source: git/modules/VertexFinderDA4D.cc@ f319c1d

/** \class VertexFinderDA4D
 *
 *  Cluster vertices from tracks using deterministic annealing and timing information
 *
 *  \authors M. Selvaggi, L. Gray
 *
 */


#include "modules/VertexFinderDA4D.h"
#include "classes/DelphesClasses.h"
#include "classes/DelphesFactory.h"
#include "classes/DelphesFormula.h"
#include "classes/DelphesPileUpReader.h"

#include "ExRootAnalysis/ExRootResult.h"
#include "ExRootAnalysis/ExRootFilter.h"
#include "ExRootAnalysis/ExRootClassifier.h"

#include "TMath.h"
#include "TString.h"
#include "TFormula.h"
#include "TRandom3.h"
#include "TObjArray.h"
#include "TDatabasePDG.h"
#include "TLorentzVector.h"
#include "TMatrixT.h"
#include "TVector3.h"

#include <utility>
#include <algorithm>
#include <stdexcept>
#include <iostream>
#include <vector>

using namespace std;

static const Double_t mm  = 1.;
static const Double_t m = 1000.*mm;
static const Double_t ns  = 1.;
static const Double_t s = 1.e+9 *ns;
static const Double_t c_light   = 2.99792458e+8 * m/s;

struct track_t
{
  double z;      // z-coordinate at point of closest approach to the beamline
  double t;      // t-coordinate at point of closest approach to the beamline
  double dz2;    // square of the error of z(pca)
  double dtz;    // covariance of z-t
  double dt2;    // square of the error of t(pca)
  Candidate *tt; // a pointer to the Candidate Track
  double Z;      // Z[i]   for DA clustering
  double pi;     // track weight
  double pt;
  double eta;
  double phi;
};

struct vertex_t
{
  double z;
  double t;
  double pk;     // vertex weight for "constrained" clustering
  // --- temporary numbers, used during update
  double ei;
  double sw;
  double swz;
  double swt;
  double se;
  // ---for Tc
  double swE;
  double Tc;
};

static bool split(double beta, std::vector<track_t> &tks, std::vector<vertex_t> &y);
static double update1(double beta, std::vector<track_t> &tks, std::vector<vertex_t> &y);
static double update2(double beta, std::vector<track_t> &tks, std::vector<vertex_t> &y, double &rho0, const double dzCutOff);
static void dump(const double beta, const std::vector<vertex_t> & y, const std::vector<track_t> & tks);
static bool merge(std::vector<vertex_t> &);
static bool merge(std::vector<vertex_t> &, double &);
static bool purge(std::vector<vertex_t> &, std::vector<track_t> & , double &, const double, const double);
static void splitAll(std::vector<vertex_t> &y);
static double beta0(const double betamax, std::vector<track_t> &tks, std::vector<vertex_t> &y, const double coolingFactor);
static double Eik(const track_t &t, const vertex_t &k);

static bool recTrackLessZ1(const track_t & tk1, const track_t & tk2)
{
  return tk1.z < tk2.z;
}

using namespace std;

//------------------------------------------------------------------------------

VertexFinderDA4D::VertexFinderDA4D() :
  fVerbose(0), fMinPT(0), fVertexSpaceSize(0), fVertexTimeSize(0),
  fUseTc(0), fBetaMax(0), fBetaStop(0), fCoolingFactor(0),
  fMaxIterations(0), fDzCutOff(0), fD0CutOff(0), fDtCutOff(0)
{
}

//------------------------------------------------------------------------------

VertexFinderDA4D::~VertexFinderDA4D()
{
}

//------------------------------------------------------------------------------

void VertexFinderDA4D::Init()
{

  fVerbose = GetBool("Verbose", 1);
  fMinPT = GetDouble("MinPT", 0.1);
  fVertexSpaceSize = GetDouble("VertexSpaceSize", 0.5); //in mm
  fVertexTimeSize = GetDouble("VertexTimeSize", 10E-12); //in s
  fUseTc         = GetBool("UseTc", 1);
  fBetaMax       = GetDouble("BetaMax ", 0.1);
  fBetaStop      = GetDouble("BetaStop", 1.0);
  fCoolingFactor = GetDouble("CoolingFactor", 0.8);
  fMaxIterations = GetInt("MaxIterations", 100);
  fDzCutOff      = GetDouble("DzCutOff", 40);  // Adaptive Fitter uses 30 mm but that appears to be a bit tight here sometimes
  fD0CutOff      = GetDouble("D0CutOff", 30);
  fDtCutOff      = GetDouble("DtCutOff", 100E-12);  // dummy

  // convert stuff in cm, ns
  fVertexSpaceSize /= 10.0;
  fVertexTimeSize *= 1E9;
  fDzCutOff       /= 10.0;   // Adaptive Fitter uses 3.0 but that appears to be a bit tight here sometimes
  fD0CutOff       /= 10.0;

  fInputArray = ImportArray(GetString("InputArray", "TrackSmearing/tracks"));
  fItInputArray = fInputArray->MakeIterator();

  fOutputArray = ExportArray(GetString("OutputArray", "tracks"));
  fVertexOutputArray = ExportArray(GetString("VertexOutputArray", "vertices"));
}

