HypoTestInverterResult.cxx

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// @(#)root/roostats:$Id$
// Author: Kyle Cranmer, Lorenzo Moneta, Gregory Schott, Wouter Verkerke
/*************************************************************************
 * Copyright (C) 1995-2008, Rene Brun and Fons Rademakers.               *
 * All rights reserved.                                                  *
 *                                                                       *
 * For the licensing terms see $ROOTSYS/LICENSE.                         *
 * For the list of contributors see $ROOTSYS/README/CREDITS.             *
 *************************************************************************/




// include header file of this class 
#include "RooStats/HypoTestInverterResult.h"

#include "RooStats/HybridResult.h"
#include "RooStats/SamplingDistribution.h"
#include "RooStats/AsymptoticCalculator.h"
#include "RooMsgService.h"
#include "RooGlobalFunc.h"

#include "TF1.h"
#include "TGraphErrors.h"
#include <cmath>
#include "Math/BrentRootFinder.h"
#include "Math/WrappedFunction.h"

#include "TCanvas.h"
#include "RooStats/SamplingDistPlot.h"

#include <algorithm>

ClassImp(RooStats::HypoTestInverterResult)

using namespace RooStats;
using namespace RooFit;
using namespace std;


// initialize static value 
double HypoTestInverterResult::fgAsymptoticMaxSigma      = 5; 


HypoTestInverterResult::HypoTestInverterResult(const char * name ) :
   SimpleInterval(name),
   fUseCLs(false),
   fIsTwoSided(false),
   fInterpolateLowerLimit(true),
   fInterpolateUpperLimit(true),
   fFittedLowerLimit(false),
   fFittedUpperLimit(false),
   fInterpolOption(kLinear),
   fLowerLimitError(-1),
   fUpperLimitError(-1),
   fCLsCleanupThreshold(0.005)
{
  // default constructor
   fLowerLimit = TMath::QuietNaN();
   fUpperLimit = TMath::QuietNaN();

   fYObjects.SetOwner();
   fExpPValues.SetOwner();
}


HypoTestInverterResult::HypoTestInverterResult( const HypoTestInverterResult& other, const char * name ) :
   SimpleInterval(other,name),
   fUseCLs(other.fUseCLs),
   fIsTwoSided(other.fIsTwoSided),
   fInterpolateLowerLimit(other.fInterpolateLowerLimit),
   fInterpolateUpperLimit(other.fInterpolateUpperLimit),
   fFittedLowerLimit(other.fFittedLowerLimit),
   fFittedUpperLimit(other.fFittedUpperLimit),
   fInterpolOption(other.fInterpolOption),
   fLowerLimitError(other.fLowerLimitError),
   fUpperLimitError(other.fUpperLimitError),
   fCLsCleanupThreshold(other.fCLsCleanupThreshold)
{
   // copy constructor
   fLowerLimit = TMath::QuietNaN();
   fUpperLimit = TMath::QuietNaN();
   int nOther = other.ArraySize();

   fXValues = other.fXValues;
   for (int i = 0; i < nOther; ++i) 
     fYObjects.Add( other.fYObjects.At(i)->Clone() );
   for (int i = 0; i <  fExpPValues.GetSize() ; ++i)
     fExpPValues.Add( other.fExpPValues.At(i)->Clone() );   

   fYObjects.SetOwner();
   fExpPValues.SetOwner();
}


HypoTestInverterResult&
HypoTestInverterResult::operator=(const HypoTestInverterResult& other)
{
  if (&other==this) {
    return *this ;
  } 

  SimpleInterval::operator = (other);
  fLowerLimit = other.fLowerLimit;
  fUpperLimit = other.fUpperLimit;
  fUseCLs = other.fUseCLs;
  fIsTwoSided = other.fIsTwoSided;
  fInterpolateLowerLimit = other.fInterpolateLowerLimit;
  fInterpolateUpperLimit = other.fInterpolateUpperLimit;
  fFittedLowerLimit = other.fFittedLowerLimit;
  fFittedUpperLimit = other.fFittedUpperLimit;
  fInterpolOption = other.fInterpolOption;
  fLowerLimitError = other.fLowerLimitError;
  fUpperLimitError = other.fUpperLimitError;
  fCLsCleanupThreshold = other.fCLsCleanupThreshold;

  int nOther = other.ArraySize();
  fXValues = other.fXValues;

  fYObjects.RemoveAll();
  for (int i=0; i < nOther; ++i) {
    fYObjects.Add( other.fYObjects.At(i)->Clone() );
  }
  fExpPValues.RemoveAll();
  for (int i=0; i <  fExpPValues.GetSize() ; ++i) {
    fExpPValues.Add( other.fExpPValues.At(i)->Clone() );   
  }

  fYObjects.SetOwner();
  fExpPValues.SetOwner();

  return *this;
}


HypoTestInverterResult::HypoTestInverterResult( const char* name,
                  const RooRealVar& scannedVariable,
                  double cl ) :
   SimpleInterval(name,scannedVariable,TMath::QuietNaN(),TMath::QuietNaN(),cl), 
   fUseCLs(false),
   fIsTwoSided(false),
   fInterpolateLowerLimit(true),
   fInterpolateUpperLimit(true),
   fFittedLowerLimit(false),
   fFittedUpperLimit(false),
   fInterpolOption(kLinear),
   fLowerLimitError(-1),
   fUpperLimitError(-1),
   fCLsCleanupThreshold(0.005)
{
   // constructor 
   fYObjects.SetOwner();
   fExpPValues.SetOwner();

   // put a cloned copy of scanned variable to set in the interval
   // to avoid I/O problem of the Result class - 
   // make the set owning the cloned copy (use clone istead of Clone to not copying all links)
   fParameters.removeAll();
   fParameters.takeOwnership();
   fParameters.addOwned(*((RooRealVar *) scannedVariable.clone(scannedVariable.GetName()) ));
}

HypoTestInverterResult::~HypoTestInverterResult()
{
   // destructor
   // explicitly empty the TLists - these contain pointers, not objects
   fYObjects.RemoveAll();
   fExpPValues.RemoveAll();

   fYObjects.Delete();
   fExpPValues.Delete();
}

int
HypoTestInverterResult::ExclusionCleanup()
{
  const int nEntries  = ArraySize();

