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

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