//------------------------------------------------------------------------------

void VertexFinderDA4D::Finish()
{
  if(fItInputArray) delete fItInputArray;
}

//------------------------------------------------------------------------------

void VertexFinderDA4D::Process()
{
  Candidate *candidate, *track;
  TObjArray *ClusterArray;
  ClusterArray = new TObjArray;
  TIterator *ItClusterArray;
  Int_t ivtx = 0;

  fInputArray->Sort();

  TLorentzVector pos, mom;
  if (fVerbose)
  {
     cout<<" start processing vertices ..."<<endl;
     cout<<" Found "<<fInputArray->GetEntriesFast()<<" input tracks"<<endl;
     //loop over input tracks
     fItInputArray->Reset();
     while((candidate = static_cast<Candidate*>(fItInputArray->Next())))
     {
        pos = candidate->InitialPosition;
        mom = candidate->Momentum;

        cout<<"pt: "<<mom.Pt()<<", eta: "<<mom.Eta()<<", phi: "<<mom.Phi()<<", z: "<<candidate->DZ/10<<endl;
     }
  }

  // clusterize tracks in Z
  clusterize(*fInputArray, *ClusterArray);

  if (fVerbose){std::cout <<  " clustering returned  "<< ClusterArray->GetEntriesFast() << " clusters  from " << fInputArray->GetEntriesFast() << " selected tracks" <<std::endl;}

  //loop over vertex candidates
  ItClusterArray = ClusterArray->MakeIterator();
  ItClusterArray->Reset();
  while((candidate = static_cast<Candidate*>(ItClusterArray->Next())))
  {

     double meantime = 0.;
     double expv_x2 = 0.;
     double normw = 0.;
     double errtime = 0;

     double meanpos = 0.;
     double meanerr2 = 0.;
     double normpos = 0.;
     double errpos = 0.;

     double sumpt2 = 0.;

     int itr = 0;

     if(fVerbose)cout<<"this vertex has: "<<candidate->GetCandidates()->GetEntriesFast()<<" tracks"<<endl;

     // loop over tracks belonging to this vertex
     TIter it1(candidate->GetCandidates());
     it1.Reset();

     while((track = static_cast<Candidate*>(it1.Next())))
     {

        itr++;
        // TBC: the time is in ns for now TBC
        double t = track->InitialPosition.T()/c_light;
        double dt = track->ErrorT/c_light;
        const double time = t;
        const double inverr = 1.0/dt;
        meantime += time*inverr;
        expv_x2  += time*time*inverr;
        normw    += inverr;

        // compute error position TBC
        const double pt = track->Momentum.Pt();
        const double z = track->DZ/10.0;
        const double err_pt = track->ErrorPT;
        const double err_z = track->ErrorDZ;

        const double wi = (pt/(err_pt*err_z))*(pt/(err_pt*err_z));
        meanpos += z*wi;

        meanerr2 += err_z*err_z*wi;
        normpos += wi;
        sumpt2 += pt*pt;

        // while we are here store cluster index in tracks
        track->ClusterIndex = ivtx;
     }

     meantime = meantime/normw;
     expv_x2 = expv_x2/normw;
     errtime = TMath::Sqrt((expv_x2 - meantime*meantime)/itr);
     meanpos = meanpos/normpos;
     meanerr2 = meanerr2/normpos;
     errpos = TMath::Sqrt(meanerr2/itr);

     candidate->Position.SetXYZT(0.0, 0.0, meanpos*10.0 , meantime*c_light);
     candidate->PositionError.SetXYZT(0.0, 0.0, errpos*10.0 , errtime*c_light);
     candidate->SumPT2 = sumpt2;
     candidate->ClusterNDF = itr;
     candidate->ClusterIndex = ivtx;

     fVertexOutputArray->Add(candidate);

     ivtx++;

     if (fVerbose){
     std::cout << "x,y,z";
       std::cout << ",t";
       std::cout << "=" << candidate->Position.X()/10.0 <<" " << candidate->Position.Y()/10.0 << " " <<  candidate->Position.Z()/10.0;
       std::cout << " " << candidate->Position.T()/c_light;

       std::cout << std::endl;
       std::cout << "sumpt2 " << candidate->SumPT2<<endl;
     
       std::cout << "ex,ey,ez";
       std::cout << ",et";
       std::cout << "=" << candidate->PositionError.X()/10.0 <<" " << candidate->PositionError.Y()/10.0 << " " <<  candidate->PositionError.Z()/10.0;
       std::cout << " " << candidate->PositionError.T()/c_light;
       std::cout << std::endl;

      }
   }// end of cluster loop


    if(fVerbose){
      std::cout << "PrimaryVertexProducerAlgorithm::vertices candidates =" << ClusterArray->GetEntriesFast() << std::endl;
    }