  // initialization
  double nsig1(1.0);
  double nsig2(2.0);
  double p[5]; 
  double q[5];
  std::vector<double> qv;
  qv.resize(11,-1.0);

  p[0] = ROOT::Math::normal_cdf(-nsig2);
  p[1] = ROOT::Math::normal_cdf(-nsig1);
  p[2] = 0.5;
  p[3] = ROOT::Math::normal_cdf(nsig1);
  p[4] = ROOT::Math::normal_cdf(nsig2);

  bool resultIsAsymptotic(false);
  if (nEntries>=1) {
    HypoTestResult* r = dynamic_cast<HypoTestResult *> ( GetResult(0) );
    assert(r!=0);
    if ( !r->GetNullDistribution() && !r->GetAltDistribution() ) { 
      resultIsAsymptotic = true;
    }
  }

  int nPointsRemoved(0);

  double CLsobsprev(1.0);
  std::vector<double>::iterator itr = fXValues.begin();

  for (; itr!=fXValues.end();) {

    double x = (*itr);
    int i = FindIndex(x);
    //HypoTestResult* oneresult = GetResult(i);

    SamplingDistribution * s = GetExpectedPValueDist(i);
    if (!s) break; 

    /////////////////////////////////////////////////////////////////////////////////////////

    const std::vector<double> & values = s->GetSamplingDistribution();
    
    /// expected p-values
    // special case for asymptotic results (cannot use TMath::quantile in that case)
    if (resultIsAsymptotic) { 
      double maxSigma = 5; // == HypoTestInverterResult::fgAsymptoticMaxSigma; // MB: HACK
      double dsig = 2.*maxSigma / (values.size() -1) ;         
      int  i0 = (int) TMath::Floor ( ( -nsig2 + maxSigma )/dsig + 0.5 );
      int  i1 = (int) TMath::Floor ( ( -nsig1 + maxSigma )/dsig + 0.5 );
      int  i2 = (int) TMath::Floor ( ( maxSigma )/dsig + 0.5 );
      int  i3 = (int) TMath::Floor ( ( nsig1 + maxSigma )/dsig + 0.5 );
      int  i4 = (int) TMath::Floor ( ( nsig2 + maxSigma )/dsig + 0.5 );
      //
      q[0] = values[i0];
      q[1] = values[i1];
      q[2] = values[i2];
      q[3] = values[i3];
      q[4] = values[i4];
    } else { 
      double * z = const_cast<double *>( &values[0] ); // need to change TMath::Quantiles
      TMath::Quantiles(values.size(), 5, z, q, p, false);
    }
    
    delete s;

    /// store useful quantities for reuse later ...
    /// http://root.cern.ch/root/html532/src/RooStats__HypoTestInverterPlot.cxx.html#197
    for (int j=0; j<5; ++j) { qv[j]=q[j]; }
    qv[5]  = CLs(i) ; //
    qv[6]  = CLsError(i) ; //
    qv[7]  = CLb(i) ; //
    qv[8]  = CLbError(i) ; //
    qv[9]  = CLsplusb(i) ; //
    qv[10] = CLsplusbError(i) ; //
    double CLsobs = qv[5];

    /////////////////////////////////////////////////////////////////////////////////////////

    bool removeThisPoint(false);

    // 1. CLs should drop, else skip this point
    if (!removeThisPoint && resultIsAsymptotic && i>=1 && CLsobs>CLsobsprev) {
      //StatToolsLogger << kERROR << "Asymptotic. CLs not dropping: " << CLsobs << ". Remove this point." << GEndl;
      removeThisPoint = true; 
    } else { CLsobsprev = CLsobs; }
      
    // 2. CLs should not spike, else skip this point
    if (!removeThisPoint && i>=1 && CLsobs>=0.9999) {
      //StatToolsLogger << kERROR << "CLs spiking at 1.0: " << CLsobs << ". Remove this point." << GEndl;      
      removeThisPoint = true; 
    }
    // 3. Not interested in CLs values that become too low.
    if (!removeThisPoint && i>=1 && qv[4]<fCLsCleanupThreshold) { removeThisPoint = true; }

    // to remove or not to remove
    if (removeThisPoint) {
      itr = fXValues.erase(itr); // returned itr has been updated.     
      fYObjects.RemoveAt(i);
      fExpPValues.RemoveAt(i);
      nPointsRemoved++;
      continue;
    } else { // keep
      CLsobsprev = CLsobs;
      itr++;
    }
  }

  // after cleanup, reset existing limits
  fFittedUpperLimit = false; 
  fFittedLowerLimit = false;
  FindInterpolatedLimit(1-ConfidenceLevel(),true);

  return nPointsRemoved;
}

bool HypoTestInverterResult::Add( const HypoTestInverterResult& otherResult   )
{
   // Merge this HypoTestInverterResult with another
   // HypoTestInverterResult passed as argument
   // The merge is done by combining the HypoTestResult when the same point value exist in both results.
   // If results exist at different points these are added in the new result
   // NOTE: Merging of the expected p-values obtained with pseudo-data.
   //  When expected p-values exist in the result (i.e. when rebuild option is used when getting the expected 
   // limit distribuiton in the HYpoTestInverter) then the expected p-values are also merged. This is equivalent 
   // at merging the pseudo-data. However there can be an inconsistency if the expected p-values have been 
   // obtained with different toys. In this case the merge is done but a warning message is printed.


   int nThis = ArraySize();
   int nOther = otherResult.ArraySize();
   if (nOther == 0) return true;
   if (nOther != otherResult.fYObjects.GetSize() ) return false; 
   if (nThis != fYObjects.GetSize() ) return false; 

   // cannot merge in case of inconsistent memebrs
   if (fExpPValues.GetSize() > 0 && fExpPValues.GetSize() != nThis ) return false;
   if (otherResult.fExpPValues.GetSize() > 0 && otherResult.fExpPValues.GetSize() != nOther ) return false;

   oocoutI(this,Eval) << "HypoTestInverterResult::Add - merging result from " << otherResult.GetName()  
                                << " in " << GetName() << std::endl;

   bool addExpPValues = (fExpPValues.GetSize() == 0 && otherResult.fExpPValues.GetSize() > 0);
   bool mergeExpPValues = (fExpPValues.GetSize() > 0 && otherResult.fExpPValues.GetSize() > 0);

   if (addExpPValues || mergeExpPValues)
      oocoutI(this,Eval) << "HypoTestInverterResult::Add - merging also the expected p-values from pseudo-data" << std::endl;


