DAClusterizerInZ.cc

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#include "RecoVertex/PrimaryVertexProducer/interface/DAClusterizerInZ.h"
#include "FWCore/MessageLogger/interface/MessageLogger.h"
#include "DataFormats/GeometryCommonDetAlgo/interface/Measurement1D.h"
#include "RecoVertex/VertexPrimitives/interface/VertexException.h"

using namespace std;

namespace {

  bool recTrackLessZ1(const DAClusterizerInZ::track_t& tk1, const DAClusterizerInZ::track_t& tk2) {
    return tk1.z < tk2.z;
  }
}  // namespace

vector<DAClusterizerInZ::track_t> DAClusterizerInZ::fill(const vector<reco::TransientTrack>& tracks) const {
  // prepare track data for clustering
  vector<track_t> tks;
  for (vector<reco::TransientTrack>::const_iterator it = tracks.begin(); it != tracks.end(); it++) {
    track_t t;
    t.z = ((*it).stateAtBeamLine().trackStateAtPCA()).position().z();
    double tantheta = tan(((*it).stateAtBeamLine().trackStateAtPCA()).momentum().theta());
    double phi = ((*it).stateAtBeamLine().trackStateAtPCA()).momentum().phi();
    //  get the beam-spot
    reco::BeamSpot beamspot = (it->stateAtBeamLine()).beamSpot();
    t.dz2 = pow((*it).track().dzError(), 2)  // track errror
            + (pow(beamspot.BeamWidthX() * cos(phi), 2) + pow(beamspot.BeamWidthY() * sin(phi), 2)) /
                  pow(tantheta, 2)  // beam-width induced
            + pow(vertexSize_, 2);  // intrinsic vertex size, safer for outliers and short lived decays
    if (d0CutOff_ > 0) {
      Measurement1D IP = (*it).stateAtBeamLine().transverseImpactParameter();       // error constains beamspot
      t.pi = 1. / (1. + exp(pow(IP.value() / IP.error(), 2) - pow(d0CutOff_, 2)));  // reduce weight for high ip tracks
    } else {
      t.pi = 1.;
    }
    t.tt = &(*it);
    t.Z = 1.;
    tks.push_back(t);
  }
  return tks;
}

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

double DAClusterizerInZ::update(double beta, vector<track_t>& tks, vector<vertex_t>& y) const {
  //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->se = 0;
    k->sw = 0;
    k->swz = 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 = 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;
        double w = k->pk * tks[i].pi * k->ei / Zi / tks[i].dz2;
        k->sw += w;
        k->swz += w * tks[i].z;
        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) {
      double znew = k->swz / k->sw;
      delta += pow(k->z - znew, 2);
      k->z = znew;
      k->Tc = 2 * k->swE / k->sw;
    } else {
      edm::LogInfo("sumw") << "invalid sum of weights in fit: " << k->sw << endl;
      if (verbose_) {
        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;
}

double DAClusterizerInZ::update(double beta, vector<track_t>& tks, vector<vertex_t>& y, double& rho0) const {
  // 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->se = 0;
    k->sw = 0;
    k->swz = 0;
    k->swE = 0;
    k->Tc = 0;
  }

  // loop over tracks
  for (unsigned int i = 0; i < nt; i++) {
    // update pik and Zi
    double Zi = rho0 * exp(-beta * dzCutOff_ * dzCutOff_);  // cut-off
    for (vector<vertex_t>::iterator k = y.begin(); k != y.end(); k++) {
      k->ei = 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;
        k->sw += w;
        k->swz += w * tks[i].z;
        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) {
      double znew = k->swz / k->sw;
      delta += pow(k->z - znew, 2);
      k->z = znew;
      k->Tc = 2 * k->swE / k->sw;
    } else {
      edm::LogInfo("sumw") << "invalid sum of weights in fit: " << k->sw << endl;
      if (verbose_) {
        cout << " a cluster melted away ?  pk=" << k->pk << " sumw=" << k->sw << endl;
      }
      k->Tc = 0;
    }
  }

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

bool DAClusterizerInZ::merge(vector<vertex_t>& y, int nt) const {
  // 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 (fabs((k + 1)->z - k->z) < 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;
      } else {
        k->z = 0.5 * (k->z + (k + 1)->z);
      }
      k->pk = rho;

