MuonSegFit.cc

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// ------------------------- //
// MuonSegFit.cc
// Created:  11.05.2015
// Based on CSCSegFit.cc
// ------------------------- //
 
#include "RecoLocalMuon/GEMSegment/plugins/MuonSegFit.h"
#include "DataFormats/GeometryVector/interface/GlobalPoint.h"
 
#include "FWCore/ParameterSet/interface/ParameterSet.h"
#include "FWCore/MessageLogger/interface/MessageLogger.h"
 
bool MuonSegFit::fit(void) {
  if (fitdone())
    return fitdone_;  // don't redo fit unnecessarily
  short n = nhits();
  if (n < 2) {
    edm::LogVerbatim("MuonSegFit") << "[MuonSegFit::fit] - cannot fit just 1 hit!!";
  } else if (n == 2) {
    fit2();
  } else if (2 * n <= MaxHits2) {
    fitlsq();
  } else {
    edm::LogVerbatim("MuonSegFit") << "[MuonSegFit::fit] - cannot fit more than " << MaxHits2 / 2 << " hits!!";
  }
  return fitdone_;
}
 
void MuonSegFit::fit2(void) {
  // Just join the two points
  // Equation of straight line between (x1, y1) and (x2, y2) in xy-plane is
  //       y = mx + c
  // with m = (y2-y1)/(x2-x1)
  // and  c = (y1*x2-x2*y1)/(x2-x1)
  //
  // Now we will make two straight lines
  // one in xz-plane, another in yz-plane
  //       x = uz + c1
  //       y = vz + c2
  // 1) Check whether hits are on the same layer
  // should be done before, so removed
  // -------------------------------------------
 
  MuonRecHitContainer::const_iterator ih = hits_.begin();
  // 2) Global Positions of hit 1 and 2 and
  //    Local  Positions of hit 1 and 2 w.r.t. reference GEM Eta Partition
  // ---------------------------------------------------------------------
  LocalPoint h1pos = (*ih)->localPosition();
  ++ih;
  LocalPoint h2pos = (*ih)->localPosition();
 
  // 3) Now make straight line between the two points in local coords
  // ----------------------------------------------------------------
  float dz = h2pos.z() - h1pos.z();
  if (dz != 0.0) {
    uslope_ = (h2pos.x() - h1pos.x()) / dz;
    vslope_ = (h2pos.y() - h1pos.y()) / dz;
  }
 
  float uintercept = (h1pos.x() * h2pos.z() - h2pos.x() * h1pos.z()) / dz;
  float vintercept = (h1pos.y() * h2pos.z() - h2pos.y() * h1pos.z()) / dz;
 
  // calculate local position (variable: intercept_)
  intercept_ = LocalPoint(uintercept, vintercept, 0.);
 
  // calculate the local direction (variable: localdir_)
  setOutFromIP();
 
  //@@ NOT SURE WHAT IS SENSIBLE FOR THESE...
  chi2_ = 0.;
  ndof_ = 0;
 
  fitdone_ = true;
}
 
void MuonSegFit::fitlsq(void) {
  // Linear least-squares fit to up to 6 GEM rechits, one per layer in a GEM chamber.
  // Comments adapted from Tim Cox' comments in the original  GEMSegAlgoSK algorithm.
 
  // Fit to the local x, y rechit coordinates in z projection
  // The strip measurement controls the precision of x
  // The wire measurement controls the precision of y.
  // Typical precision: u (strip, sigma~200um), v (wire, sigma~1cm)
 
  // Set up the normal equations for the least-squares fit as a matrix equation
 
  // We have a vector of measurements m, which is a 2n x 1 dim matrix
  // The transpose mT is (u1, v1, u2, v2, ..., un, vn) where
  // ui is the strip-associated measurement and
  // vi is the wire-associated measurement
  // for a given rechit i.
 
  // The fit is to
  // u = u0 + uz * z
  // v = v0 + vz * z
  // where u0, uz, v0, vz are the parameters to be obtained from the fit.
 
  // These are contained in a vector p which is a 4x1 dim matrix, and
  // its transpose pT is (u0, v0, uz, vz). Note the ordering!
 
  // The covariance matrix for each pair of measurements is 2 x 2 and
  // the inverse of this is the error matrix E.
  // The error matrix for the whole set of n measurements is a diagonal
  // matrix with diagonal elements the individual 2 x 2 error matrices
  // (because the inverse of a diagonal matrix is a diagonal matrix
  // with each element the inverse of the original.)
 
