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FITEXY

FITEXY

## Purpose

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

## Explanation

Linear Least-squares approximation in one-dimension (y = a + b*x),
when both x and y data have errors
CALLING EXAMPLE:
FITEXY, x, y, A, B, X_SIG= , Y_SIG= , [sigma_A_B, chi_sq, q, TOL=]

## Inputs

x = array of values for independent variable.
y = array of data values assumed to be linearly dependent on x.
REQUIRED INPUT KEYWORDS:
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.

## Outputs

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:
common fitexy, communicates the data for computation of chi-square.

## Procedure Calls

CHISQ_FITEXY() ;Included in this file
MINF_BRACKET, MINF_PARABOLIC, ZBRENT ;In IDL Astronomy Library
MOMENT(), CHISQR_PDF() ;In standard IDL distribution

## Procedure

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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