLikeMod2(mu,x,y,em,e,sde,tau,b,what);
      case 3: return EvalLikeMod3(mu,x,bm,em,e,sde,sdb,b,what);
      case 4: return EvalLikeMod4(mu,x,y,tau,b,what);
      case 5: return EvalLikeMod5(mu,x,bm,sdb,b,what);
      case 6: return EvalLikeMod6(mu,x,z,e,b,m,what);
      case 7: return EvalLikeMod7(mu,x,em,e,sde,b,what);
   }

   return 0;
}

//_________________________________________________________________________
Double_t MRolke::EvalLikeMod1(Double_t mu, Int_t x, Int_t y, Int_t z, Double_t e, Double_t tau, Double_t b, Int_t m, Int_t what)
{
   // Calculates the Profile Likelihood for MODEL 1:
   //  Poisson background/ Binomial Efficiency
   // what = 1: Maximum likelihood estimate is returned
   // what = 2: Profile Likelihood of Maxmimum Likelihood estimate is returned.
   // what = 3: Profile Likelihood of Test hypothesis is returned
   // otherwise parameters as described in the beginning of the class)

   Double_t f  = 0;
   Double_t zm = Double_t(z)/m;

   if (what == 1) {
      f = (x-y/tau)/zm;
   }

   if (what == 2) {
      mu = (x-y/tau)/zm;
      b  = y/tau;
      Double_t e = zm;
      f = LikeMod1(mu,b,e,x,y,z,tau,m);
   }

   if (what == 3) {
      if (mu == 0){
         b = (x+y)/(1.0+tau);
         e = zm;
         f = LikeMod1(mu,b,e,x,y,z,tau,m);
      } else {
         MRolke g;
         g.ProfLikeMod1(mu,b,e,x,y,z,tau,m);
         f = LikeMod1(mu,b,e,x,y,z,tau,m);
      }
   }

   return f;
}

//________________________________________________________________________
Double_t MRolke::LikeMod1(Double_t mu,Double_t b, Double_t e, Int_t x, Int_t y, Int_t z, Double_t tau, Int_t m)
{
   // Profile Likelihood function for MODEL 1:
   // Poisson background/ Binomial Efficiency

   return 2*(x*TMath::Log(e*mu+b)-(e*mu +b)-LogFactorial(x)+y*TMath::Log(tau*b)-tau*b-LogFactorial(y) + TMath::Log(TMath::Factorial(m)) - TMath::Log(TMath::Factorial(m-z)) - TMath::Log(TMath::Factorial(z))+ z * TMath::Log(e) + (m-z)*TMath::Log(1-e));
}

//________________________________________________________________________
void MRolke::ProfLikeMod1(Double_t mu,Double_t &b,Double_t &e,Int_t x,Int_t y, Int_t z,Double_t tau,Int_t m)
{
   // Void needed to calculate estimates of efficiency and background for model 1

   Double_t med = 0.0,fmid;
   Int_t maxiter =1000;
   Double_t acc = 0.00001;
   Double_t emin = ((m+mu*tau)-TMath::Sqrt((m+mu*tau)*(m+mu*tau)-4 * mu* tau * z))/2/mu/tau;

   Double_t low  = TMath::Max(1e-10,emin+1e-10);
   Double_t high = 1 - 1e-10;

   for(Int_t i = 0; i < maxiter; i++) {
      med = (low+high)/2.;

      fmid = LikeGradMod1(med,mu,x,y,z,tau,m);

      if(high < 0.5) acc = 0.00001*high;
      else           acc = 0.00001*(1-high);

      if ((high - low) < acc*high) break;

      if(fmid > 0) low  = med;
      else         high = med;
   }

   e = med;
   Double_t eta = Double_t(z)/e -Double_t(m-z)/(1-e);

   b = Double_t(y)/(tau -eta/mu);
}

//___________________________________________________________________________
Double_t MRolke::LikeGradMod1(Double_t e, Double_t mu, Int_t x,Int_t y,Int_t z,Double_t tau,Int_t m)
{
   //gradient model
   Double_t eta, etaprime, bprime,f;
   eta = static_cast<double>(z)/e - static_cast<double>(m-z)/(1.0 - e);
   etaprime = (-1) * (static_cast<double>(m-z)/((1.0 - e)*(1.0 - e)) + static_cast<double>(z)/(e*e));
   Double_t b = y/(tau - eta/mu);
   bprime = (b*b * etaprime)/mu/y;
   f =  (mu + bprime) * (x/(e * mu + b) - 1)+(y/b - tau) * bprime + eta;
   return f;
}