   // case current result is empty
   // just make a simple copy of the other result
   if (nThis == 0) {
      fXValues = otherResult.fXValues;
      for (int i = 0; i < nOther; ++i) 
         fYObjects.Add( otherResult.fYObjects.At(i)->Clone() );
      for (int i = 0; i <  fExpPValues.GetSize() ; ++i)
         fExpPValues.Add( otherResult.fExpPValues.At(i)->Clone() );
   }
   // now do teh real mergemerge combining point with same value or adding extra ones 
   else { 
      for (int i = 0; i < nOther; ++i) {
         double otherVal = otherResult.fXValues[i];
         HypoTestResult * otherHTR = (HypoTestResult*) otherResult.fYObjects.At(i);
         if (otherHTR == 0) continue;
         bool sameXFound = false;
         for (int j = 0; j < nThis; ++j) {
            double thisVal = fXValues[j];
            
            // if same value merge the result 
            if ( (std::abs(otherVal) < 1  && TMath::AreEqualAbs(otherVal, thisVal,1.E-12) ) || 
                 (std::abs(otherVal) >= 1 && TMath::AreEqualRel(otherVal, thisVal,1.E-12) ) ) {
               HypoTestResult * thisHTR = (HypoTestResult*) fYObjects.At(j);               
               thisHTR->Append(otherHTR);
               sameXFound = true;
               if (mergeExpPValues) { 
                  ((SamplingDistribution*) fExpPValues.At(j))->Add( (SamplingDistribution*)otherResult.fExpPValues.At(i) );
                  // check if same toys have been used for the test statistic distribution
                  int thisNToys = (thisHTR->GetNullDistribution() ) ? thisHTR->GetNullDistribution()->GetSize() : 0;
                  int otherNToys = (otherHTR->GetNullDistribution() ) ? otherHTR->GetNullDistribution()->GetSize() : 0;
                  if (thisNToys != otherNToys ) 
                     oocoutW(this,Eval) << "HypoTestInverterResult::Add expexted p values have been generated with different toys " << thisNToys << " , " << otherNToys << std::endl;                  
               }
               break;
            }
         }
         if (!sameXFound) { 
            // add the new result 
            fYObjects.Add(otherHTR->Clone() );
            fXValues.push_back( otherVal);
         }
         // add in any case also when same x found
         if (addExpPValues)  
            fExpPValues.Add( otherResult.fExpPValues.At(i)->Clone() );
         
         
      }
   }
   
   if (ArraySize() > nThis) 
      oocoutI(this,Eval) << "HypoTestInverterResult::Add  - new number of points is " << fXValues.size()
                         << std::endl;
   else 
      oocoutI(this,Eval) << "HypoTestInverterResult::Add  - new toys/point is " 
                         <<  ((HypoTestResult*) fYObjects.At(0))->GetNullDistribution()->GetSize() 
                         << std::endl;

   // reset cached limit values
   fLowerLimit = TMath::QuietNaN();
   fUpperLimit = TMath::QuietNaN();
   
   return true;
}

bool HypoTestInverterResult::Add (Double_t x, const HypoTestResult & res) 
{
   // Add a single point result (an HypoTestResult)
   int i= FindIndex(x);
   if (i<0) {
      fXValues.push_back(x);
      fYObjects.Add(res.Clone());
   } else {
      HypoTestResult* r= GetResult(i);
      if (!r) return false;
      r->Append(&res);
   }

   // reset cached limit values
   fLowerLimit = TMath::QuietNaN();
   fUpperLimit = TMath::QuietNaN();
   
   return true;
}


 
double HypoTestInverterResult::GetXValue( int index ) const
{
 // function to return the value of the parameter of interest for the i^th entry in the results
  if ( index >= ArraySize() || index<0 ) {
    oocoutE(this,InputArguments) << "Problem: You are asking for an impossible array index value\n";
    return -999;
  }

  return fXValues[index];
}

double HypoTestInverterResult::GetYValue( int index ) const
{
    // function to return the value of the confidence level for the i^th entry in the results
    auto result = GetResult(index);
    if ( !result ) { 
      return -999;
    }

    if (fUseCLs) { 
      return result->CLs();
    } else { 
      return result->CLsplusb();  // CLs+b
    }
}

double HypoTestInverterResult::GetYError( int index ) const
{
    // function to return the estimated error on the value of the confidence level for the i^th entry in the results
    auto result = GetResult(index);
    if ( !result ) {
        return -999;
    }

    if (fUseCLs) { 
        return result->CLsError();
    } else {
        return result->CLsplusbError();
    }
}

double HypoTestInverterResult::CLb( int index ) const
{
    // function to return the observed CLb value  for the i-th entry
    auto result = GetResult(index);
    if ( !result ) {
        return -999;
    }
    return result->CLb();  
}

double HypoTestInverterResult::CLsplusb( int index ) const
{
    // function to return the observed CLs+b value  for the i-th entry
    auto result = GetResult(index);
    if ( !result) { 
        return -999;
    }
    return result->CLsplusb();  
}
double HypoTestInverterResult::CLs( int index ) const
{
    // function to return the observed CLs value  for the i-th entry
    auto result = GetResult(index);
    if ( !result ) { 
        return -999;
    }
    return result->CLs();  
}

double HypoTestInverterResult::CLbError( int index ) const
{
    // function to return the error on the observed CLb value  for the i-th entry
    auto result = GetResult(index);
    if ( !result ) { 
        return -999;
    }
    return result->CLbError();  
}

double HypoTestInverterResult::CLsplusbError( int index ) const
{
    // function to return the error on the observed CLs+b value  for the i-th entry
    auto result = GetResult(index);
    if ( ! result ) {
        return -999;
    }
    return result->CLsplusbError();  
}

double HypoTestInverterResult::CLsError( int index ) const
{
    // function to return the error on the observed CLs value  for the i-th entry
    auto result = GetResult(index);
    if ( ! result ){ 
        return -999;
    }
    return result->CLsError();  
}