    //TBC maybe this can be done later
    // sort vertices by pt**2  vertex (aka signal vertex tagging)
    /*if(pvs.size()>1){
      sort(pvs.begin(), pvs.end(), VertexHigherPtSquared());
    }
     */

  delete ClusterArray;

}

//------------------------------------------------------------------------------

void VertexFinderDA4D::clusterize(const TObjArray &tracks, TObjArray &clusters)
{
  if(fVerbose) {
    cout << "###################################################" << endl;
    cout << "# VertexFinderDA4D::clusterize   nt="<<tracks.GetEntriesFast() << endl;
    cout << "###################################################" << endl;
  }

  vector< Candidate* > pv = vertices();

  if(fVerbose){ cout << "# VertexFinderDA4D::clusterize   pv.size="<<pv.size() << endl;  }
  if (pv.size()==0){ return;  }

  // convert into vector of candidates
  //TObjArray *ClusterArray = pv.begin()->GetCandidates();
  //Candidate *aCluster = static_cast<Candidate*>(&(pv.at(0)));
   Candidate *aCluster = pv.at(0);

  // fill into clusters and merge


  if( fVerbose ) {
      std::cout << '\t' << 0;
      std::cout << ' ' << (*pv.begin())->Position.Z()/10.0 << ' ' << (*pv.begin())->Position.T()/c_light << std::endl;
    }

  for(vector<Candidate*>::iterator k=pv.begin()+1; k!=pv.end(); k++){
    if( fVerbose ) {
      std::cout << '\t' << std::distance(pv.begin(),k);
      std::cout << ' ' << (*k)->Position.Z() << ' ' << (*k)->Position.T() << std::endl;
    }


    // TBC - check units here
    if ( std::abs((*k)->Position.Z() - (*(k-1))->Position.Z())/10.0 > (2*fVertexSpaceSize) ||
         std::abs((*k)->Position.T() - (*(k-1))->Position.Z())/c_light > 2*0.010 ) {
      // close a cluster
      clusters.Add(aCluster);
      //aCluster.clear();
    }
    //for(unsigned int i=0; i<k->GetCandidates().GetEntriesFast(); i++){
      aCluster = *k;
    //}

  }
  clusters.Add(aCluster);

  if(fVerbose) { std::cout << "# VertexFinderDA4D::clusterize clusters.size="<<clusters.GetEntriesFast() << std::endl; }

}

//------------------------------------------------------------------------------

vector< Candidate* > VertexFinderDA4D::vertices()
{
  Candidate *candidate;
  UInt_t clusterIndex = 0;
  vector< Candidate* > clusters;

  vector<track_t> tks;
  track_t tr;
  Double_t z, dz, t, l, dt, d0, d0error;

  // loop over input tracks
  fItInputArray->Reset();
  while((candidate = static_cast<Candidate*>(fItInputArray->Next())))
  {
    //TBC everything in cm
    z = candidate->DZ/10;
    tr.z = z;
    dz = candidate->ErrorDZ/10;
    tr.dz2 = dz*dz          // track error
    //TBC: beamspot size induced error, take 0 for now.
    // + (std::pow(beamspot.BeamWidthX()*cos(phi),2.)+std::pow(beamspot.BeamWidthY()*sin(phi),2.))/std::pow(tantheta,2.) // beam-width induced
    + fVertexSpaceSize*fVertexSpaceSize; // intrinsic vertex size, safer for outliers and short lived decays

    // TBC: the time is in ns for now TBC
    //t = candidate->Position.T()/c_light;
    t = candidate->InitialPosition.T()/c_light;
    l = candidate->L/c_light;
    double pt = candidate->Momentum.Pt();
    double eta = candidate->Momentum.Eta();
    double phi = candidate->Momentum.Phi();

    tr.pt = pt;
    tr.eta = eta;
    tr.phi = phi;
    tr.t = t; //
    tr.dtz = 0.;
    dt = candidate->ErrorT/c_light;
    tr.dt2 = dt*dt + fVertexTimeSize*fVertexTimeSize;   // the ~injected~ timing error plus a small minimum vertex size in time
    if(fD0CutOff>0)
    {

      d0 = TMath::Abs(candidate->D0)/10.0;
      d0error = candidate->ErrorD0/10.0;

      tr.pi=1./(1.+exp((d0*d0)/(d0error*d0error) - fD0CutOff*fD0CutOff));  // reduce weight for high ip tracks

    }
    else
    {
      tr.pi=1.;
    }
    tr.tt=&(*candidate);
    tr.Z=1.;

    // TBC now putting track selection here (> fPTMin)
    if(tr.pi > 1e-3 && tr.pt > fMinPT)
    {
      tks.push_back(tr);
    }
  }