  // In function 'weightMatrix()', the variable 'matrix' is filled with this
  // block-diagonal overall covariance matrix. Then 'matrix' is inverted to the
  // block-diagonal error matrix, and returned.
 
  // Define the matrix A as
  //    1   0   z1  0
  //    0   1   0   z1
  //    1   0   z2  0
  //    0   1   0   z2
  //    ..  ..  ..  ..
  //    1   0   zn  0
  //    0   1   0   zn
 
  // This matrix A is set up and returned by function 'derivativeMatrix()'.
 
  // Then the normal equations are described by the matrix equation
  //
  //    (AT E A)p = (AT E)m
  //
  // where AT is the transpose of A.
 
  // Call the combined matrix on the LHS, M, and that on the RHS, B:
  //     M p = B
 
  // We solve this for the parameter vector, p.
  // The elements of M and B then involve sums over the hits
 
  // The covariance matrix of the parameters is obtained by
  // (AT E A)^-1 calculated in 'covarianceMatrix()'.
 
  SMatrix4 M;  // 4x4, init to 0
  SVector4 B;  // 4x1, init to 0;
 
  MuonRecHitContainer::const_iterator ih = hits_.begin();
 
  // Loop over the GEMRecHits and make small (2x2) matrices used to fill the blockdiagonal covariance matrix E^-1
  for (ih = hits_.begin(); ih != hits_.end(); ++ih) {
    LocalPoint lp = (*ih)->localPosition();
    // Local position of hit w.r.t. chamber
    double u = lp.x();
    double v = lp.y();
    double z = lp.z();
 
    // Covariance matrix of local errors
    SMatrixSym2 IC;  // 2x2, init to 0
 
    IC(0, 0) = (*ih)->localPositionError().xx();
    IC(1, 1) = (*ih)->localPositionError().yy();
    //@@ NOT SURE WHICH OFF-DIAGONAL ELEMENT MUST BE DEFINED BUT (1,0) WORKS
    //@@ (and SMatrix enforces symmetry)
    IC(1, 0) = (*ih)->localPositionError().xy();
    // IC(0,1) = IC(1,0);
 
#ifdef EDM_ML_DEBUG
    edm::LogVerbatim("MuonSegFitMatrixDetails")
        << "[MuonSegFit::fit] 2x2 covariance matrix for this GEMRecHit :: [[" << IC(0, 0) << ", " << IC(0, 1) << "]["
        << IC(1, 0) << "," << IC(1, 1) << "]]";
#endif
 
    // Invert covariance matrix (and trap if it fails!)
    bool ok = IC.Invert();
    if (!ok) {
      edm::LogVerbatim("MuonSegFitMatrixDetails") << "[MuonSegFit::fit] Failed to invert covariance matrix: \n" << IC;
      //      return ok;  //@@ SHOULD PASS THIS BACK TO CALLER?
    }
 
    M(0, 0) += IC(0, 0);
    M(0, 1) += IC(0, 1);
    M(0, 2) += IC(0, 0) * z;
    M(0, 3) += IC(0, 1) * z;
    B(0) += u * IC(0, 0) + v * IC(0, 1);
 
    M(1, 0) += IC(1, 0);
    M(1, 1) += IC(1, 1);
    M(1, 2) += IC(1, 0) * z;
    M(1, 3) += IC(1, 1) * z;
    B(1) += u * IC(1, 0) + v * IC(1, 1);
 
    M(2, 0) += IC(0, 0) * z;
    M(2, 1) += IC(0, 1) * z;
    M(2, 2) += IC(0, 0) * z * z;
    M(2, 3) += IC(0, 1) * z * z;
    B(2) += (u * IC(0, 0) + v * IC(0, 1)) * z;
 
    M(3, 0) += IC(1, 0) * z;
    M(3, 1) += IC(1, 1) * z;
    M(3, 2) += IC(1, 0) * z * z;
    M(3, 3) += IC(1, 1) * z * z;
    B(3) += (u * IC(1, 0) + v * IC(1, 1)) * z;
  }
 
  SVector4 p;
  bool ok = M.Invert();
  if (!ok) {
    edm::LogVerbatim("MuonSegFitMatrixDetails") << "[MuonSegFit::fit] Failed to invert matrix: \n" << M;
    //    return ok; //@@ SHOULD PASS THIS BACK TO CALLER?
  } else {
    p = M * B;
  }
 