HypoTestResult* HypoTestInverterResult::GetResult( int index ) const
{
   // get the HypoTestResult object at the given index point
   if ( index >= ArraySize() || index<0 ) {
      oocoutE(this,InputArguments) << "Problem: You are asking for an impossible array index value\n";
      return 0;
   }
   
   return ((HypoTestResult*) fYObjects.At(index));
}

int HypoTestInverterResult::FindIndex(double xvalue) const
{
   // find the index corresponding at the poi value xvalue
   // If no points is found return -1
   // Note that a tolerance is used of 10^-12 to find the closest point
  const double tol = 1.E-12;
  for (int i=0; i<ArraySize(); i++) {
     double xpoint = fXValues[i];
     if ( (std::abs(xvalue) > 1 && TMath::AreEqualRel( xvalue, xpoint, tol) ) ||
          (std::abs(xvalue) < 1 && TMath::AreEqualAbs( xvalue, xpoint, tol) ) )
        return i; 
  }
  return -1;
}


struct InterpolatedGraph { 
   InterpolatedGraph( const TGraph & g, double target, const char * interpOpt) : 
      fGraph(g), fTarget(target), fInterpOpt(interpOpt) {}

   // return interpolated value for x - target
   double operator() (double x) const { 
      return fGraph.Eval(x, (TSpline*) 0, fInterpOpt) - fTarget;
   }
   const TGraph & fGraph;
   double fTarget;
   TString fInterpOpt;
}; 

double HypoTestInverterResult::GetGraphX(const TGraph & graph, double y0, bool lowSearch, double &axmin, double &axmax) const  {
   // return the X value of the given graph for the target value y0
   // the graph is evaluated using linear interpolation by default. 
   // if option = "S" a TSpline3 is used 



   TString opt = "";
   if (fInterpolOption == kSpline)  opt = "S";

   InterpolatedGraph f(graph,y0,opt);
   ROOT::Math::BrentRootFinder brf;
   ROOT::Math::WrappedFunction<InterpolatedGraph> wf(f);

   // find reasanable xmin and xmax if not given
   const double * y = graph.GetY(); 
   int n = graph.GetN();
   if (n < 2) {
      ooccoutE(this,Eval) << "HypoTestInverterResult::GetGraphX - need at least 2 points for interpolation (n=" << n << ")\n";
      return (n>0) ?  y[0] : 0;
   } 

   double varmin = - TMath::Infinity();
   double varmax = TMath::Infinity();
   const RooRealVar* var = dynamic_cast<RooRealVar*>( fParameters.first() );
   if (var) { 
       varmin = var->getMin();
       varmax = var->getMax();
   }

   double xmin = axmin; 
   double xmax = axmax;
   // case no range is given check if need to extrapolate to lower/upper values 
   if (axmin >= axmax ) { 
      xmin = graph.GetX()[0];
      xmax = graph.GetX()[n-1];

      // case we have lower/upper limits

      // find ymin and ymax  and corresponding values 
      double ymin = TMath::MinElement(n,y);
      double ymax = TMath::MaxElement(n,y);
      // do lower extrapolation 
      if ( (ymax < y0 && !lowSearch) || ( ymin > y0 && lowSearch) ) { 
         xmin = varmin;
      }
      // do upper extrapolation  
      if ( (ymax < y0 && lowSearch) || ( ymin > y0 && !lowSearch) ) { 
         xmax = varmax;
      }
   }
   else { 

#ifdef ISREALLYNEEDED //??
      // in case of range given, check if all points are above or below the confidence level line
      bool isCross = false;
      bool first = true; 
      double prod = 1;
      for (int i = 0; i< n; ++i) { 
         double xp, yp;
         graph.GetPoint(i,xp,yp);
         if (xp >= xmin && xp <= xmax) { 
            prod *= TMath::Sign(1., (yp - y0) );
            if (prod < 0 && !first) { 
               isCross = true; 
               break;
            }
            first = false;
         }
      }
      if (!isCross) { 
         return (lowSearch) ? varmin : varmax;
      }
#endif
   }
   


   brf.SetFunction(wf,xmin,xmax);
   brf.SetNpx(20);
   bool ret = brf.Solve(100, 1.E-16, 1.E-6);
   if (!ret) { 
      ooccoutE(this,Eval) << "HypoTestInverterResult - interpolation failed - return inf" << std::endl;
         return TMath::Infinity();
   }
   double limit =  brf.Root();

//#define DO_DEBUG
#ifdef DO_DEBUG
   if (lowSearch) std::cout << "lower limit search : ";
   else std::cout << "Upper limit search :  ";
   std::cout << "do interpolation between " << xmin << "  and " << xmax << " limit is " << limit << std::endl;
#endif

   // look in case if a new interseption exists
   // only when boundaries are not given
   if (axmin >= axmax) { 
      int index = TMath::BinarySearch(n, graph.GetX(), limit);
#ifdef DO_DEBUG
   std::cout << "do new interpolation dividing from " << index << "  and " << y[index] << std::endl;
#endif

      if (lowSearch && index >= 1 && (y[0] - y0) * ( y[index]- y0) < 0) { 
         //search if  another interseption exists at a lower value
         limit =  GetGraphX(graph, y0, lowSearch, graph.GetX()[0], graph.GetX()[index] );
      }
      else if (!lowSearch && index < n-2 && (y[n-1] - y0) * ( y[index+1]- y0) < 0) { 
         // another interseption exists at an higher value
         limit =  GetGraphX(graph, y0, lowSearch, graph.GetX()[index+1], graph.GetX()[n-1] );
      }
   }
   // return also xmin, xmax values
   axmin = xmin;
   axmax = xmax;

   return limit;
}

double HypoTestInverterResult::FindInterpolatedLimit(double target, bool lowSearch, double xmin, double xmax )
{
   // interpolate to find a limit value
   // Use a linear or a spline interpolation depending on the interpolation option

   ooccoutD(this,Eval) << "HypoTestInverterResult - " 
                       << "Interpolate the upper limit between the 2 results closest to the target confidence level" 
                       << std::endl;

   // variable minimum and maximum
   double varmin = - TMath::Infinity();
   double varmax = TMath::Infinity();
   const RooRealVar* var = dynamic_cast<RooRealVar*>( fParameters.first() );
   if (var) { 
       varmin = var->getMin();
       varmax = var->getMax();
   }

   if (ArraySize()<2) {
      double val =  (lowSearch) ? xmin : xmax;
      oocoutW(this,Eval) << "HypoTestInverterResult::FindInterpolatedLimit" 
                         << " - not enough points to get the inverted interval - return " 
                         <<  val << std::endl;
      fLowerLimit = varmin;
      fUpperLimit = varmax; 
      return (lowSearch) ? fLowerLimit : fUpperLimit;  
   }

