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      Best straight-line fit to data with errors in both coordinates


      Linear Least-squares approximation in one-dimension (y = a + b*x),
      when both x and y data have errors Users might be interested in
      Michael Williams MPFITEXY routines which include a number of
      enhancements to FITEXY.
      ( http://user.astro.columbia.edu/~williams/mpfitexy/ )
      FITEXY, x, y, A, B, X_SIG= , Y_SIG= , [sigma_A_B, chi_sq, q, TOL=]


      x = array of values for independent variable.
      y = array of data values assumed to be linearly dependent on x.
      X_SIGMA = scalar or array specifying the standard deviation of x data.
      Y_SIGMA = scalar or array specifying the standard deviation of y data.

Optional Input Keyword

      TOLERANCE = desired accuracy of minimum & zero location, default=1.e-3.


      A_intercept = constant parameter result of linear fit,
      B_slope = slope parameter, so that:
                      ( A_intercept + B_slope * x ) approximates the y data.

Optional Output

      sigma_A_B = two element array giving standard deviation of
                A_intercept and B_slope parameters, respectively.
                The standard deviations are not meaningful if (i) the
                fit is poor (see parameter q), or (ii) b is so large that
                the data are consistent with a vertical (infinite b) line.
                If the data are consistent with *all* values of b, then
                sigma_A_B = [1e33,e33]
      chi_sq = resulting minimum Chi-Square of Linear fit, scalar
      q - chi-sq probability, scalar (0-1) giving the probability that
              a correct model would give a value equal or larger than the
              observed chi squared. A small value of q indicates a poor
              fit, perhaps because the errors are underestimated. As
              discussed by Tremaine et al. (2002, ApJ, 574, 740) an
              underestimate of the errors (e.g. due to an intrinsic dispersion)
              can lead to a bias in the derived slope, and it may be worth
              enlarging the error bars to get a reduced chi_sq ~ 1
      common fitexy, communicates the data for computation of chi-square.

Procedure Calls

      CHISQ_FITEXY() ;Included in this file
      MOMENT(), CHISQR_PDF() ;In standard IDL distribution


      From "Numerical Recipes" column by Press and Teukolsky:
      in "Computer in Physics", May, 1992 Vol.6 No.3
      Also see the 2nd edition of the book "Numerical Recipes" by Press et al.
      In order to avoid problems with data sets where X and Y are of very
      different order of magnitude the data are normalized before the fitting
      process is started. The following normalization is used:
      xx = (x - xm) / xs and sigx = x_sigma / xs
                            where xm = MEAN(x) and xs = STDDEV(x)
      yy = (y - ym) / ys and sigy = y_sigma / ys
                            where ym = MEAN(y) and ys = STDDEV(y)

Modification History

      Written, Frank Varosi NASA/GSFC September 1992.
      Now returns q rather than 1-q W. Landsman December 1992
      Use CHISQR_PDF, MOMENT instead of STDEV,CHI_SQR1 W. Landsman April 1998
      Fixed typo for initial guess of slope, this error was nearly
            always insignificant W. Landsman March 2000
      Normalize X,Y before calculation (from F. Holland) W. Landsman Nov 2006

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