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Jason Friedman
02-08-2006, 06:08 PM
In regards to the question of Petter Bottcher and least-square fitting
from the 2nd February:

I think you have come across several of the limitations of least-squares
fittings. However, there are some solutions.

Firstly, regarding the difference between the "average" slope and the
"optimal" slope, there is a technique known as robust least-squares
fitting which may solve this problem. This is where you would like to
give less weight to outliers. The matlab function robustfit will do
this, a good reference on this topic is the book:
Numerical recipes in C, by Press et al. You can read the book online,
see chapter 15.7
http://www.library.cornell.edu/nr/cbookcpdf.html

Regarding fitting circles and spheres, this is also a well-studied
problem. The standard criteria of least squares is generally not
equivalent to minimizing the sum of squares of the distances
from the data points to the shape being fit. Non-linear techniques
should instead be used. I recommend looking at this paper for
suggested algorithms:

W. Gander, G.H. Golub, and R. Strebel.
Least-squares fitting of circles and ellipses.
BIT, 34:558-578, 1994.

The topic is explored in more detail in the PhD thesis
of Sung Joon Ahn, titled "Least Squares Orthogonal Distance Fitting
of Curves and Surfaces in Space", which is published as a book by
Springer.
You may be able to view it online if you institution has the relevant
subscription at:
http://www.springerlink.com/openurl.asp?
genre=issue&issn=0302-9743&volume=3151

Jason Friedman
--
Jason Friedman
Ph.D. Student
Department of Computer Science and Applied Mathematics
Weizmann Institute of Science, Rehovot, Israel.
Home page: http://www.wisdom.weizmann.ac.il/~jason