Editing Curve fitting (section)

{{Short description|Process of constructing a curve that has the best fit to a series of data points}}
{{Redirect|Best fit|placing ("fitting") variable-sized objects in storage|Fragmentation (computing)}}
[[File:Regression pic assymetrique.gif|thumb|upright=1.5|Fitting of a noisy curve by an asymmetrical peak model, with an iterative process ([[Gauss–Newton algorithm]] with variable damping factor α).]] 
'''Curve fitting'''<ref>Sandra Lach Arlinghaus, PHB Practical Handbook of Curve Fitting. CRC Press, 1994.</ref><ref>William M. Kolb. [https://books.google.com/books?id=ZiLYAAAAMAAJ&q=%22Curve+fitting%22 Curve Fitting for Programmable Calculators]. Syntec, Incorporated, 1984.</ref> is the process of constructing a [[curve]], or [[function (mathematics)|mathematical function]], that has the best fit to a series of [[data points]],<ref>S.S. Halli, K.V. Rao. 1992. Advanced Techniques of Population Analysis. {{ISBN|0306439972}} Page 165 (''cf''. ... functions are fulfilled if we have a good to moderate fit for the observed data.)</ref> possibly subject to constraints.<ref>[https://books.google.com/books?id=SI-VqAT4_hYC ''The Signal and the Noise: Why So Many Predictions Fail-but Some Don't.''] By Nate Silver</ref><ref>[https://books.google.com/books?id=hhdVr9F-JfAC Data Preparation for Data Mining]: Text. By Dorian Pyle.</ref> Curve fitting can involve either [[interpolation]],<ref>Numerical Methods in Engineering with MATLAB®. By Jaan Kiusalaas. Page 24.</ref><ref>[https://books.google.com/books?id=YlkgAwAAQBAJ&q=%22curve+fitting%22 Numerical Methods in Engineering with Python 3]. By Jaan Kiusalaas. Page 21.</ref> where an exact fit to the data is required, or [[smoothing]],<ref>[https://books.google.com/books?id=UjnB0FIWv_AC&q=smoothing Numerical Methods of Curve Fitting]. By P. G. Guest, Philip George Guest. Page 349.</ref><ref>See also: [[Mollifier]]</ref> in which a "smooth" function is constructed that approximately fits the data.  A related topic is [[regression analysis]],<ref>[https://books.google.com/books?id=g1FO9pquF3kC&q=%22regression+analysis%22 Fitting Models to Biological Data Using Linear and Nonlinear Regression]. By Harvey Motulsky, Arthur Christopoulos.</ref><ref>[https://books.google.com/books?id=Us4YE8lJVYMC&q=%22regression+analysis%22 Regression Analysis] By Rudolf J. Freund, William J. Wilson, Ping Sa. Page 269.</ref> which focuses more on questions of [[statistical inference]] such as how much uncertainty is present in a curve that is fitted to data observed with random errors. Fitted curves can be used as an aid for data visualization,<ref>Visual Informatics. Edited by Halimah Badioze Zaman, Peter Robinson, Maria Petrou, Patrick Olivier, Heiko Schröder. Page 689.</ref><ref>[https://books.google.com/books?id=rdJvXG1k3HsC&q=%22Curve+fitting%22 Numerical Methods for Nonlinear Engineering Models]. By John R. Hauser. Page 227.</ref> to infer values of a function where no data are available,<ref>Methods of Experimental Physics: Spectroscopy, Volume 13, Part 1. By Claire Marton. Page 150.</ref> and to summarize the relationships among two or more variables.<ref>Encyclopedia of Research Design, Volume 1. Edited by Neil J. Salkind. Page 266.</ref> [[Extrapolation]] refers to the use of a fitted curve beyond the [[range (statistics)|range]] of the observed data,<ref>[https://books.google.com/books?id=ba0hAQAAQBAJ&q=%22Curve+fitting%22+OR+extrapolation Community Analysis and Planning Techniques]. By Richard E. Klosterman. Page 1.</ref> and is subject to a [[Uncertainty|degree of uncertainty]]<ref>An Introduction to Risk and Uncertainty in the Evaluation of Environmental Investments. DIANE Publishing. [https://books.google.com/books?id=rJ23LWaZAqsC&pg=PA69 Pg 69]</ref> since it may reflect the method used to construct the curve as much as it reflects the observed data.

For linear-algebraic analysis of data, "fitting" usually means trying to find the curve that minimizes the vertical (''y''-axis) displacement of a point from the curve (e.g., [[ordinary least squares]]). However, for graphical and image applications, geometric fitting seeks to provide the best visual fit; which usually means trying to minimize the [[orthogonal distance]] to the curve (e.g., [[total least squares]]), or to otherwise include both axes of displacement of a point from the curve. Geometric fits are not popular because they usually require non-linear and/or iterative calculations, although they have the advantage of a more aesthetic and geometrically accurate result.<ref>{{citation |first=Sung-Joon |last=Ahn |title=Geometric Fitting of Parametric Curves and Surfaces |journal=Journal of Information Processing Systems |volume=4 |issue=4 |pages=153–158 |date=December 2008 |doi=10.3745/JIPS.2008.4.4.153 |url=http://jips-k.org/dlibrary/JIPS_v04_no4_paper4.pdf |url-status=dead |archiveurl=https://web.archive.org/web/20140313084307/http://jips-k.org/dlibrary/JIPS_v04_no4_paper4.pdf |archivedate=2014-03-13 }}</ref><ref>{{citation |first1=N. |last1=Chernov |first2=H. |last2=Ma |year=2011 |contribution=Least squares fitting of quadratic curves and surfaces |title=Computer Vision |editor-first=Sota R. |editor-last=Yoshida |publisher=Nova Science Publishers |isbn=9781612093994
 |pages=285–302 |url=<!-- http://people.cas.uab.edu/~mosya/papers/CM1nova.pdf No indication of copyright --> }}</ref><ref>{{citation |first1=Yang |last1=Liu |first2=Wenping |last2=Wang |year=2008 |contribution=A Revisit to Least Squares Orthogonal Distance Fitting of Parametric Curves and Surfaces |editor1-first=F. |editor1-last=Chen |editor2-first=B. |editor2-last=Juttler |title=Advances in Geometric Modeling and Processing |series=Lecture Notes in Computer Science |volume=4975 |pages=384–397 |doi=10.1007/978-3-540-79246-8_29
 |isbn=978-3-540-79245-1|citeseerx=10.1.1.306.6085 }}</ref>