//___________________________________________________________________________
Double_t MRolke::EvalLikeMod2(Double_t mu, Int_t x, Int_t y, Double_t em, Double_t e,Double_t sde, Double_t tau, Double_t b, Int_t what)
{
   // Calculates the Profile Likelihood for MODEL 2:
   //  Poisson background/ Gauss Efficiency
   // what = 1: Maximum likelihood estimate is returned
   // what = 2: Profile Likelihood of Maxmimum Likelihood estimate is returned.
   // what = 3: Profile Likelihood of Test hypothesis is returned
   // otherwise parameters as described in the beginning of the class)

   Double_t v =  sde*sde;
   Double_t coef[4],roots[3];
   Double_t f = 0;

   if (what == 1) {
      f = (x-y/tau)/em;
   }

   if (what == 2) {
      mu = (x-y/tau)/em;
      b = y/tau;
      e = em;
      f = LikeMod2(mu,b,e,x,y,em,tau,v);
   }

   if (what == 3) {
      if (mu == 0 ) {
         b = (x+y)/(1+tau);
         f = LikeMod2(mu,b,e,x,y,em,tau,v);
      } else {
         coef[3] = mu;
         coef[2] = mu*mu*v-2*em*mu-mu*mu*v*tau;
         coef[1] = ( - x)*mu*v - mu*mu*mu*v*v*tau - mu*mu*v*em + em*mu*mu*v*tau + em*em*mu - y*mu*v;
         coef[0] = x*mu*mu*v*v*tau + x*em*mu*v - y*mu*mu*v*v + y*em*mu*v;

         TMath::RootsCubic(coef,roots[0],roots[1],roots[2]);

         e = roots[1];
         b = y/(tau + (em - e)/mu/v);
         f = LikeMod2(mu,b,e,x,y,em,tau,v);
      }
   }

   return f;
}

//_________________________________________________________________________
Double_t MRolke::LikeMod2(Double_t mu, Double_t b, Double_t e,Int_t x,Int_t y,Double_t em,Double_t tau, Double_t v)
{
   // Profile Likelihood function for MODEL 2:
   // Poisson background/Gauss Efficiency

   return 2*(x*TMath::Log(e*mu+b)-(e*mu+b)-LogFactorial(x)+y*TMath::Log(tau*b)-tau*b-LogFactorial(y)-0.9189385-TMath::Log(v)/2-(em-e)*(em-e)/v/2);
}

//_____________________________________________________________________

Double_t MRolke::EvalLikeMod3(Double_t mu, Int_t x, Double_t bm, Double_t em, Double_t e, Double_t sde, Double_t sdb, Double_t b, Int_t what)
{
   // Calculates the Profile Likelihood for MODEL 3:
   // Gauss  background/ Gauss Efficiency
   // what = 1: Maximum likelihood estimate is returned
   // what = 2: Profile Likelihood of Maxmimum Likelihood estimate is returned.
   // what = 3: Profile Likelihood of Test hypothesis is returned
   // otherwise parameters as described in the beginning of the class)

   Double_t f = 0.;
   Double_t  v = sde*sde;
   Double_t  u = sdb*sdb;

   if (what == 1) {
      f = (x-bm)/em;
   }


   if (what == 2) {
      mu = (x-bm)/em;
      b  = bm;
      e  = em;
      f  = LikeMod3(mu,b,e,x,bm,em,u,v);
   }


   if(what == 3) {
      if(mu == 0.0){
         b = ((bm-u)+TMath::Sqrt((bm-u)*(bm-u)+4*x*u))/2.;
         e = em;
         f = LikeMod3(mu,b,e,x,bm,em,u,v);
      } else {
         Double_t temp[3];
         temp[0] = mu*mu*v+u;
         temp[1] = mu*mu*mu*v*v+mu*v*u-mu*mu*v*em+mu*v*bm-2*u*em;
         temp[2] = mu*mu*v*v*bm-mu*v*u*em-mu*v*bm*em+u*em*em-mu*mu*v*v*x;
         e = (-temp[1]+TMath::Sqrt(temp[1]*temp[1]-4*temp[0]*temp[2]))/2/temp[0];
         b = bm-(u*(em-e))/v/mu;
         f = LikeMod3(mu,b,e,x,bm,em,u,v);
      }
   }

   return f;
}

//____________________________________________________________________
Double_t MRolke::LikeMod3(Double_t mu,Double_t b,Double_t e,Int_t x,Double_t bm,Double_t em,Double_t u,Double_t v)
{
   // Profile Likelihood function for MODEL 3:
   // Gauss background/Gauss Efficiency

   return 2*(x * TMath::Log(e*mu+b)-(e*mu+b)-LogFactorial(x)-1.837877-TMath::Log(u)/2-(bm-b)*(bm-b)/u/2-TMath::Log(v)/2-(em-e)*(em-e)/v/2);
}