   // sort the values in x 
   int n = ArraySize();
   std::vector<unsigned int> index(n );
   TMath::SortItr(fXValues.begin(), fXValues.end(), index.begin(), false); 
   // make a graph with the sorted point
   TGraph graph(n);
   for (int i = 0; i < n; ++i) 
      graph.SetPoint(i, GetXValue(index[i]), GetYValue(index[i] ) );


   //std::cout << " search for " << lowSearch << std::endl;
   

   // search first for min/max in the given range
   if (xmin >= xmax) {
      

      // search for maximum between the point
      double * itrmax = std::max_element(graph.GetY() , graph.GetY() +n);
      double ymax = *itrmax; 
      int iymax = itrmax - graph.GetY(); 
      double xwithymax = graph.GetX()[iymax];

      //std::cout << " max of y " << iymax << "  " << xwithymax << "  " << ymax << " target is " << target << std::endl;

      // look if maximum is above/belove target
      if (ymax > target) {
         if (lowSearch)  {
            if ( iymax > 0) { 
                  // low search (minimmum is first point or minimum range)
               xmin = ( graph.GetY()[0] <= target ) ? graph.GetX()[0] : varmin;  
               xmax = xwithymax;
               } 
            else { 
               // no room for lower limit
               fLowerLimit = varmin; 
               lowSearch = false; // search now for upper limits
            }
         }
         if (!lowSearch ) {
            // up search 
            if ( iymax < n-1 ) { 
               xmin = xwithymax; 
               xmax = ( graph.GetY()[n-1] <= target ) ? graph.GetX()[n-1] : varmax;
            }
            else { 
               // no room for upper limit
               fUpperLimit = varmax; 
               lowSearch = true; // search now for lower limits
               xmin = varmin; 
               xmax = xwithymax;
               }
         }
      }
      else { 
         // in case is below the target
         // find out if is a lower or upper search
         if (iymax <= (n-1)/2 ) { 
            lowSearch = false; 
            fLowerLimit = varmin; 
         }
         else { 
            lowSearch = true; 
            fUpperLimit = varmax;
         }
      }

#ifdef DO_DEBUG   
      std::cout << " found xmin, xmax  = " << xmin << "  " << xmax << " for search " << lowSearch << std::endl;
#endif

      // now come here if I have already found a lower/upper limit 
      // i.e. I am calling routine for the second time
#ifdef ISNEEDED
      // should not really come here
      if (lowSearch &&  fUpperLimit < varmax) {
         xmin = fXValues[ index.front() ];
         // find xmax (is first point before upper limit)
         int upI = FindClosestPointIndex(target, 2, fUpperLimit);
         if (upI < 1) return xmin; 
         xmax = GetXValue(upI);
      }
      else if (!lowSearch && fLowerLimit > varmin ) {
         // find xmin (is first point after lower limit)
         int lowI = FindClosestPointIndex(target, 3, fLowerLimit);
         if (lowI >= n-1) return xmax;
         xmin = GetXValue(lowI);
         xmax = fXValues[ index.back() ];
      }
#endif
   }

#ifdef DO_DEBUG   
   std::cout << "finding " << lowSearch << " limit betweem " << xmin << "  " << xmax << endl;
#endif

   double limit =  GetGraphX(graph, target, lowSearch, xmin, xmax);
   if (lowSearch) fLowerLimit = limit; 
   else fUpperLimit = limit;
   CalculateEstimatedError( target, lowSearch, xmin, xmax);

#ifdef DO_DEBUG
   std::cout << "limit is " << limit << std::endl;
#endif

   if (lowSearch && !TMath::IsNaN(fUpperLimit)) return fLowerLimit;
   if (!lowSearch && !TMath::IsNaN(fLowerLimit)) return fUpperLimit;

   // now perform the opposite search on the complement interval
   if (lowSearch) { 
      xmin = xmax;
      xmax = varmax;         
   } else { 
      xmax = xmin;         
      xmin = varmin;
   }
   double limit2 =  GetGraphX(graph, target, !lowSearch, xmin, xmax);
   if (!lowSearch) fLowerLimit = limit2; 
   else fUpperLimit = limit2;

   CalculateEstimatedError( target, !lowSearch, xmin, xmax);

#ifdef DO_DEBUG
   std::cout << "other limit is " << limit2 << std::endl;
#endif
 
   return (lowSearch) ? fLowerLimit : fUpperLimit;

}


int HypoTestInverterResult::FindClosestPointIndex(double target, int mode, double xtarget)
{
   // if mode = 0 
   // find closest point to target in Y, the object closest to the target which is 3 sigma from the target 
   // and has smaller error
   // if mode = 1
   // find 2 closest point to target in X and between these two take the one closer to the target
   // if mode = 2  as in mode = 1 but return the lower point not the closest one
   // if mode = 3  as in mode = 1 but return the upper point not the closest one

  int bestIndex = -1;
  int closestIndex = -1;
  if (mode == 0) { 
     double smallestError = 2; // error must be < 1
     double bestValue = 2;
     for (int i=0; i<ArraySize(); i++) {
        double dist = fabs(GetYValue(i)-target);
        if ( dist <3 *GetYError(i) ) { // less than 1 sigma from target CL
           if (GetYError(i) < smallestError ) { 
              smallestError = GetYError(i);
              bestIndex = i; 
           }
        }
        if ( dist < bestValue) { 
           bestValue = dist;  
           closestIndex = i; 
        }
     }
     if (bestIndex >=0) return bestIndex;
     // if no points found just return the closest one to the target
     return closestIndex;
  }
  // else mode = 1,2,3
  // find the two closest points to limit value
  // sort the array first 
  int n = fXValues.size();
  std::vector<unsigned int> indx(n);
  TMath::SortItr(fXValues.begin(), fXValues.end(), indx.begin(), false); 
  std::vector<double> xsorted( n);
  for (int i = 0; i < n; ++i) xsorted[i] = fXValues[indx[i] ];
  int index1 = TMath::BinarySearch( n, &xsorted[0], xtarget);

