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ROBUST_LINEFIT

## Name

ROBUST_LINEFIT

## Purpose

An outlier-resistant two-variable linear regression.

## Explanation

Either Y on X or, for the case in which there is no true independent
variable, the bisecting line of Y vs X and X vs Y is calculated. No
knowledge of the errors of the input points is assumed.

## Calling Sequence

COEFF = ROBUST_LINEFIT( X, Y, YFIT, SIG, COEF_SIG, [ /BISECT,
BiSquare_Limit = , Close_factor = , NumIT = ] )

## Inputs

X = Independent variable vector, floating-point or double-precision
Y = Dependent variable vector

## Outputs

Function result = coefficient vector.
If = 0.0 (scalar), no fit was possible.
If vector has more than 2 elements (the last=0) then the fit is dubious.

## Optional Output Parameters

YFIT = Vector of calculated y's
SIG = The "standard deviation" of the fit's residuals. If BISECTOR
is set, this will be smaller by ~ sqrt(2).
COEF_SIG = The estimated standard deviations of the coefficients. If
BISECTOR is set, however, this becomes the vector of fit
residuals measured orthogonal to the line.

## Optional Input Keywords

NUMIT = the number of iterations allowed. Default = 25
BISECT if set, the bisector of the "Y vs X" and "X vs Y" fits is
determined. The distance PERPENDICULAR to this line is used
in calculating weights. This is better when the uncertainties
in X and Y are comparable, so there is no true independent
variable. Bisquare_Limit Limit used for calculation of
bisquare weights. In units of outlier-resistant standard
deviations. Default: 6.
Smaller limit ==>more resistant, less efficient
Close_Factor - Factor used to determine when the calculation has converged.
Convergence if the computed standard deviation changes by less
than Close_Factor * ( uncertainty of the std dev of a normal
distribution ). Default: 0.03.

## Subroutine Calls

ROB_CHECKFIT
ROBUST_SIGMA, to calculate a robust analog to the std. deviation

## Procedure

For the initial estimate, the data is sorted by X and broken into 2
groups. A line is fitted to the x and y medians of each group.
Bisquare ("Tukey's Biweight") weights are then calculated, using the
a limit of 6 outlier-resistant standard deviations.
This is done iteratively until the standard deviation changes by less
than CLOSE_ENOUGH = CLOSE_FACTOR * {uncertainty of the standard
deviation of a normal distribution}

## Revision History

Written, H. Freudenreich, STX, 4/91.
4/13/93 to return more realistic SS's HF
2/94 --more error-checking, changed convergence criterion HF