//____________________________________________________________________
Double_t MRolke::EvalLikeMod4(Double_t mu, Int_t x, Int_t y, Double_t tau, Double_t b, Int_t what)
{
   // Calculates the Profile Likelihood for MODEL 4:
   // Poiss  background/Efficiency known
   // what = 1: Maximum likelihood estimate is returned
   // what = 2: Profile Likelihood of Maxmimum Likelihood estimate is returned.
   // what = 3: Profile Likelihood of Test hypothesis is returned
   // otherwise parameters as described in the beginning of the class)

   Double_t f = 0.0;

   if (what == 1) f = x-y/tau;
   if (what == 2) {
      mu = x-y/tau;
      b  = Double_t(y)/tau;
      f  = LikeMod4(mu,b,x,y,tau);
   }
   if (what == 3) {
      if (mu == 0.0) {
         b = Double_t(x+y)/(1+tau);
         f = LikeMod4(mu,b,x,y,tau);
      } else {
         b = (x+y-(1+tau)*mu+sqrt((x+y-(1+tau)*mu)*(x+y-(1+tau)*mu)+4*(1+tau)*y*mu))/2/(1+tau);
         f = LikeMod4(mu,b,x,y,tau);
      }
   }
   return f;
}

//___________________________________________________________________
Double_t MRolke::LikeMod4(Double_t mu,Double_t b,Int_t x,Int_t y,Double_t tau)
{
   // Profile Likelihood function for MODEL 4:
   // Poiss background/Efficiency known

   return 2*(x*TMath::Log(mu+b)-(mu+b)-LogFactorial(x)+y*TMath::Log(tau*b)-tau*b-LogFactorial(y) );
}

//___________________________________________________________________
Double_t MRolke::EvalLikeMod5(Double_t mu, Int_t x, Double_t bm, Double_t sdb, Double_t b, Int_t what)
{
   // Calculates the Profile Likelihood for MODEL 5:
   // Gauss  background/Efficiency known
   // what = 1: Maximum likelihood estimate is returned
   // what = 2: Profile Likelihood of Maxmimum Likelihood estimate is returned.
   // what = 3: Profile Likelihood of Test hypothesis is returned
   // otherwise parameters as described in the beginning of the class)

   Double_t u=sdb*sdb;
   Double_t f = 0;

   if(what == 1) {
      f = x - bm;
   }
   if(what == 2) {
      mu = x-bm;
      b  = bm;
      f  = LikeMod5(mu,b,x,bm,u);
   }

   if (what == 3) {
      b = ((bm-u-mu)+TMath::Sqrt((bm-u-mu)*(bm-u-mu)-4*(mu*u-mu*bm-u*x)))/2;
      f = LikeMod5(mu,b,x,bm,u);
   }
   return f;
}

//_______________________________________________________________________
Double_t MRolke::LikeMod5(Double_t mu,Double_t b,Int_t x,Double_t bm,Double_t u)
{
   // Profile Likelihood function for MODEL 5:
   // Gauss background/Efficiency known

   return 2*(x*TMath::Log(mu+b)-(mu + b)-LogFactorial(x)-0.9189385-TMath::Log(u)/2-((bm-b)*(bm-b))/u/2);
}

//_______________________________________________________________________
Double_t MRolke::EvalLikeMod6(Double_t mu, Int_t x, Int_t z, Double_t e, Double_t b, Int_t m, Int_t what)
{
   // Calculates the Profile Likelihood for MODEL 6:
   // Gauss  known/Efficiency binomial
   // what = 1: Maximum likelihood estimate is returned
   // what = 2: Profile Likelihood of Maxmimum Likelihood estimate is returned.
   // what = 3: Profile Likelihood of Test hypothesis is returned
   // otherwise parameters as described in the beginning of the class)

