Goodness-of-fit New Technique
Mohamed Al-Dabbagh
After 6 months of intensive work, I put between your hands a new
project of research about the Relative Goodness-of-fit. This project
addresses an important problem of how to choose the best fit from
amongst a lot of available models in hand. The outcome is η-Index
which is a score ranging between 100 for the exact fit and 0 for the
trivial fit.
The η-algorithm uses pairwise comparisons of the shifted logarithms
of absolute residuals. It does not depend on dealing with the squared
residuals, thereby it will not be biased towards minimizing least
squares of the residuals for the same models (Chi square method is
always biased to the least squares). So if you have four line models
(y= a*x+b) coming from Least Squares, Minimax fitting, Orthogonal
Distance fitting and Least Absolute Deviation fitting, the algorithm
will give the right judgment for every one of these lines without
bias.
The algorithm opens the door to deal with the problem using
eigenvector scaling method invented by Dr. Thomas Saaty to rate the
goodness-of-fit when other factors (other than residuals) are
considered in a complex decision. Mathematica source code examples are
provided for immediate use by every Mathematica user. For details see:
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LEAST SQUARES FITTING: Actually I wanted to complete the benefit by
discussing in details the constrained (forced through some point or
slope or parameter or all) and unconstrained least squares curve
fitting. For those who are interested to see how we derive formulas of
least squares curve-fitting with the help of Mathematica algebraic
system, please go through this page (and be patient please.. It is
lengthy!):
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MINIMAX FITTING: Some examples were provided in the form of
Mathematica source code to explain method. I also discussed a method
to find Chebyshev line with the help of Convex Hull:
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LEAST ABSOLUTE DEVIATION FITTING: Being in the shadow, this method is
a very good one when we have some outlier points, and it is really
handy, and can easily beat the least squares (for the same models). I
wanted to give the chance for those not aware of it to have the
Mathematica source code. You can use it right now! The details are
here:
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ORTHOGONAL DISTANCE LEAST SQUARES FITTING: Though this method of
fitting is not popular due to the difficulty of dealing with it, I
tried to include some information about it. I dealt with the subject
in more details when I explained the line equation. I also derived a
formula for ODLSF line which is forced through a point (Constrained
ODLSF Line). Using Mathematica algebraic system, I derived a some
assisting formulas for the orthogonal fit of the second degree
polynomial, the thing that you cannot find it in any reference. The
details along with the Mathematica source code can be found here:
For convenience, I have provided Mathematica notebooks for every web
page. The web pages where all done using the typesetting of
Mathematica. They are all available for download. Also, Acrobat PDF
files are provided and ready to download. Both Mathematica NB and
Acrobat PDF are compressed in ZIP archives, so you have to decompress
them first using winzip or winrar application software.
I hope that this effort will be of use to you all. Enjoy Curve-
Fitting!
Mohamed Al-Dabbagh (B.Sc. Mech. Eng.)
Mathematics and Physics Teacher
Pakistan School Sanaa
Republic of Yemen
Fax: +967-1-243-673
Mobile: +967-7117-0-4224