  //print out input tracks

  if(fVerbose)
  {
    std::cout<<" start processing vertices ..."<<std::endl;
    std::cout<<" Found "<<tks.size()<<" input tracks"<<std::endl;
    //loop over input tracks


   for(std::vector<track_t>::const_iterator it=tks.begin(); it!=tks.end(); it++){
     double z = it->z;
     double pt=it->pt;
     double eta=it->eta;
     double phi=it->phi;
     double t = it->t;

     std::cout<<"pt: "<<pt<<", eta: "<<eta<<", phi: "<<phi<<", z: "<<z<<", t: "<<t<<std::endl;
   }
  }

  unsigned int nt=tks.size();
  double rho0=0.0;  // start with no outlier rejection

  if (tks.empty()) return clusters;

  vector<vertex_t> y; // the vertex prototypes

  // initialize:single vertex at infinite temperature
  vertex_t vstart;
  vstart.z=0.;
  vstart.t=0.;
  vstart.pk=1.;
  y.push_back(vstart);
  int niter=0;      // number of iterations

  // estimate first critical temperature
  double beta=beta0(fBetaMax, tks, y, fCoolingFactor);
  niter=0; while((update1(beta, tks,y)>1.e-6)  && (niter++ < fMaxIterations)){ }

  // annealing loop, stop when T<Tmin  (i.e. beta>1/Tmin)
  while(beta<fBetaMax){

    if(fUseTc){
      update1(beta, tks,y);
      while(merge(y,beta)){update1(beta, tks,y);}
      split(beta, tks,y);
      beta=beta/fCoolingFactor;
    }else{
      beta=beta/fCoolingFactor;
      splitAll(y);
    }

   // make sure we are not too far from equilibrium before cooling further
   niter=0; while((update1(beta, tks,y)>1.e-6)  && (niter++ < fMaxIterations)){ }

  }

  if(fUseTc){
    // last round of splitting, make sure no critical clusters are left
    update1(beta, tks,y);
    while(merge(y,beta)){update1(beta, tks,y);}
    unsigned int ntry=0;
    while( split(beta, tks,y) && (ntry++<10) ){
      niter=0;
      while((update1(beta, tks,y)>1.e-6)  && (niter++ < fMaxIterations)){}
      merge(y,beta);
      update1(beta, tks,y);
    }
  }else{
    // merge collapsed clusters
    while(merge(y,beta)){update1(beta, tks,y);}
    if(fVerbose ){ cout << "dump after 1st merging " << endl;  dump(beta,y,tks);}
  }

  // switch on outlier rejection
  rho0=1./nt; for(vector<vertex_t>::iterator k=y.begin(); k!=y.end(); k++){ k->pk =1.; }  // democratic
  niter=0; while((update2(beta, tks,y,rho0, fDzCutOff) > 1.e-8)  && (niter++ < fMaxIterations)){  }
  if(fVerbose  ){ cout << "rho0=" << rho0 <<   " niter=" << niter <<  endl; dump(beta,y,tks);}


  // merge again  (some cluster split by outliers collapse here)
  while(merge(y)){}
  if(fVerbose  ){ cout << "dump after 2nd merging " << endl;  dump(beta,y,tks);}


  // continue from freeze-out to Tstop (=1) without splitting, eliminate insignificant vertices
  while(beta<=fBetaStop){
    while(purge(y,tks,rho0, beta, fDzCutOff)){
      niter=0; while((update2(beta, tks, y, rho0, fDzCutOff) > 1.e-6)  && (niter++ < fMaxIterations)){  }
    }
    beta/=fCoolingFactor;
    niter=0; while((update2(beta, tks, y, rho0, fDzCutOff) > 1.e-6)  && (niter++ < fMaxIterations)){  }
  }


//   // new, one last round of cleaning at T=Tstop
//   while(purge(y,tks,rho0, beta)){
//     niter=0; while((update2(beta, tks,y,rho0, fDzCutOff) > 1.e-6)  && (niter++ < fMaxIterations)){  }
//   }


  if(fVerbose){
   cout << "Final result, rho0=" << rho0 << endl;
   dump(beta,y,tks);
  }


  // select significant tracks and use a TransientVertex as a container
  //GlobalError dummyError;

  // ensure correct normalization of probabilities, should make double assginment reasonably impossible
  for(unsigned int i=0; i<nt; i++){
    tks[i].Z=rho0*exp(-beta*( fDzCutOff*fDzCutOff));
    for(vector<vertex_t>::iterator k=y.begin(); k!=y.end(); k++){
      tks[i].Z += k->pk * exp(-beta*Eik(tks[i],*k));
    }
  }

  for(vector<vertex_t>::iterator k=y.begin(); k!=y.end(); k++){

    DelphesFactory *factory = GetFactory();
    candidate = factory->NewCandidate();