#ifdef EDM_ML_DEBUG
  LogTrace("MuonSegFitMatrixDetails") << "[MuonSegFit::fit] p = " << p(0) << ", " << p(1) << ", " << p(2) << ", "
                                      << p(3);
#endif
 
  // fill member variables  (note origin has local z = 0)
  //  intercept_
  intercept_ = LocalPoint(p(0), p(1), 0.);
 
  // localdir_ - set so segment points outwards from IP
  uslope_ = p(2);
  vslope_ = p(3);
  setOutFromIP();
 
  // calculate chi2 of fit
  setChi2();
 
  // flag fit has been done
  fitdone_ = true;
}
 
void MuonSegFit::setChi2(void) {
  double chsq = 0.;
 
  MuonRecHitContainer::const_iterator ih;
 
  for (ih = hits_.begin(); ih != hits_.end(); ++ih) {
    LocalPoint lp = (*ih)->localPosition();
 
    double u = lp.x();
    double v = lp.y();
    double z = lp.z();
 
    double du = intercept_.x() + uslope_ * z - u;
    double dv = intercept_.y() + vslope_ * z - v;
 
    SMatrixSym2 IC;  // 2x2, init to 0
 
    IC(0, 0) = (*ih)->localPositionError().xx();
    //    IC(0,1) = (*ih)->localPositionError().xy();
    IC(1, 0) = (*ih)->localPositionError().xy();
    IC(1, 1) = (*ih)->localPositionError().yy();
    //    IC(1,0) = IC(0,1);
 
#ifdef EDM_ML_DEBUG
    edm::LogVerbatim("MuonSegFitMatrixDetails") << "[MuonSegFit::setChi2] IC before = \n" << IC;
#endif
 
    // Invert covariance matrix
    bool ok = IC.Invert();
    if (!ok) {
      edm::LogVerbatim("MuonSegFitMatrixDetails") << "[MuonSegFit::setChi2] Failed to invert covariance matrix: \n"
                                                  << IC;
      //      return ok;
    }
    chsq += du * du * IC(0, 0) + 2. * du * dv * IC(0, 1) + dv * dv * IC(1, 1);
 
#ifdef EDM_ML_DEBUG
    edm::LogVerbatim("MuonSegFitMatrixDetails") << "[MuonSegFit::setChi2] IC after = \n" << IC;
    edm::LogVerbatim("MuonSegFit")
        << "[GEM RecHit ] Contribution to Chi^2 of this hit :: du^2*Cov(0,0) + 2*du*dv*Cov(0,1) + dv^2*IC(1,1) = "
        << du * du << "*" << IC(0, 0) << " + 2.*" << du << "*" << dv << "*" << IC(0, 1) << " + " << dv * dv << "*"
        << IC(1, 1) << " = " << chsq;
#endif
  }
 
  // fill member variables
  chi2_ = chsq;
  ndof_ = 2. * hits_.size() - 4;
 
#ifdef EDM_ML_DEBUG
  edm::LogVerbatim("MuonSegFit") << "[MuonSegFit::setChi2] chi2/ndof = " << chi2_ << "/" << ndof_;
  edm::LogVerbatim("MuonSegFit") << "-----------------------------------------------------";
 
  // check fit quality ... maybe write a separate function later on
  // that is there only for debugging issues
 
  edm::LogVerbatim("MuonSegFit") << "[GEM Segment with Local Direction = " << localdir_ << " and Local Position "
                                 << intercept_ << "] can be written as:";
  edm::LogVerbatim("MuonSegFit") << "[ x ] = " << localdir_.x() << " * t + " << intercept_.x();
  edm::LogVerbatim("MuonSegFit") << "[ y ] = " << localdir_.y() << " * t + " << intercept_.y();
  edm::LogVerbatim("MuonSegFit") << "[ z ] = " << localdir_.z() << " * t + " << intercept_.z();
  edm::LogVerbatim("MuonSegFit")
      << "Now extrapolate to each of the GEMRecHits XY plane (so constant z = RH LP.z()) to obtain [x1,y1]";
#endif
}
 
MuonSegFit::SMatrixSym12 MuonSegFit::weightMatrix() {
  bool ok = true;
 