#ifdef DO_DEBUG
  std::cout << "finding closest point to " << xtarget << " is " << index1 << "  " << indx[index1] << std::endl; 
#endif

   // case xtarget is outside the range (bbefore or afterwards)
  if (index1 < 0) return indx[0]; 
  if (index1 >= n-1) return indx[n-1];  
  int index2 = index1 +1;
  
  if (mode == 2) return (GetXValue(indx[index1]) < GetXValue(indx[index2])) ? indx[index1] : indx[index2];
  if (mode == 3) return (GetXValue(indx[index1]) > GetXValue(indx[index2])) ? indx[index1] : indx[index2];
  // get smaller point of the two (mode == 1)
  if (fabs(GetYValue(indx[index1])-target) <= fabs(GetYValue(indx[index2])-target) )
     return indx[index1];
  return indx[index2];

}
        

Double_t HypoTestInverterResult::LowerLimit()
{
  if (fFittedLowerLimit) return fLowerLimit;
  //std::cout << "finding point with cl = " << 1-(1-ConfidenceLevel())/2 << endl;
  if ( fInterpolateLowerLimit ) { 
     // find both lower/upper limit
     if (TMath::IsNaN(fLowerLimit) )  FindInterpolatedLimit(1-ConfidenceLevel(),true);
  } else {
     //LM: I think this is never called
     fLowerLimit = GetXValue( FindClosestPointIndex((1-ConfidenceLevel())) );
  }
  return fLowerLimit;
}

Double_t HypoTestInverterResult::UpperLimit()
{
   //std::cout << "finding point with cl = " << (1-ConfidenceLevel())/2 << endl;
  if (fFittedUpperLimit) return fUpperLimit;
  if ( fInterpolateUpperLimit ) {
     if (TMath::IsNaN(fUpperLimit) )  FindInterpolatedLimit(1-ConfidenceLevel(),false);
  } else {
     //LM: I think this is never called
     fUpperLimit = GetXValue( FindClosestPointIndex((1-ConfidenceLevel())) );
  }
  return fUpperLimit;
}

Double_t HypoTestInverterResult::CalculateEstimatedError(double target, bool lower, double xmin, double xmax)
{
  // Return an error estimate on the upper(lower) limit.  This is the error on
  // either CLs or CLsplusb divided by an estimate of the slope at this
  // point.
   
  if (ArraySize()==0) {
     oocoutW(this,Eval) << "HypoTestInverterResult::CalculateEstimateError" 
                        << "Empty result \n";
    return 0;
  }

  if (ArraySize()<2) {
     oocoutW(this,Eval) << "HypoTestInverterResult::CalculateEstimateError" 
                        << " only  points - return its error\n"; 
     return GetYError(0);
  }

  // it does not make sense in case of asymptotic which do not have point errors 
  if (!GetNullTestStatDist(0) ) return 0; 
 
  TString type = (!lower) ? "upper" : "lower";

#ifdef DO_DEBUG
  std::cout << "calculate estimate error " << type << " between " << xmin << " and " << xmax << std::endl;
  std::cout << "computed limit is " << ( (lower) ? fLowerLimit : fUpperLimit ) << std::endl;
#endif

    // make a TGraph Errors with the sorted points
  std::vector<unsigned int> indx(fXValues.size());
  TMath::SortItr(fXValues.begin(), fXValues.end(), indx.begin(), false); 
  // make a graph with the sorted point
  TGraphErrors graph;
  int ip = 0, np = 0;
  for (int i = 0; i < ArraySize(); ++i) {
     if ( (xmin < xmax) && ( GetXValue(indx[i]) >= xmin && GetXValue(indx[i]) <= xmax) ) {
        np++;
        // exclude points with zero or very small errors 
        if (GetYError(indx[i] ) > 1.E-6) {  
           graph.SetPoint(ip, GetXValue(indx[i]), GetYValue(indx[i] ) );
           graph.SetPointError(ip, 0.,  GetYError(indx[i]) );
           ip++;
        }
     }
  }
  if (graph.GetN() < 2) {
     if (np >= 2) oocoutW(this,Eval) << "HypoTestInverterResult::CalculateEstimatedError - no valid points - cannot estimate  the " << type << " limit error " << std::endl;
     return 0; 
  }

  double minX = xmin;
  double maxX = xmax;
  if (xmin >= xmax) {
     minX = fXValues[ indx.front() ];
     maxX = fXValues[ indx.back() ];
  }



  TF1 fct("fct", "exp([0] * (x - [2] ) + [1] * (x-[2])**2)", minX, maxX);
  double scale = maxX-minX; 
  if (lower) { 
     fct.SetParameters( 2./scale, 0.1/scale, graph.GetX()[0] ); 
     fct.SetParLimits(0,0,100./scale); 
     fct.SetParLimits(1,0, 10./scale); }
  else  { 
     fct.SetParameters( -2./scale, -0.1/scale ); 
     fct.SetParLimits(0,-100./scale, 0); 
     fct.SetParLimits(1,-100./scale, 0); }

  if (graph.GetN() < 3) fct.FixParameter(1,0.);

  // find the point closest to the limit
  double limit = (!lower) ? fUpperLimit : fLowerLimit;
  if (TMath::IsNaN(limit)) return 0; // cannot do if limit not computed  


#ifdef DO_DEBUG
  TCanvas * c1 = new TCanvas(); 
  std::cout << "fitting for limit " << type << "between " << minX << " , " << maxX << " points considered " << graph.GetN() <<  std::endl;
  int fitstat = graph.Fit(&fct," EX0");
  graph.SetMarkerStyle(20);
  graph.Draw("AP");
  graph.Print(); 
  c1->SaveAs(TString::Format("graphFit_%s.pdf",type.Data()) );
  delete c1; 
#else
  int fitstat = graph.Fit(&fct,"Q EX0");
#endif 

  int index = FindClosestPointIndex(target, 1, limit);
  double theError = 0;
  if (fitstat == 0) { 
     double errY = GetYError(index);
     if (errY >  0) {
        double m = fct.Derivative( GetXValue(index) );
        theError = std::min(fabs( GetYError(index) / m), maxX-minX);     
     }
  }
  else { 
     oocoutW(this,Eval) << "HypoTestInverterResult::CalculateEstimatedError - cannot estimate  the " << type << " limit error " << std::endl;
     theError = 0; 
  }
  if (lower) 
     fLowerLimitError = theError;
  else
     fUpperLimitError = theError;