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

  return false;
}

bool DAClusterizerInZ::merge(vector<vertex_t>& y, double& beta) const {
  // 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 (fabs((k + 1)->z - k->z) < 2.e-3) {
      double rho = k->pk + (k + 1)->pk;
      double swE = k->swE + (k + 1)->swE - k->pk * (k + 1)->pk / rho * pow((k + 1)->z - k->z, 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;
        } else {
          k->z = 0.5 * (k->z + (k + 1)->z);
        }
        k->pk = rho;
        k->sw += (k + 1)->sw;
        k->swE = swE;
        k->Tc = Tc;
        y.erase(k + 1);
        return true;
      }
    }
  }

  return false;
}

bool DAClusterizerInZ::purge(vector<vertex_t>& y, vector<track_t>& tks, double& rho0, const double beta) const {
  // 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 * 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()) {
    if (verbose_) {
      cout << "eliminating prototype at " << k0->z << " with sump=" << sumpmin << endl;
    }
    //rho0+=k0->pk;
    y.erase(k0);
    return true;
  } else {
    return false;
  }
}

double DAClusterizerInZ::beta0(double betamax, vector<track_t>& tks, vector<vertex_t>& y) const {
  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 sumw = 0;
    for (unsigned int i = 0; i < nt; i++) {
      double w = tks[i].pi / tks[i].dz2;
      sumwz += w * tks[i].z;
      sumw += w;
    }
    k->z = sumwz / 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 w = tks[i].pi / tks[i].dz2;
      a += w * pow(dx, 2) / tks[i].dz2;
      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(log(T0 * betamax) / log(coolingFactor_)) - 1);
  } else {
    // ensure at least one annealing step
    return betamax / coolingFactor_;
  }
}

bool DAClusterizerInZ::split(double beta, vector<track_t>& tks, vector<vertex_t>& y, double threshold) const {
  // 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

  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));
    }
  }
  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, w1 = 0;
    double p2 = 0, z2 = 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;
        if (tks[i].z < y[ik].z) {
          p1 += p;
          z1 += w * tks[i].z;
          w1 += w;
        } else {
          p2 += p;
          z2 += w * tks[i].z;
          w2 += w;
        }
      }
    }
    if (w1 > 0) {
      z1 = z1 / w1;
    } else {
      z1 = y[ik].z - epsilon;
    }
    if (w2 > 0) {
      z2 = z2 / w2;
    } else {
      z2 = y[ik].z + 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);
    }
    if ((ik + 1 < y.size()) && (y[ik + 1].z <= z2)) {
      z2 = 0.5 * (y[ik].z + y[ik + 1].z);
    }

    // split if the new subclusters are significantly separated
    if ((z2 - z1) > 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;
      y[ik].z = z2;
      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 DAClusterizerInZ::splitAll(vector<vertex_t>& y) const {
  double epsilon = 1e-3;      // split all single vertices by 10 um
  double zsep = 2 * epsilon;  // split vertices that are isolated by at least zsep (vertices that haven't collapsed)
  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;
      (*k).z = k->z + 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.push_back(*k);
    } else {
      y1.back().z -= epsilon;
      k->z += epsilon;
      y1.push_back(*k);
    }
  }  // vertex loop

  y = y1;
}

DAClusterizerInZ::DAClusterizerInZ(const edm::ParameterSet& conf) {
  // some defaults to avoid uninitialized variables
  verbose_ = conf.getUntrackedParameter<bool>("verbose", false);
  useTc_ = true;
  betamax_ = 0.1;
  betastop_ = 1.0;
  coolingFactor_ = 0.8;
  maxIterations_ = 100;
  vertexSize_ = 0.05;  // 0.5 mm
  dzCutOff_ = 4.0;     // Adaptive Fitter uses 3.0 but that appears to be a bit tight here sometimes

  // configure

  double Tmin = conf.getParameter<double>("Tmin");
  vertexSize_ = conf.getParameter<double>("vertexSize");
  coolingFactor_ = conf.getParameter<double>("coolingFactor");
  d0CutOff_ = conf.getParameter<double>("d0CutOff");
  dzCutOff_ = conf.getParameter<double>("dzCutOff");
  maxIterations_ = 100;
  if (Tmin == 0) {
    cout << "DAClusterizerInZ: invalid Tmin" << Tmin << "  reset do default " << 1. / betamax_ << endl;
  } else {
    betamax_ = 1. / Tmin;
  }

  // for testing, negative cooling factor: revert to old splitting scheme
  if (coolingFactor_ < 0) {
    coolingFactor_ = -coolingFactor_;
    useTc_ = false;
  }
}

void DAClusterizerInZ::dump(const double beta,
                            const vector<vertex_t>& y,
                            const vector<track_t>& tks0,
                            int verbosity) const {
  // 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);
  }
  stable_sort(tks.begin(), tks.end(), recTrackLessZ1);

  cout << "-----DAClusterizerInZ::dump ----" << endl;
  cout << "beta=" << beta << "   betamax= " << betamax_ << 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=" << 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;

  if (verbosity > 0) {
    double E = 0, F = 0;
    cout << endl;
    cout << "----       z +/- dz                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;
      cout << setw(3) << i << ")" << setw(8) << fixed << setprecision(4) << tz << " +/-" << setw(6) << sqrt(tks[i].dz2);

      if (tks[i].tt->track().quality(reco::TrackBase::highPurity)) {
        cout << " *";
      } else {
        cout << "  ";
      }
      if (tks[i].tt->track().hitPattern().hasValidHitInPixelLayer(PixelSubdetector::SubDetector::PixelBarrel, 1)) {
        cout << "+";
      } else {
        cout << "-";
      }
      cout << setw(1)
           << tks[i]
                  .tt->track()
                  .hitPattern()
                  .pixelBarrelLayersWithMeasurement();  // see DataFormats/TrackReco/interface/HitPattern.h
      cout << setw(1) << tks[i].tt->track().hitPattern().pixelEndcapLayersWithMeasurement();
      cout << setw(1) << hex
           << tks[i].tt->track().hitPattern().trackerLayersWithMeasurement() -
                  tks[i].tt->track().hitPattern().pixelLayersWithMeasurement()
           << dec;
      cout << "=" << setw(1) << hex
           << tks[i].tt->track().hitPattern().numberOfLostHits(reco::HitPattern::MISSING_OUTER_HITS) << dec;