    //cout<<"new vertex"<<endl;
    //GlobalPoint pos(0, 0, k->z);
    double time = k->t;
    double z = k->z;
    //vector< reco::TransientTrack > vertexTracks;
    //double max_track_time_err2 = 0;
    double mean = 0.;
    double expv_x2 = 0.;
    double normw = 0.;
    for(unsigned int i=0; i<nt; i++){
      const double invdt = 1.0/std::sqrt(tks[i].dt2);
      if(tks[i].Z>0){
  double p = k->pk * exp(-beta*Eik(tks[i],*k)) / tks[i].Z;
  if( (tks[i].pi>0) && ( p > 0.5 ) ){
          //std::cout << "pushing back " << i << ' ' << tks[i].tt << std::endl;
          //vertexTracks.push_back(*(tks[i].tt)); tks[i].Z=0;

          candidate->AddCandidate(tks[i].tt); tks[i].Z=0;

          mean     += tks[i].t*invdt*p;
          expv_x2  += tks[i].t*tks[i].t*invdt*p;
          normw    += invdt*p;
        } // setting Z=0 excludes double assignment
      }
    }

    mean = mean/normw;
    expv_x2 = expv_x2/normw;
    const double time_var = expv_x2 - mean*mean;
    const double crappy_error_guess = std::sqrt(time_var);
    /*GlobalError dummyErrorWithTime(0,
                                   0,0,
                                   0,0,0,
                                   0,0,0,crappy_error_guess);*/
    //TransientVertex v(pos, time, dummyErrorWithTime, vertexTracks, 5);


    candidate->ClusterIndex = clusterIndex++;;
    candidate->Position.SetXYZT(0.0, 0.0, z*10.0 , time*c_light);

    // TBC - fill error later ...
    candidate->PositionError.SetXYZT(0.0, 0.0, 0.0 , crappy_error_guess*c_light);

    clusterIndex++;
    clusters.push_back(candidate);
  }


  return clusters;

}

//------------------------------------------------------------------------------

static double Eik(const track_t & t, const vertex_t &k)
{
  return std::pow(t.z-k.z,2.)/t.dz2 + std::pow(t.t - k.t,2.)/t.dt2;
}

//------------------------------------------------------------------------------

static void dump(const double beta, const vector<vertex_t> &y, const vector<track_t> &tks0)
{
  // copy and sort for nicer printout
  vector<track_t> tks;
  for(vector<track_t>::const_iterator t=tks0.begin(); t!=tks0.end(); t++){tks.push_back(*t); }
  std::stable_sort(tks.begin(), tks.end(), recTrackLessZ1);

  cout << "-----DAClusterizerInZT::dump ----" << endl;
  cout << " beta=" << beta << endl;
  cout << "                                                               z= ";
  cout.precision(4);
  for(vector<vertex_t>::const_iterator k=y.begin(); k!=y.end(); k++){
    //cout  <<  setw(8) << fixed << k->z;
  }
  cout << endl << "                                                               t= ";
  for(vector<vertex_t>::const_iterator k=y.begin(); k!=y.end(); k++){
    //cout  <<  setw(8) << fixed << k->t;
  }
  //cout << endl << "T=" << setw(15) << 1./beta <<"                                             Tc= ";
  for(vector<vertex_t>::const_iterator k=y.begin(); k!=y.end(); k++){
    //cout  <<  setw(8) << fixed << k->Tc ;
  }

  cout << endl << "                                                              pk=";
  double sumpk=0;
  for(vector<vertex_t>::const_iterator k=y.begin(); k!=y.end(); k++){
    //cout <<  setw(8) <<  setprecision(3) <<  fixed << k->pk;
    sumpk+=k->pk;
  }
  cout  << endl;

  double E=0, F=0;
  cout << endl;
  cout << "----       z +/- dz        t +/- dt        ip +/-dip       pt    phi  eta    weights  ----" << endl;
  cout.precision(4);
  for(unsigned int i=0; i<tks.size(); i++){
    if (tks[i].Z>0){  F-=log(tks[i].Z)/beta;}
    double tz= tks[i].z;
    double tt= tks[i].t;
    //cout <<  setw (3)<< i << ")" <<  setw (8) << fixed << setprecision(4)<<  tz << " +/-" <<  setw (6)<< sqrt(tks[i].dz2)
    //     << setw(8) << fixed << setprecision(4) << tt << " +/-" << setw(6) << std::sqrt(tks[i].dt2)  ;

    double sump=0.;
    for(vector<vertex_t>::const_iterator k=y.begin(); k!=y.end(); k++){
    if((tks[i].pi>0)&&(tks[i].Z>0)){
    //double p=pik(beta,tks[i],*k);
    double p=k->pk * std::exp(-beta*Eik(tks[i],*k)) / tks[i].Z;
    if( p > 0.0001){
      //cout <<  setw (8) <<  setprecision(3) << p;
    }else{
      cout << "    .   ";
    }
    E+=p*Eik(tks[i],*k);
    sump+=p;
  }else{
      cout << "        ";
  }
      }
      cout << endl;
    }
    cout << endl << "T=" << 1/beta  << " E=" << E << " n="<< y.size() << "  F= " << F <<  endl <<  "----------" << endl;
}

