CSCSegFit.cc

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// CSCSegFit.cc
// Last mod: 03.02.2015

#include "RecoLocalMuon/CSCSegment/src/CSCSegFit.h"

#include "DataFormats/GeometryVector/interface/GlobalPoint.h"

#include "Geometry/CSCGeometry/interface/CSCLayer.h"

#include "FWCore/ParameterSet/interface/ParameterSet.h"
#include "FWCore/MessageLogger/interface/MessageLogger.h"

void CSCSegFit::fit(void) {
  if (fitdone())
    return;  // don't redo fit unnecessarily
  short n = nhits();
  switch (n) {
    case 1:
      edm::LogVerbatim("CSCSegFit") << "[CSCSegFit::fit] - cannot fit just 1 hit!!";
      break;
    case 2:
      fit2();
      break;
    case 3:
    case 4:
    case 5:
    case 6:
      fitlsq();
      break;
    default:
      edm::LogVerbatim("CSCSegFit") << "[CSCSegFit::fit] - cannot fit more than 6 hits!!";
  }
}

void CSCSegFit::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)

  CSCSetOfHits::const_iterator ih = hits_.begin();
  int il1 = (*ih)->cscDetId().layer();
  const CSCRecHit2D& h1 = (**ih);
  ++ih;
  int il2 = (*ih)->cscDetId().layer();
  const CSCRecHit2D& h2 = (**ih);

  // Skip if on same layer, but should be impossible :)
  if (il1 == il2) {
    edm::LogVerbatim("CSCSegFit") << "[CSCSegFit:fit]2 - 2 hits on same layer!!";
    return;
  }

  const CSCLayer* layer1 = chamber()->layer(il1);
  const CSCLayer* layer2 = chamber()->layer(il2);

  GlobalPoint h1glopos = layer1->toGlobal(h1.localPosition());
  GlobalPoint h2glopos = layer2->toGlobal(h2.localPosition());

  // We want hit wrt chamber (and local z will be != 0)
  LocalPoint h1pos = chamber()->toLocal(h1glopos);
  LocalPoint h2pos = chamber()->toLocal(h2glopos);

  float dz = h2pos.z() - h1pos.z();

  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;
  intercept_ = LocalPoint(uintercept, vintercept, 0.);

  setOutFromIP();

  //@@ NOT SURE WHAT IS SENSIBLE FOR THESE...
  chi2_ = 0.;
  ndof_ = 0;

  fitdone_ = true;
}

void CSCSegFit::fitlsq(void) {
  // Linear least-squares fit to up to 6 CSC rechits, one per layer in a CSC.
  // Comments adapted from  mine in original  CSCSegAlgoSK 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()'.

  // NOTE
  // We need local position of a RecHit w.r.t. the CHAMBER
  // and the RecHit itself only knows its local position w.r.t.
  // the LAYER, so we must explicitly transform global position.

  SMatrix4 M;  // 4x4, init to 0
  SVector4 B;  // 4x1, init to 0;

  CSCSetOfHits::const_iterator ih = hits_.begin();

  for (ih = hits_.begin(); ih != hits_.end(); ++ih) {
    const CSCRecHit2D& hit = (**ih);
    const CSCLayer* layer = chamber()->layer(hit.cscDetId().layer());
    GlobalPoint gp = layer->toGlobal(hit.localPosition());
    LocalPoint lp = chamber()->toLocal(gp);

    // 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) = hit.localPositionError().xx();
    IC(1, 1) = hit.localPositionError().yy();
    //@@ NOT SURE WHICH OFF-DIAGONAL ELEMENT MUST BE DEFINED BUT (1,0) WORKS
    //@@ (and SMatrix enforces symmetry)
    IC(1, 0) = hit.localPositionError().xy();
    // IC(0,1) = IC(1,0);

    // Invert covariance matrix (and trap if it fails!)
    bool ok = IC.Invert();
    if (!ok) {
      edm::LogVerbatim("CSCSegment|CSCSegFit") << "[CSCSegFit::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("CSCSegment|CSCSegFit") << "[CSCSegFit::fit] Failed to invert matrix: \n" << M;
    //    return ok; //@@ SHOULD PASS THIS BACK TO CALLER?
  } else {
    p = M * B;
  }