#ifdef DO_DEBUG
  std::cout << "closes point to the limit is " << index << "  " << GetXValue(index) << " and has error " << GetYError(index) << std::endl;
#endif

  return theError;
}


Double_t HypoTestInverterResult::LowerLimitEstimatedError()
{
   // need to have compute first lower limit
   if (TMath::IsNaN(fLowerLimit) ) LowerLimit();
   if (fLowerLimitError >= 0) return fLowerLimitError;    
   // try to recompute the error
   return CalculateEstimatedError(1-ConfidenceLevel(), true);
}


Double_t HypoTestInverterResult::UpperLimitEstimatedError()
{
   if (TMath::IsNaN(fUpperLimit) ) UpperLimit();
   if (fUpperLimitError >= 0) return fUpperLimitError;
   // try to recompute the error
   return CalculateEstimatedError(1-ConfidenceLevel(), false);
}

SamplingDistribution *  HypoTestInverterResult::GetBackgroundTestStatDist(int index ) const { 
   // get the background test statistic distribution

   HypoTestResult * firstResult = (HypoTestResult*) fYObjects.At(index);
   if (!firstResult) return 0;
   return firstResult->GetBackGroundIsAlt() ? firstResult->GetAltDistribution() : firstResult->GetNullDistribution();
}

SamplingDistribution *  HypoTestInverterResult::GetSignalAndBackgroundTestStatDist(int index) const { 
   // get the signal and background test statistic distribution
   HypoTestResult * result = (HypoTestResult*) fYObjects.At(index);
   if (!result) return 0;
   return !result->GetBackGroundIsAlt() ? result->GetAltDistribution() : result->GetNullDistribution();       
}

SamplingDistribution *  HypoTestInverterResult::GetExpectedPValueDist(int index) const { 
   // get the expected p-value distribution at the scanned point index

   if (index < 0 || index >=  ArraySize() ) return 0; 

   if (fExpPValues.GetSize() == ArraySize()  ) {
      return (SamplingDistribution*)  fExpPValues.At(index)->Clone();
   }
   
   static bool useFirstB = false;
   // get the S+B distribution 
   int bIndex = (useFirstB) ? 0 : index;
 
   SamplingDistribution * bDistribution = GetBackgroundTestStatDist(bIndex);
   SamplingDistribution * sbDistribution = GetSignalAndBackgroundTestStatDist(index);

   HypoTestResult * result = (HypoTestResult*) fYObjects.At(index);

   if (bDistribution && sbDistribution) { 

      // create a new HypoTestResult
      HypoTestResult tempResult; 
      tempResult.SetPValueIsRightTail( result->GetPValueIsRightTail() );
      tempResult.SetBackgroundAsAlt( true);
      // ownership of SamplingDistribution is in HypoTestResult class now
      tempResult.SetNullDistribution( new SamplingDistribution(*sbDistribution) );
      tempResult.SetAltDistribution( new SamplingDistribution(*bDistribution ) );
      
      std::vector<double> values(bDistribution->GetSize()); 
      for (int i = 0; i < bDistribution->GetSize(); ++i) { 
         tempResult.SetTestStatisticData( bDistribution->GetSamplingDistribution()[i] );
         values[i] = (fUseCLs) ? tempResult.CLs() : tempResult.CLsplusb();
      }
      return new SamplingDistribution("expected values","expected values",values);
   }  
   // in case b abs sbDistribution are null assume is coming from the asymptotic calculator 
   // hard -coded this value (no really needed to be used by user)
   fgAsymptoticMaxSigma = 5; 
   const int npoints = 11;
   const double smax = fgAsymptoticMaxSigma;
   const double dsig = 2* smax/ (npoints-1) ;
   std::vector<double> values(npoints);
   for (int i = 0; i < npoints; ++i) { 
      double nsig = -smax + dsig*i; 
      double pval = AsymptoticCalculator::GetExpectedPValues( result->NullPValue(), result->AlternatePValue(), nsig, fUseCLs, !fIsTwoSided);
      if (pval < 0) { return 0;}
      values[i] = pval;
   }
   return new SamplingDistribution("Asymptotic expected values","Asymptotic expected values",values);
   
}



SamplingDistribution *  HypoTestInverterResult::GetLimitDistribution(bool lower ) const { 
   // get the limit distribution (lower/upper depending on the flag)
   // by interpolating  the expected p values for each point

   
   if (ArraySize()<2) {
      oocoutE(this,Eval) << "HypoTestInverterResult::GetLimitDistribution" 
                         << " not  enought points -  return 0 " << std::endl; 
      return 0; 
   }
   
   ooccoutD(this,Eval) << "HypoTestInverterResult - computing  limit distribution...." << std::endl;

      
   // find optimal size by looking at the PValue distribution obtained
   int npoints = ArraySize();
   std::vector<SamplingDistribution*> distVec( npoints );
   double sum = 0;
   for (unsigned int i = 0; i < distVec.size(); ++i) {
      distVec[i] =  GetExpectedPValueDist(i);
      // sort the distributions
      // hack (by calling InverseCDF(0) will sort the sampling distribution
      if (distVec[i] ) { 
         distVec[i]->InverseCDF(0);
         sum += distVec[i]->GetSize();     
      }
   }
   int  size =  int( sum/ npoints);
  
   if (size < 10) { 
      ooccoutW(this,InputArguments) << "HypoTestInverterResult - set a minimum size of 10 for limit distribution" <<   std::endl;
      size = 10;
   }


  double target = 1-fConfidenceLevel;

  // vector with the quantiles of the p-values for each scanned poi point
  std::vector< std::vector<double>  > quantVec(npoints );
  for (int i = 0; i <  npoints; ++i) {

     if (!distVec[i]) continue;