//------------------------------------------------------------------------------

static double update1(double beta, vector<track_t> &tks, vector<vertex_t> &y)
{
  //update weights and vertex positions
  // mass constrained annealing without noise
  // returns the squared sum of changes of vertex positions

  unsigned int nt=tks.size();

  //initialize sums
  double sumpi=0;
  for(vector<vertex_t>::iterator k=y.begin(); k!=y.end(); ++k){
    k->sw=0.; k->swz=0.; k->swt = 0.; k->se=0.;
    k->swE=0.;  k->Tc=0.;
  }


  // loop over tracks
  for(unsigned int i=0; i<nt; i++){

    // update pik and Zi
    double Zi = 0.;
    for(vector<vertex_t>::iterator k=y.begin(); k!=y.end(); ++k){
      k->ei = std::exp(-beta*Eik(tks[i],*k));// cache exponential for one track at a time
      Zi   += k->pk * k->ei;
    }
    tks[i].Z=Zi;

    // normalization for pk
    if (tks[i].Z>0){
      sumpi += tks[i].pi;
      // accumulate weighted z and weights for vertex update
      for(vector<vertex_t>::iterator k=y.begin(); k!=y.end(); ++k){
  k->se  += tks[i].pi* k->ei / Zi;
  const double w = k->pk * tks[i].pi* k->ei / ( Zi * ( tks[i].dz2 * tks[i].dt2 ) );
  k->sw  += w;
  k->swz += w * tks[i].z;
        k->swt += w * tks[i].t;
  k->swE += w * Eik(tks[i],*k);
      }
    }else{
      sumpi += tks[i].pi;
    }


  } // end of track loop


  // now update z and pk
  double delta=0;
  for(vector<vertex_t>::iterator k=y.begin(); k!=y.end(); k++){
    if ( k->sw > 0){
      const double znew = k->swz/k->sw;
      const double tnew = k->swt/k->sw;
      delta += std::pow(k->z-znew,2.) + std::pow(k->t-tnew,2.);
      k->z  = znew;
      k->t  = tnew;
      k->Tc = 2.*k->swE/k->sw;
    }else{
      // cout << " a cluster melted away ?  pk=" << k->pk <<  " sumw=" << k->sw <<  endl
      k->Tc=-1;
    }

    k->pk = k->pk * k->se / sumpi;
  }

  // return how much the prototypes moved
  return delta;
}

//------------------------------------------------------------------------------

static double update2(double beta, vector<track_t> &tks, vector<vertex_t> &y, double &rho0, double dzCutOff)
{
  // MVF style, no more vertex weights, update tracks weights and vertex positions, with noise
  // returns the squared sum of changes of vertex positions

  unsigned int nt=tks.size();

  //initialize sums
  for(vector<vertex_t>::iterator k=y.begin(); k!=y.end(); k++){
    k->sw = 0.;   k->swz = 0.; k->swt = 0.; k->se = 0.;
    k->swE = 0.;  k->Tc=0.;
  }


  // loop over tracks
  for(unsigned int i=0; i<nt; i++){

    // update pik and Zi and Ti
    double Zi = rho0*std::exp(-beta*(dzCutOff*dzCutOff));// cut-off (eventually add finite size in time)
    //double Ti = 0.; // dt0*std::exp(-beta*fDtCutOff);
    for(vector<vertex_t>::iterator k=y.begin(); k!=y.end(); k++){
      k->ei = std::exp(-beta*Eik(tks[i],*k));// cache exponential for one track at a time
      Zi   += k->pk * k->ei;
    }
    tks[i].Z=Zi;

    // normalization
    if (tks[i].Z>0){
       // accumulate weighted z and weights for vertex update
      for(vector<vertex_t>::iterator k=y.begin(); k!=y.end(); k++){
  k->se += tks[i].pi* k->ei / Zi;
  double w = k->pk * tks[i].pi * k->ei /( Zi * ( tks[i].dz2 * tks[i].dt2 ) );
  k->sw  += w;
  k->swz += w * tks[i].z;
        k->swt += w * tks[i].t;
  k->swE += w * Eik(tks[i],*k);
      }
    }

  } // end of track loop

  // now update z
  double delta=0;
  for(vector<vertex_t>::iterator k=y.begin(); k!=y.end(); k++){
    if ( k->sw > 0){
      const double znew=k->swz/k->sw;
      const double tnew=k->swt/k->sw;
      delta += std::pow(k->z-znew,2.) + std::pow(k->t-tnew,2.);
      k->z   = znew;
      k->t   = tnew;
      k->Tc  = 2*k->swE/k->sw;
    }else{
      // cout << " a cluster melted away ?  pk=" << k->pk <<  " sumw=" << k->sw <<  endl;
      k->Tc = 0;
    }