   Double_t coef[4],roots[3];
   Double_t f = 0.;
   Double_t zm = Double_t(z)/m;

   if(what==1){
      f = (x-b)/zm;
   }

   if(what==2){
      mu = (x-b)/zm;
      e  = zm;
      f  = LikeMod6(mu,b,e,x,z,m);
   }
   if(what == 3){
      if(mu==0){
         e = zm;
      } else {
         coef[3] = mu*mu;
         coef[2] = mu * b - mu * x - mu*mu - mu * m;
         coef[1] = mu * x - mu * b + mu * z - m * b;
         coef[0] = b * z;
         TMath::RootsCubic(coef,roots[0],roots[1],roots[2]);
         e = roots[1];
      }
      f =LikeMod6(mu,b,e,x,z,m);
   }
   return f;
}

//_______________________________________________________________________
Double_t MRolke::LikeMod6(Double_t mu,Double_t b,Double_t e,Int_t x,Int_t z,Int_t m)
{
   // Profile Likelihood function for MODEL 6:
   // background known/ Efficiency binomial

   Double_t f = 0.0;

   if (z > 100 || m > 100) {
      f = 2*(x*TMath::Log(e*mu+b)-(e*mu+b)-LogFactorial(x)+(m*TMath::Log(m)  - m)-(z*TMath::Log(z) - z)  - ((m-z)*TMath::Log(m-z) - m + z)+z*TMath::Log(e)+(m-z)*TMath::Log(1-e));
   } else {
      f = 2*(x*TMath::Log(e*mu+b)-(e*mu+b)-LogFactorial(x)+TMath::Log(TMath::Factorial(m))-TMath::Log(TMath::Factorial(z))-TMath::Log(TMath::Factorial(m-z))+z*TMath::Log(e)+(m-z)*TMath::Log(1-e));
   }
   return f;
}


//___________________________________________________________________________
Double_t MRolke::EvalLikeMod7(Double_t mu, Int_t x, Double_t em, Double_t e, Double_t sde, Double_t b, Int_t what)
{
   // Calculates the Profile Likelihood for MODEL 7:
   // background known/Efficiency Gauss
   // what = 1: Maximum likelihood estimate is returned
   // what = 2: Profile Likelihood of Maxmimum Likelihood estimate is returned.
   // what = 3: Profile Likelihood of Test hypothesis is returned
   // otherwise parameters as described in the beginning of the class)

   Double_t v=sde*sde;
   Double_t f = 0.;

   if(what ==  1) {
      f = (x-b)/em;
   }

   if(what == 2) {
      mu = (x-b)/em;
      e  = em;
      f  = LikeMod7(mu, b, e, x, em, v);
   }

   if(what == 3) {
      if(mu==0) {
         e = em;
      } else {
         e = ( -(mu*em-b-mu*mu*v)-TMath::Sqrt((mu*em-b-mu*mu*v)*(mu*em-b-mu*mu*v)+4*mu*(x*mu*v-mu*b*v + b * em)))/( - mu)/2;
      }
      f = LikeMod7(mu, b, e, x, em, v);
   }

   return f;
}

//___________________________________________________________________________
Double_t MRolke::LikeMod7(Double_t mu,Double_t b,Double_t e,Int_t x,Double_t em,Double_t v)
{
   // Profile Likelihood function for MODEL 6:
   // background known/ Efficiency binomial

   return 2*(x*TMath::Log(e*mu+b)-(e*mu + b)-LogFactorial(x)-0.9189385-TMath::Log(v)/2-(em-e)*(em-e)/v/2);
}

//______________________________________________________________________
Double_t MRolke::EvalPolynomial(Double_t x, const Int_t  coef[], Int_t N)
{
  // evaluate polynomial

   const Int_t   *p;
   p = coef;
   Double_t ans = *p++;
   Int_t i = N;

   do
      ans = ans * x  +  *p++;
   while( --i );

   return ans;
}

//______________________________________________________________________
Double_t MRolke::EvalMonomial(Double_t x, const Int_t coef[], Int_t N)
{
   // evaluate mononomial

   Double_t ans;
   const Int_t   *p;

   p   = coef;
   ans = x + *p++;
   Int_t i = N-1;

   do
      ans = ans * x  + *p++;
   while( --i );

   return ans;
}
//--------------------------------------------------------------------------
//
// Uses Stirling-Wells formula: ln(N!) ~ N*ln(N) - N + 0.5*ln(2piN)
// if N exceeds 70, otherwise use the TMath functions
//
//
Double_t MRolke::LogFactorial(Int_t n)
{
   if (n>69)
       return n*TMath::Log(n)-n + 0.5*TMath::Log(TMath::TwoPi()*n);
   else
       return TMath::Log(TMath::Factorial(n));
}