     // make quantiles from the sampling distributions of the expected p values 
     std::vector<double> pvalues = distVec[i]->GetSamplingDistribution();
     delete distVec[i];  distVec[i] = 0;
     std::sort(pvalues.begin(), pvalues.end());
     // find the quantiles of the distribution 
     double p[1] = {0}; 
     double q[1] = {0};

     quantVec[i] = std::vector<double>(size);
     for (int ibin = 0; ibin < size; ++ibin) { 
        // exclude for a bug in TMath::Quantiles last item 
        p[0] = std::min( (ibin+1) * 1./double(size), 1.0);
        // use the type 1 which give the point value
        TMath::Quantiles(pvalues.size(), 1, &pvalues[0], q, p, true, 0, 1 );
        (quantVec[i])[ibin] = q[0]; 
     } 
  
  }

  // sort the values in x 
  std::vector<unsigned int> index( npoints );
  TMath::SortItr(fXValues.begin(), fXValues.end(), index.begin(), false); 

  // SamplingDistribution * dist0 = distVec[index[0]];
  // SamplingDistribution * dist1 = distVec[index[1]];

  
  std::vector<double> limits(size);
  // loop on the p values and find the limit for each expected point in the quantiles vector  
  for (int j = 0; j < size; ++j ) {

     TGraph g;
     for (int k = 0; k < npoints ; ++k) { 
        if (quantVec[index[k]].size()  > 0 )
           g.SetPoint(g.GetN(), GetXValue(index[k]), (quantVec[index[k]])[j] );
     }

     limits[j] = GetGraphX(g, target, lower);

  }


  if (lower) 
     return new SamplingDistribution("Expected lower Limit","Expected lower limits",limits);
  else
     return new SamplingDistribution("Expected upper Limit","Expected upper limits",limits);
  
}


double  HypoTestInverterResult::GetExpectedLowerLimit(double nsig, const char * opt ) const {
   // Get the expected lower limit  
   // nsig is used to specify which expected value of the UpperLimitDistribution
   // For example 
   // nsig = 0 (default value) returns the expected value 
   // nsig = -1 returns the lower band value at -1 sigma 
   // nsig + 1 return the upper value
   // opt = "" (default) : compute limit by interpolating all the p values, find the corresponding limit distribution
   //                         and then find the quantiles in the limit distribution 
   // ioption = "P" is the method used for plotting. One Finds the corresponding nsig quantile in the p values and then
   // interpolates them
   
   return GetExpectedLimit(nsig, true, opt);
} 

double  HypoTestInverterResult::GetExpectedUpperLimit(double nsig, const char * opt ) const {
   // Get the expected upper limit  
   // nsig is used to specify which expected value of the UpperLimitDistribution
   // For example 
   // nsig = 0 (default value) returns the expected value 
   // nsig = -1 returns the lower band value at -1 sigma 
   // nsig + 1 return the upper value
   // opt is an option specifying the type of method used for computing the upper limit
   // opt = "" (default) : compute limit by interpolating all the p values, find the corresponding limit distribution
   //                         and then find the quantiles in the limit distribution 
   // ioption = "P" is the method used for plotting. One Finds the corresponding nsig quantile in the p values and then
   // interpolates them
   //              
   
   return GetExpectedLimit(nsig, false, opt);
} 


   
double  HypoTestInverterResult::GetExpectedLimit(double nsig, bool lower, const char * opt ) const {
   // get expected limit (lower/upper) depending on the flag 
   // for asymptotic is a special case (the distribution is generated an step in sigma values)
   // distringuish asymptotic looking at the hypotest results
   // if option = "P" get expected limit using directly quantiles of p value distribution 
   // else (default) find expected limit by obtaining first a full limit distributions
   // The last one is in general more correct

   const int nEntries = ArraySize();
   if (nEntries <= 0)  return (lower) ? 1 : 0;  // return 1 for lower, 0 for upper
   
   HypoTestResult * r = dynamic_cast<HypoTestResult *> (fYObjects.First() );
   assert(r != 0);
   if (!r->GetNullDistribution() && !r->GetAltDistribution() ) { 
      // we are in the asymptotic case 
      // get the limits obtained at the different sigma values 
      SamplingDistribution * limitDist = GetLimitDistribution(lower); 
      if (!limitDist) return 0;
      const std::vector<double> & values = limitDist->GetSamplingDistribution();
      if (values.size() <= 1) return 0; 
      double dsig = 2* fgAsymptoticMaxSigma/ (values.size() -1) ;
      int  i = (int) TMath::Floor ( (nsig +  fgAsymptoticMaxSigma)/dsig + 0.5);
      return values[i];
   }

   double p[1] = {0};
   double q[1] = {0};
   p[0] = ROOT::Math::normal_cdf(nsig,1);

   // for CLs+b can get the quantiles of p-value distribution and 
   // interpolate them
   // In the case of CLs (since it is not a real p-value anymore but a ratio) 
   // then it is needed to get first all limit distribution values and then the quantiles 

   TString option(opt);
   option.ToUpper();
   if (option.Contains("P")) {

      TGraph  g;   

      // sort the arrays based on the x values
      std::vector<unsigned int> index(nEntries);
      TMath::SortItr(fXValues.begin(), fXValues.end(), index.begin(), false);

      for (int j=0; j<nEntries; ++j) {
         int i = index[j]; // i is the order index 
         SamplingDistribution * s = GetExpectedPValueDist(i);
         if (!s) { 
            ooccoutI(this,Eval) << "HypoTestInverterResult - cannot compute expected p value distribution for point, x = " 
                                << GetXValue(i)  << " skip it " << std::endl;
            continue;
         } 
         const std::vector<double> & values = s->GetSamplingDistribution();
         double * x = const_cast<double *>(&values[0]); // need to change TMath::Quantiles
         TMath::Quantiles(values.size(), 1, x,q,p,false);
         g.SetPoint(g.GetN(), fXValues[i], q[0] );
         delete s;
      }
      if (g.GetN() < 2) { 
         ooccoutE(this,Eval) << "HypoTestInverterResult - cannot compute limits , not enough points, n =  " << g.GetN() << std::endl;
         return 0;
      }

      // interpolate the graph to obtain the limit
      double target = 1-fConfidenceLevel;
      return GetGraphX(g, target, lower);

   }
   // here need to use the limit distribution 
   SamplingDistribution * limitDist = GetLimitDistribution(lower); 
   if (!limitDist) return 0;
   const std::vector<double> & values = limitDist->GetSamplingDistribution();
   double * x = const_cast<double *>(&values[0]); // need to change TMath::Quantiles
   TMath::Quantiles(values.size(), 1, x,q,p,false);
   return q[0];
   
}

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