  }

  // return how much the prototypes moved
  return delta;
}

//------------------------------------------------------------------------------

static bool merge(vector<vertex_t> &y)
{
  // merge clusters that collapsed or never separated, return true if vertices were merged, false otherwise

  if(y.size()<2)  return false;

  for(vector<vertex_t>::iterator k=y.begin(); (k+1)!=y.end(); k++){
    if( std::abs( (k+1)->z - k->z ) < 1.e-3 &&
        std::abs( (k+1)->t - k->t ) < 1.e-3    ){  // with fabs if only called after freeze-out (splitAll() at highter T)
      double rho = k->pk + (k+1)->pk;
      if(rho>0){
        k->z = ( k->pk * k->z + (k+1)->z * (k+1)->pk)/rho;
        k->t = ( k->pk * k->t + (k+1)->t * (k+1)->pk)/rho;
      }else{
        k->z = 0.5*(k->z + (k+1)->z);
        k->t = 0.5*(k->t + (k+1)->t);
      }
      k->pk = rho;

      y.erase(k+1);
      return true;
    }
  }

  return false;
}

//------------------------------------------------------------------------------

static bool merge(vector<vertex_t> &y, double &beta)
{
  // merge clusters that collapsed or never separated,
  // only merge if the estimated critical temperature of the merged vertex is below the current temperature
  // return true if vertices were merged, false otherwise
  if(y.size()<2)  return false;

  for(vector<vertex_t>::iterator k=y.begin(); (k+1)!=y.end(); k++){
    if ( std::abs((k+1)->z - k->z) < 2.e-3 &&
         std::abs((k+1)->t - k->t) < 2.e-3    ) {
      double rho=k->pk + (k+1)->pk;
      double swE=k->swE+(k+1)->swE - k->pk * (k+1)->pk / rho * ( std::pow((k+1)->z - k->z,2.) +
                                                                 std::pow((k+1)->t - k->t,2.)   );
      double Tc=2*swE/(k->sw+(k+1)->sw);

      if(Tc*beta<1){
  if(rho>0){
    k->z = ( k->pk * k->z + (k+1)->z * (k+1)->pk)/rho;
          k->t = ( k->pk * k->t + (k+1)->t * (k+1)->pk)/rho;
  }else{
    k->z = 0.5*(k->z + (k+1)->z);
          k->t = 0.5*(k->t + (k+1)->t);
  }
  k->pk  = rho;
  k->sw += (k+1)->sw;
  k->swE = swE;
  k->Tc  = Tc;
  y.erase(k+1);
  return true;
      }
    }
  }

  return false;
}

//------------------------------------------------------------------------------

static bool purge(vector<vertex_t> &y, vector<track_t> &tks, double & rho0, const double beta, const double dzCutOff)
{
  // eliminate clusters with only one significant/unique track
  if(y.size()<2)  return false;

  unsigned int nt=tks.size();
  double sumpmin=nt;
  vector<vertex_t>::iterator k0=y.end();
  for(vector<vertex_t>::iterator k=y.begin(); k!=y.end(); k++){
    int nUnique=0;
    double sump=0;
    double pmax=k->pk/(k->pk+rho0*exp(-beta*dzCutOff*dzCutOff));
    for(unsigned int i=0; i<nt; i++){
      if(tks[i].Z > 0){
  double p = k->pk * std::exp(-beta*Eik(tks[i],*k)) / tks[i].Z ;
  sump+=p;
  if( (p > 0.9*pmax) && (tks[i].pi>0) ){ nUnique++; }
      }
    }

    if((nUnique<2)&&(sump<sumpmin)){
      sumpmin=sump;
      k0=k;
    }
  }

  if(k0!=y.end()){
    //cout << "eliminating prototype at " << k0->z << "," << k0->t << " with sump=" << sumpmin << endl;
    //rho0+=k0->pk;
    y.erase(k0);
    return true;
  }else{
    return false;
  }
}

//------------------------------------------------------------------------------

static double beta0(double betamax, vector<track_t> &tks, vector<vertex_t> &y, const double coolingFactor)
{

  double T0=0;  // max Tc for beta=0
  // estimate critical temperature from beta=0 (T=inf)
  unsigned int nt=tks.size();

  for(vector<vertex_t>::iterator k=y.begin(); k!=y.end(); k++){

    // vertex fit at T=inf
    double sumwz=0.;
    double sumwt=0.;
    double sumw=0.;
    for(unsigned int i=0; i<nt; i++){
      double w = tks[i].pi/(tks[i].dz2 * tks[i].dt2);
      sumwz += w*tks[i].z;
      sumwt += w*tks[i].t;
      sumw  += w;
    }
    k->z = sumwz/sumw;
    k->t = sumwt/sumw;

    // estimate Tcrit, eventually do this in the same loop
    double a=0, b=0;
    for(unsigned int i=0; i<nt; i++){
      double dx = tks[i].z-(k->z);
      double dt = tks[i].t-(k->t);
      double w  = tks[i].pi/(tks[i].dz2 * tks[i].dt2);
      a += w*(std::pow(dx,2.)/tks[i].dz2 + std::pow(dt,2.)/tks[i].dt2);
      b += w;
    }
    double Tc= 2.*a/b;  // the critical temperature of this vertex
    if(Tc>T0) T0=Tc;
  }// vertex loop (normally there should be only one vertex at beta=0)

  if (T0>1./betamax){
    return betamax/pow(coolingFactor, int(std::log(T0*betamax)/std::log(coolingFactor))-1 );
  }else{
    // ensure at least one annealing step
    return betamax/coolingFactor;
  }
}

