Polynomial and rational function modeling: Difference between revisions

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::one would need to select four representative points, and perform a linear fit on the model

:::<math>

y = A_0 + A_1x + \ldots + A_{p_n}x^{p_n} - B_1xy - \ldots - B_{p_d}xB_2x^{p_d}y2y .,

</math>

::Here,which ''p_n''is andderived ''p_d'' arefrom the degreesprevious ofequation theby numeratorclearing andthe denominator,. respectivelyHere, and the ''x'' and ''y'' contain the subset of points, not the full data set. The estimated coefficients from this linear fit are used as the starting values for fitting the nonlinear model to the full data set.

::This type of fit, with the response variable appearing on both sides of the function, should only be used to obtain starting values for the nonlinear fit. The statistical properties of fits like this are not well understood.

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==See also==

* [[Response surface methodology]]

* [[Padé approximant|Pade Approximant]]

==Bibliography==

* {{cite book |author=[http://stats.lse.ac.uk/atkinson/ Atkinson, A. C.] and [http://www.maths.manchester.ac.uk/~adonev/ |author2=Donev, A. N.] and [http://support.sas.com/publishing/bbu/companion_site/index_author.html#tobias |author3=Tobias, R. D.]|title=Optimum Experimental Designs, with SAS| url=https://books.google.com/books?id=oIHsrw6NBmoC| publisher=Oxford University Press|year=2007 |pages=511+xvi |isbn=978-0-19-929660-6 |oclc= |doi=}}

* Box, G. E. P. and Draper, Norman. 2007. ''Response Surfaces, Mixtures, and Ridge Analyses'', Second Edition [of ''Empirical Model-Building and Response Surfaces'', 1987], Wiley.

* {{cite book |authorlinkauthor-link=Jack Kiefer (statistician)| last=Kiefer| first=Jack Carl| title=Collected Papers III Design of Experiments |editorlinkeditor-link=Lawrence D. Brown| editor=L. D. Brown|publisher=Springer-Verlag|year=1985|isbn=978-0-387-96004-X3|display-editors=etal}}

* R. H. Hardin and [[Neil Sloane|N. J. A. Sloane]], [http://neilsloane.com/doc/design.pdf "A New Approach to the Construction of Optimal Designs", ''Journal of Statistical Planning and Inference'', vol. 37, 1993, pp. 339-369]

* R. H. Hardin and [[Neil Sloane|N. J. A. Sloane]], [http://neilsloane.com/doc/doeh.pdf "Computer-Generated Minimal (and Larger) Response Surface Designs: (I) The Sphere"]

* R. H. Hardin and [[Neil Sloane|N. J. A. Sloane]], [http://neilsloane.com/doc/meatball.pdf "Computer-Generated Minimal (and Larger) Response Surface Designs: (II) The Cube"]

* {{Cite book| title=Design and Analysis of Experiments | series=Handbook of Statistics| volume=13|editor=Ghosh, S. |editor2=[[Calyampudi Radhakrishna Rao|Rao, C. R.]] |editor2-link=Calyampudi Radhakrishna Rao | publisher=North-Holland| year=1996| isbn=978-0-444-82061-27}}

** {{Cite book|author1=Draper, Norman |author2=Lin, Dennis K. J. |lastauthorampname-list-style=yesamp | chapter=Response Surface Designs |pages=343–375}}

** {{Cite book|author1=Gaffke, N. |author2=Heiligers, B |lastauthorampname-list-style=yesamp | chapter=Approximate Designs for [[Linear regression|Polynomial Regression]]: [[Invariant estimator|Invariance]], [[Admissible decision rule|Admissibility]], and [[Optimal design|Optimality]] |pages=1149–1199}}

* {{cite book |author=Melas, Viatcheslav B.|title=Functional Approach to Optimal Experimental Design |series=Lecture Notes in Statistics| volume=184 | publisher=Springer-Verlag | year=2006 |isbn=978-0-387-98741-X5}} (Modeling with rational functions)

===Historical===

*{{cite journal

|title=Application de la méthode des moindre quarrés a l'interpolation des suites<!-- [The application of the method of least squares to the interpolation of sequences] -->

|author=[[Joseph Diaz Gergonne|Gergonne, J. D.]]

|journal=[[Annales de mathématiques pures et appliquées]]

|volume=6

|year=1815

|pages=242–252

|author-link=Joseph Diaz Gergonne

}}

*{{cite journal

|title=The application of the method of least squares to the interpolation of sequences

|author=[[Joseph Diaz Gergonne|Gergonne, J. D.]]

|journal=Historia Mathematica

|volume=1

|issue=4 <!-- |month=November -->

|year=1974 |origyearorig-year=1815

|pages=439–447

|edition=Translated by Ralph St. John and [[Stephen M. Stigler|S. M. Stigler]] from the 1815 French

|doi=10.1016/0315-0860(74)90034-2

|author-link=Joseph Diaz Gergonne

|url=http://www.sciencedirect.com/science/article/B6WG9-4D7JMHH-20/2/df451ec5fbb7c044d0f4d900af80ec86

|doi-access=free

}}

*{{cite journal

|title=Gergonne's 1815 paper on the design and analysis of polynomial regression experiments

|author=[[Stephen M. Stigler|Stigler, Stephen M.]]

|journal=[[Historia Mathematica]]

|volume=1

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|pages=431–439

|doi=10.1016/0315-0860(74)90033-0

|author-link=Stephen M. Stigler

|url=http://www.sciencedirect.com/science/article/B6WG9-4D7JMHH-1Y/2/680c7ada0198761e9866197d53512ab4}}

|doi-access=free

}}

* {{cite journal

|author=Smith, Kirstine

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|issue=1/2

|pages=1–85

|jstor=2331929}}

|url=http://biomet.oxfordjournals.org/cgi/content/citation/12/1-2/1

|doi=10.1093/biomet/12.1-2.1}}

|jstor=2331929}}

==External links==