      Measurement1D IP = tks[i].tt->stateAtBeamLine().transverseImpactParameter();
      cout << setw(8) << IP.value() << "+/-" << setw(6) << IP.error();
      cout << " " << setw(6) << setprecision(2) << tks[i].tt->track().pt() * tks[i].tt->track().charge();
      cout << " " << setw(5) << setprecision(2) << tks[i].tt->track().phi() << " " << setw(5) << setprecision(2)
           << tks[i].tt->track().eta();

      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 * 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;
  }
}

vector<TransientVertex> DAClusterizerInZ::vertices(const vector<reco::TransientTrack>& tracks,
                                                   const int verbosity) const {
  vector<track_t> tks = fill(tracks);
  unsigned int nt = tracks.size();
  double rho0 = 0.0;  // start with no outlier rejection

  vector<TransientVertex> clusters;
  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.pk = 1.;
  y.push_back(vstart);
  int niter = 0;  // number of iterations

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

  // annealing loop, stop when T<Tmin  (i.e. beta>1/Tmin)
  while (beta < betamax_) {
    if (useTc_) {
      update(beta, tks, y);
      while (merge(y, beta)) {
        update(beta, tks, y);
      }
      split(beta, tks, y, 1.);
      beta = beta / coolingFactor_;
    } else {
      beta = beta / coolingFactor_;
      splitAll(y);
    }

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

  if (useTc_) {
    // last round of splitting, make sure no critical clusters are left
    update(beta, tks, y);
    while (merge(y, beta)) {
      update(beta, tks, y);
    }
    unsigned int ntry = 0;
    while (split(beta, tks, y, 1.) && (ntry++ < 10)) {
      niter = 0;
      while ((update(beta, tks, y) > 1.e-6) && (niter++ < maxIterations_)) {
      }
      merge(y, beta);
      update(beta, tks, y);
    }
  } else {
    // merge collapsed clusters
    while (merge(y, beta)) {
      update(beta, tks, y);
    }
    if (verbose_) {
      cout << "dump after 1st merging " << endl;
      dump(beta, y, tks, 2);
    }
  }

  // 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 ((update(beta, tks, y, rho0) > 1.e-8) && (niter++ < maxIterations_)) {
  }
  if (verbose_) {
    cout << "rho0=" << rho0 << " niter=" << niter << endl;
    dump(beta, y, tks, 2);
  }

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

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

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

  if (verbose_) {
    cout << "Final result, rho0=" << rho0 << endl;
    dump(beta, y, tks, 2);
  }

  // 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 * dzCutOff_ * dzCutOff_);
    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++) {
    GlobalPoint pos(0, 0, k->z);
    vector<reco::TransientTrack> vertexTracks;
    for (unsigned int i = 0; i < nt; i++) {
      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)) {
          vertexTracks.push_back(*(tks[i].tt));
          tks[i].Z = 0;
        }  // setting Z=0 excludes double assignment
      }
    }
    TransientVertex v(pos, dummyError, vertexTracks, 5);
    clusters.push_back(v);
  }

  return clusters;
}

vector<vector<reco::TransientTrack> > DAClusterizerInZ::clusterize(const vector<reco::TransientTrack>& tracks) const {
  if (verbose_) {
    cout << "###################################################" << endl;
    cout << "# DAClusterizerInZ::clusterize   nt=" << tracks.size() << endl;
    cout << "###################################################" << endl;
  }

  vector<vector<reco::TransientTrack> > clusters;
  vector<TransientVertex> pv = vertices(tracks);

  if (verbose_) {
    cout << "# DAClusterizerInZ::clusterize   pv.size=" << pv.size() << endl;
  }
  if (pv.empty()) {
    return clusters;
  }

  // fill into clusters and merge
  vector<reco::TransientTrack> aCluster = pv.begin()->originalTracks();

  for (vector<TransientVertex>::iterator k = pv.begin() + 1; k != pv.end(); k++) {
    if (fabs(k->position().z() - (k - 1)->position().z()) > (2 * vertexSize_)) {
      // close a cluster
      clusters.push_back(aCluster);
      aCluster.clear();
    }
    for (unsigned int i = 0; i < k->originalTracks().size(); i++) {
      aCluster.push_back(k->originalTracks().at(i));
    }
  }
  clusters.push_back(aCluster);

  return clusters;
}