//------------------------------------------------------------------------------

static bool split(double beta, vector<track_t> &tks, vector<vertex_t> &y)
{
  // split only critical vertices (Tc >~ T=1/beta   <==>   beta*Tc>~1)
  // an update must have been made just before doing this (same beta, no merging)
  // returns true if at least one cluster was split

  const double epsilon = 1e-3; // split all single vertices by 10 um
  bool split = false;

  // avoid left-right biases by splitting highest Tc first

  std::vector<std::pair<double, unsigned int> > critical;
  for(unsigned int ik=0; ik<y.size(); ik++){
    if (beta*y[ik].Tc > 1.){
      critical.push_back( make_pair(y[ik].Tc, ik));
    }
  }
  std::stable_sort(critical.begin(), critical.end(), std::greater<std::pair<double, unsigned int> >() );

  for(unsigned int ic=0; ic<critical.size(); ic++){
    unsigned int ik=critical[ic].second;
    // estimate subcluster positions and weight
    double p1=0, z1=0, t1=0, w1=0;
    double p2=0, z2=0, t2=0, w2=0;
    //double sumpi=0;
    for(unsigned int i=0; i<tks.size(); i++){
      if(tks[i].Z>0){
  //sumpi+=tks[i].pi;
  double p=y[ik].pk * exp(-beta*Eik(tks[i],y[ik])) / tks[i].Z*tks[i].pi;
  double w=p/(tks[i].dz2 * tks[i].dt2);
  if(tks[i].z < y[ik].z){
    p1+=p; z1+=w*tks[i].z; t1+=w*tks[i].t; w1+=w;
  }else{
    p2+=p; z2+=w*tks[i].z; t2+=w*tks[i].t; w2+=w;
  }
      }
    }
    if(w1>0){  z1=z1/w1; t1=t1/w1;} else{ z1=y[ik].z-epsilon; t1=y[ik].t-epsilon; }
    if(w2>0){  z2=z2/w2; t2=t2/w2;} else{ z2=y[ik].z+epsilon; t2=y[ik].t+epsilon;}

    // reduce split size if there is not enough room
    if( ( ik   > 0       ) && ( y[ik-1].z>=z1 ) ){ z1=0.5*(y[ik].z+y[ik-1].z); t1=0.5*(y[ik].t+y[ik-1].t); }
    if( ( ik+1 < y.size()) && ( y[ik+1].z<=z2 ) ){ z2=0.5*(y[ik].z+y[ik+1].z); t2=0.5*(y[ik].t+y[ik+1].t); }

    // split if the new subclusters are significantly separated
    if( (z2-z1)>epsilon || std::abs(t2-t1) > epsilon){
      split=true;
      vertex_t vnew;
      vnew.pk = p1*y[ik].pk/(p1+p2);
      y[ik].pk= p2*y[ik].pk/(p1+p2);
      vnew.z  = z1;
      vnew.t  = t1;
      y[ik].z = z2;
      y[ik].t = t2;
      y.insert(y.begin()+ik, vnew);

     // adjust remaining pointers
      for(unsigned int jc=ic; jc<critical.size(); jc++){
        if (critical[jc].second>ik) {critical[jc].second++;}
      }
    }
  }

  //  stable_sort(y.begin(), y.end(), clusterLessZ);
  return split;
}

//------------------------------------------------------------------------------

void splitAll(vector<vertex_t> &y)
{


  const double epsilon=1e-3;      // split all single vertices by 10 um
  const double zsep=2*epsilon;    // split vertices that are isolated by at least zsep (vertices that haven't collapsed)
  const double tsep=2*epsilon;    // check t as well

  vector<vertex_t> y1;

  for(vector<vertex_t>::iterator k=y.begin(); k!=y.end(); k++){
    if ( ( (k==y.begin())|| (k-1)->z < k->z - zsep) && (((k+1)==y.end()  )|| (k+1)->z > k->z + zsep)) {
      // isolated prototype, split
      vertex_t vnew;
      vnew.z  = k->z - epsilon;
      vnew.t  = k->t - epsilon;
      (*k).z  = k->z + epsilon;
      (*k).t  = k->t + epsilon;
      vnew.pk= 0.5* (*k).pk;
      (*k).pk= 0.5* (*k).pk;
      y1.push_back(vnew);
      y1.push_back(*k);

    }else if( y1.empty() || (y1.back().z < k->z -zsep) || (y1.back().t < k->t - tsep) ){
      y1.push_back(*k);
    }else{
      y1.back().z -= epsilon;
      y1.back().t -= epsilon;
      k->z += epsilon;
      k->t += epsilon;
      y1.push_back(*k);
    }
  }// vertex loop

  y=y1;
}