  //  LogTrace("CSCSegFit") << "[CSCSegFit::fit] p = "
  //        << p(0) << ", " << p(1) << ", " << p(2) << ", " << p(3);

  // 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 CSCSegFit::setChi2(void) {
  double chsq = 0.;

  CSCSetOfHits::const_iterator ih;
  for (ih = hits_.begin(); ih != hits_.end(); ++ih) {
    const CSCRecHit2D& hit = (**ih);
    const CSCLayer* layer = chamber()->layer(hit.cscDetId().layer());
    GlobalPoint gp = layer->toGlobal(hit.localPosition());
    LocalPoint lp = chamber()->toLocal(gp);

    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;

    //    LogTrace("CSCSegFit") << "[CSCSegFit::setChi2] u, v, z = " << u << ", " << v << ", " << z;

    SMatrixSym2 IC;  // 2x2, init to 0

    IC(0, 0) = hit.localPositionError().xx();
    //    IC(0,1) = hit.localPositionError().xy();
    IC(1, 0) = hit.localPositionError().xy();
    IC(1, 1) = hit.localPositionError().yy();
    //    IC(1,0) = IC(0,1);

    //    LogTrace("CSCSegFit") << "[CSCSegFit::setChi2] IC before = \n" << IC;

    // Invert covariance matrix
    bool ok = IC.Invert();
    if (!ok) {
      edm::LogVerbatim("CSCSegment|CSCSegFit") << "[CSCSegFit::setChi2] Failed to invert covariance matrix: \n" << IC;
      //      return ok;
    }
    //    LogTrace("CSCSegFit") << "[CSCSegFit::setChi2] IC after = \n" << IC;
    chsq += du * du * IC(0, 0) + 2. * du * dv * IC(0, 1) + dv * dv * IC(1, 1);
  }

  // fill member variables
  chi2_ = chsq;
  ndof_ = 2. * hits_.size() - 4;

  //  LogTrace("CSCSegFit") << "[CSCSegFit::setChi2] chi2 = " << chi2_ << "/" << ndof_ << " dof";
}

CSCSegFit::SMatrixSym12 CSCSegFit::weightMatrix() {
  bool ok = true;

  SMatrixSym12 matrix = ROOT::Math::SMatrixIdentity();  // 12x12, init to 1's on diag

  int row = 0;

  for (CSCSetOfHits::const_iterator it = hits_.begin(); it != hits_.end(); ++it) {
    const CSCRecHit2D& hit = (**it);

    // Note scaleXError allows rescaling the x error if necessary

    matrix(row, row) = scaleXError() * hit.localPositionError().xx();
    matrix(row, row + 1) = hit.localPositionError().xy();
    ++row;
    matrix(row, row - 1) = hit.localPositionError().xy();
    matrix(row, row) = hit.localPositionError().yy();
    ++row;
  }

  ok = matrix.Invert();  // invert in place
  if (!ok) {
    edm::LogVerbatim("CSCSegment|CSCSegFit") << "[CSCSegFit::weightMatrix] Failed to invert matrix: \n" << matrix;
    //    return ok; //@@ SHOULD PASS THIS BACK TO CALLER?
  }
  return matrix;
}

CSCSegFit::SMatrix12by4 CSCSegFit::derivativeMatrix() {
  SMatrix12by4 matrix;  // 12x4, init to 0
  int row = 0;

  for (CSCSetOfHits::const_iterator it = hits_.begin(); it != hits_.end(); ++it) {
    const CSCRecHit2D& hit = (**it);
    const CSCLayer* layer = chamber()->layer(hit.cscDetId().layer());
    GlobalPoint gp = layer->toGlobal(hit.localPosition());
    LocalPoint lp = chamber()->toLocal(gp);
    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 CSCSegFit::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 sometimes needs a sign flip
  // Examine its direction and origin in global z: to point outward
  // the localDir should always have same sign as global z...