  SMatrixSym12 matrix = ROOT::Math::SMatrixIdentity();  // 12x12, init to 1's on diag
 
  int row = 0;
 
  for (MuonRecHitContainer::const_iterator it = hits_.begin(); it != hits_.end(); ++it) {
    // Note scaleXError allows rescaling the x error if necessary
    matrix(row, row) = scaleXError() * (*it)->localPositionError().xx();
    matrix(row, row + 1) = (*it)->localPositionError().xy();
    ++row;
    matrix(row, row - 1) = (*it)->localPositionError().xy();
    matrix(row, row) = (*it)->localPositionError().yy();
    ++row;
  }
 
  ok = matrix.Invert();  // invert in place
  if (!ok) {
    edm::LogVerbatim("MuonSegFitMatrixDetails") << "[MuonSegFit::weightMatrix] Failed to invert matrix: \n" << matrix;
    //    return ok; //@@ SHOULD PASS THIS BACK TO CALLER?
  }
  return matrix;
}
 
MuonSegFit::SMatrix12by4 MuonSegFit::derivativeMatrix() {
  SMatrix12by4 matrix;  // 12x4, init to 0
  int row = 0;
 
  for (MuonRecHitContainer::const_iterator it = hits_.begin(); it != hits_.end(); ++it) {
    LocalPoint lp = (*it)->localPosition();
 
    float z = lp.z();
 
    matrix(row, 0) = 1.;
    matrix(row, 2) = z;
    ++row;
    matrix(row, 1) = 1.;
    matrix(row, 3) = z;
    ++row;
  }
  return matrix;
}
 
void MuonSegFit::setOutFromIP() {
  // Set direction of segment to point from IP outwards
  // (Incorrect for particles not coming from IP, of course.)
 
  double dxdz = uslope_;
  double dydz = vslope_;
  double dz = 1. / sqrt(1. + dxdz * dxdz + dydz * dydz);
  double dx = dz * dxdz;
  double dy = dz * dydz;
  LocalVector localDir(dx, dy, dz);
 
  localdir_ = (localDir).unit();
#ifdef EDM_ML_DEBUG
  edm::LogVerbatim("MuonSegFit") << "[MuonSegFit::setOutFromIP] :: dxdz = uslope_ = " << std::setw(9) << uslope_
                                 << " dydz = vslope_ = " << std::setw(9) << vslope_ << " local dir = " << localDir;
  edm::LogVerbatim("MuonSegFit") << "[MuonSegFit::setOutFromIP] ::  ==> local dir = " << localdir_
                                 << " localdir.phi = " << localdir_.phi();
#endif
}
 
AlgebraicSymMatrix MuonSegFit::covarianceMatrix() {
  SMatrixSym12 weights = weightMatrix();
  SMatrix12by4 A = derivativeMatrix();
#ifdef EDM_ML_DEBUG
  edm::LogVerbatim("MuonSegFitMatrixDetails") << "[MuonSegFit::covarianceMatrix] weights matrix W: \n" << weights;
  edm::LogVerbatim("MuonSegFitMatrixDetails") << "[MuonSegFit::covarianceMatrix] derivatives matrix A: \n" << A;
#endif
 
  // (AT W A)^-1
  // e.g. See http://www.phys.ufl.edu/~avery/fitting.html, part I
 
  bool ok;
  SMatrixSym4 result = ROOT::Math::SimilarityT(A, weights);
#ifdef EDM_ML_DEBUG
  edm::LogVerbatim("MuonSegFitMatrixDetails") << "[MuonSegFit::covarianceMatrix] (AT W A): \n" << result;
#endif
 
  ok = result.Invert();  // inverts in place
  if (!ok) {
    edm::LogVerbatim("MuonSegFitMatrixDetails") << "[MuonSegFit::calculateError] Failed to invert matrix: \n" << result;
    //    return ok;  //@@ SHOULD PASS THIS BACK TO CALLER?
  }
#ifdef EDM_ML_DEBUG
  edm::LogVerbatim("MuonSegFitMatrixDetails") << "[MuonSegFit::covarianceMatrix] (AT W A)^-1: \n" << result;
#endif
 
