Fit a circle using data points with errors

[ajdobbs]

Hi,

I am attempting to fit a circle to 4 or 5 data points using Minuit, based on the example at https://root.cern.ch/doc/v608/fitCircle_8C.html. I have adapted the example to take the input data from vectors, rather than a TGraph:

bool FitCircle(const std::vector<double>& x, const std::vector<double>& y,
               MAUS::SimpleCircle& circ, TMatrixD& cov_matrix) {

  auto Chi2Function = [&x, &y](const double *par) {
    // Minimisation function computing the sum of squares of residuals
    // looping over the points
    double f = 0.0;
    for (size_t i = 0; i < x.size(); i++) {
        double u = x[i] - par[0];
        double v = y[i] - par[1];
        double dr = par[2] - std::sqrt(u*u+v*v);
        f += dr*dr;
    }
    return f;
  };

  // Wrap chi2 funciton in a function object for the fit
  // 3 is the number of fit parameters (size of array par)
  ROOT::Math::Functor fcn(Chi2Function, 3);
  ROOT::Fit::Fitter fitter;
  double pStart[3] = {0, 0, 1};
  fitter.SetFCN(fcn, pStart);
  fitter.Config().ParSettings(0).SetName("x0");
  fitter.Config().ParSettings(1).SetName("y0");
  fitter.Config().ParSettings(2).SetName("R");
  
  // do the fit 
  bool ok = fitter.FitFCN();
  if (!ok) {
    return ok;
  }   
  const ROOT::Fit::FitResult & result = fitter.Result();
  result.Print(std::cout);
  
  // Create a circle object with the results
  circ = MAUS::SimpleCircle(result.Parameter(0), result.Parameter(1), 
                            result.Parameter(2));
  circ.set_x0_err(result.Error(0));
  circ.set_y0_err(result.Error(1));
  circ.set_R_err(result.Error(2));
  circ.set_chisq(result.Chi2());
  result.GetCovarianceMatrix(cov_matrix);
  
  return true;
}

It works well but, after some searching, I cannot see any obvious way of supplying the errors on the input data points to the fit. Could someone give me some pointers please? Thanks.

[moneta2]

HI,

You would need to weight the distance between the point and the circle by using the inverse of the point uncertainty. You might need to propagate the uncertainty in x and y of the data points along the distance between the circle and the points

Best Regards

Lorenzo