  double globalZpos = (chamber()->toGlobal(intercept_)).z();
  double globalZdir = (chamber()->toGlobal(localDir)).z();
  double directionSign = globalZpos * globalZdir;
  localdir_ = (directionSign * localDir).unit();
}

AlgebraicSymMatrix CSCSegFit::covarianceMatrix() {
  SMatrixSym12 weights = weightMatrix();
  SMatrix12by4 A = derivativeMatrix();
  //  LogTrace("CSCSegFit") << "[CSCSegFit::covarianceMatrix] weights matrix W: \n" << weights;
  //  LogTrace("CSCSegFit") << "[CSCSegFit::covarianceMatrix] derivatives matrix A: \n" << A;

  // (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);
  //  LogTrace("CSCSegFit") << "[CSCSegFit::covarianceMatrix] (AT W A): \n" << result;
  ok = result.Invert();  // inverts in place
  if (!ok) {
    edm::LogVerbatim("CSCSegment|CSCSegFit") << "[CSCSegFit::calculateError] Failed to invert matrix: \n" << result;
    //    return ok;  //@@ SHOULD PASS THIS BACK TO CALLER?
  }
  //  LogTrace("CSCSegFit") << "[CSCSegFit::covarianceMatrix] (AT W A)^-1: \n" << result;

  // reorder components to match TrackingRecHit interface (CSCSegment isa TrackingRecHit)
  // i.e. slopes first, then positions
  AlgebraicSymMatrix flipped = flipErrors(result);

  return flipped;
}

AlgebraicSymMatrix CSCSegFit::flipErrors(const SMatrixSym4& a) {
  // The CSCSegment needs the error matrix re-arranged to match
  // parameters in order (uz, vz, u0, v0)
  // where uz, vz = slopes, u0, v0 = intercepts

  //  LogTrace("CSCSegFit") << "[CSCSegFit::flipErrors] input: \n" << a;

  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
    }
  }

  //  LogTrace("CSCSegFit") << "[CSCSegFit::flipErrors] after copy:";
  //  LogTrace("CSCSegFit") << "(" << hold(1,1) << "  " << hold(1,2) << "  " << hold(1,3) << "  " << hold(1,4);
  //  LogTrace("CSCSegFit") << " " << hold(2,1) << "  " << hold(2,2) << "  " << hold(2,3) << "  " << hold(2,4);
  //  LogTrace("CSCSegFit") << " " << hold(3,1) << "  " << hold(3,2) << "  " << hold(3,3) << "  " << hold(3,4);
  //  LogTrace("CSCSegFit") << " " << hold(4,1) << "  " << hold(4,2) << "  " << hold(4,3) << "  " << hold(4,4) << ")";

  // 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)

  //  LogTrace("CSCSegFit") << "[CSCSegFit::flipErrors] after flip:";
  //  LogTrace("CSCSegFit") << "(" << hold(1,1) << "  " << hold(1,2) << "  " << hold(1,3) << "  " << hold(1,4);
  //  LogTrace("CSCSegFit") << " " << hold(2,1) << "  " << hold(2,2) << "  " << hold(2,3) << "  " << hold(2,4);
  //  LogTrace("CSCSegFit") << " " << hold(3,1) << "  " << hold(3,2) << "  " << hold(3,3) << "  " << hold(3,4);
  //  LogTrace("CSCSegFit") << " " << hold(4,1) << "  " << hold(4,2) << "  " << hold(4,3) << "  " << hold(4,4) << ")";

  return hold;
}

float CSCSegFit::xfit(float z) const {
  //@@ ADD THIS TO EACH ACCESSOR OF FIT RESULTS?
  //  if ( !fitdone() ) fit();
  return intercept_.x() + uslope_ * z;
}

float CSCSegFit::yfit(float z) const { return intercept_.y() + vslope_ * z; }

float CSCSegFit::xdev(float x, float z) const { return intercept_.x() + uslope_ * z - x; }

float CSCSegFit::ydev(float y, float z) const { return intercept_.y() + vslope_ * z - y; }

float CSCSegFit::Rdev(float x, float y, float z) const {
  return sqrt(xdev(x, z) * xdev(x, z) + ydev(y, z) * ydev(y, z));
}