  // reorder components to match TrackingRecHit interface (GEMSegment isa TrackingRecHit)
  // i.e. slopes first, then positions
  AlgebraicSymMatrix flipped = flipErrors(result);
 
  return flipped;
}
 
AlgebraicSymMatrix MuonSegFit::flipErrors(const SMatrixSym4& a) {
  // The GEMSegment needs the error matrix re-arranged to match
  // parameters in order (uz, vz, u0, v0)
  // where uz, vz = slopes, u0, v0 = intercepts
 
#ifdef EDM_ML_DEBUG
  edm::LogVerbatim("MuonSegFitMatrixDetails") << "[MuonSegFit::flipErrors] input: \n" << a;
#endif
 
  AlgebraicSymMatrix hold(4, 0.);
 
  for (short j = 0; j != 4; ++j) {
    for (short i = 0; i != 4; ++i) {
      hold(i + 1, j + 1) = a(i, j);  // SMatrix counts from 0, AlgebraicMatrix from 1
    }
  }
 
#ifdef EDM_ML_DEBUG
  edm::LogVerbatim("MuonSegFitMatrixDetails") << "[MuonSegFit::flipErrors] after copy:";
  edm::LogVerbatim("MuonSegFitMatrixDetails")
      << "(" << hold(1, 1) << "  " << hold(1, 2) << "  " << hold(1, 3) << "  " << hold(1, 4);
  edm::LogVerbatim("MuonSegFitMatrixDetails")
      << " " << hold(2, 1) << "  " << hold(2, 2) << "  " << hold(2, 3) << "  " << hold(2, 4);
  edm::LogVerbatim("MuonSegFitMatrixDetails")
      << " " << hold(3, 1) << "  " << hold(3, 2) << "  " << hold(3, 3) << "  " << hold(3, 4);
  edm::LogVerbatim("MuonSegFitMatrixDetails")
      << " " << hold(4, 1) << "  " << hold(4, 2) << "  " << hold(4, 3) << "  " << hold(4, 4) << ")";
#endif
 
  // errors on slopes into upper left
  hold(1, 1) = a(2, 2);
  hold(1, 2) = a(2, 3);
  hold(2, 1) = a(3, 2);
  hold(2, 2) = a(3, 3);
 
  // errors on positions into lower right
  hold(3, 3) = a(0, 0);
  hold(3, 4) = a(0, 1);
  hold(4, 3) = a(1, 0);
  hold(4, 4) = a(1, 1);
 
  // must also interchange off-diagonal elements of off-diagonal 2x2 submatrices
  hold(4, 1) = a(1, 2);
  hold(3, 2) = a(0, 3);
  hold(2, 3) = a(3, 0);  // = a(0,3)
  hold(1, 4) = a(2, 1);  // = a(1,2)
 
#ifdef EDM_ML_DEBUG
  edm::LogVerbatim("MuonSegFitMatrixDetails") << "[MuonSegFit::flipErrors] after flip:";
  edm::LogVerbatim("MuonSegFitMatrixDetails")
      << "(" << hold(1, 1) << "  " << hold(1, 2) << "  " << hold(1, 3) << "  " << hold(1, 4);
  edm::LogVerbatim("MuonSegFitMatrixDetails")
      << " " << hold(2, 1) << "  " << hold(2, 2) << "  " << hold(2, 3) << "  " << hold(2, 4);
  edm::LogVerbatim("MuonSegFitMatrixDetails")
      << " " << hold(3, 1) << "  " << hold(3, 2) << "  " << hold(3, 3) << "  " << hold(3, 4);
  edm::LogVerbatim("MuonSegFitMatrixDetails")
      << " " << hold(4, 1) << "  " << hold(4, 2) << "  " << hold(4, 3) << "  " << hold(4, 4) << ")";
#endif
 
  return hold;
}
 
float MuonSegFit::xfit(float z) const {
  //@@ ADD THIS TO EACH ACCESSOR OF FIT RESULTS?
  //  if ( !fitdone() ) fit();
  return intercept_.x() + uslope_ * z;
}
 
float MuonSegFit::yfit(float z) const { return intercept_.y() + vslope_ * z; }
 
float MuonSegFit::xdev(float x, float z) const { return intercept_.x() + uslope_ * z - x; }
 
float MuonSegFit::ydev(float y, float z) const { return intercept_.y() + vslope_ * z - y; }
 
float MuonSegFit::Rdev(float x, float y, float z) const {
  return sqrt(xdev(x, z) * xdev(x, z) + ydev(y, z) * ydev(y, z));
}