Re: linear fitting: different errors on X data and Y data

In article <nyupnd769lae.5ec9pfwfrt3c.dlg@xxxxxxxxxx>,
Cannellino <fagiolino@xxxxxxxxxxxxx> wrote:

Hi,
I'm looking for a software that can perform linear fitting with weights
both on X data and Y data. I mean tha most general linear fitting, where
the x1, x2, x3, ... values have different errors, and the same for
y1,y2,y3,.. values.
Thanks in advance and sorry for my english

This problem has been studied for a long time. (Some early examples:

R. J. Ad***, "Note on the method of least squares," The Analyst (Des
Moines), vol. 4, 183-184 (1877)

R. J. Ad***, "A problem in least squares," The Analyst (Des Moines),
vol. 5, 53-54 (1878)

K. Pearson, "On lines and planes of closest fit to systems of points in
space," Phil. Mag. vol. 2, 559-572 (1901) )

I don't know whether you can easily find software that treats this case
(strange, since the problem has been well-known for a long time, and can
easily be handled with a computer). I hope this summary will be of some
help.

The equations are inherently nonlinear even for straight-line fitting,
so implementation of the theory was difficult in the pre-computer era.
One of the best references from the period just before the development
of digital computers is a book (well worth reading):

W. E. Deming, "Statistical Adjustment of Data," Wiley, 1943.

There was a flurry of interest within the last few decades. Here is a
sampling:

D. York, Can. J. Phys vol.44 1079 (1966)--Just the straight line, but a
good analysis.

M. Lybanon, "A better least-squares method when both variables have
uncertainties," Am. J. Phys. vol 52, 22-26 (1984)--This method works for
straight lines or more complicated functions.

M. Lybanon, "A simple generalized least-squares algorithm," Computers &
Geosciences vol. 11, no. 4, 501-508 (1985)--A program implementing the
algorithm of the preceding paper.

S. D. Christian, E. E. Tucker, and E. Enwall, "east squares analysis: A
primer," American Laboratory, vol. 8, no. 6, 41-49 (1986)--Good
formulation of the straight line problem (also some material on more
general functions).

P. L. Jolivette, "Least-squares fits when there are errors in X,"
Computers in Physics vol. 7 no. 2, 208-212 (1993)--Jolivette offered the
program via e-mail, but the same e-mail address
(jolivette@xxxxxxxxxxxxxxxx) may no longer be a valid address.

J. R. Macdonald and W. J. Thompson, "Least-squares fitting when both
variables contain errors: Pitfalls and possibilities," Am. J. Phys.
vol. 60, 66-72 (1992)--This article has a section on straight-line
fitting, and includes a survey of available (at the time) software.

W. H. Jefferys, M. J. Fitzpatrick, and B. E. McArthur, "GaussFit--a
system for least squares and robust estimation," Celestial Mech. vo..
41, 39-49 (1988)--The software,, developed for the Hubble Space
Telescope program, used to be available from Jefferys at the University
of Texas Department of Astronomy. Jefferys published some earlier
papers, but this article is a compilation, puls some new material.

B. P. Miller and He. E. Dunn, "Orthogonal least-squares line fit with
variable scaling," Computers in Physics July/Aug. 1988 (not sure of the
volume number)--An interesting geometrical interpretation, applied
specifically to the straight line. The authors offered software.

M. Lybanon and K. C. Messa, Jr., "Genetic Algorithm Model Fitting,"
Chapter 8 (pp. 269-345) in L. D. Chambers (ed.), Practical Handbook of
Genetic Algorithms, Complex Coding Systems, Volume III, CRC Press, ISBN
0-8493-2539-0 (1999)--The chapter discusses application of the method to
the case of arbitrary uncertainties in both variables, and includes a
program listing. The software was (and may still be) available from CRC
Press.
.

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