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Curve fitting

"Best fit" redirects here. For placing ("fitting") variable-sized objects in storage, see Fragmentation (computing).
Fitting of a noisy curve by an asymmetrical peak model, with an iterative process (Gauss–Newton algorithm with variable damping factor α).

Curve fitting[1][2] is the process of constructing a curve, or mathematical function, that has the best fit to a series of data points,[3] possibly subject to constraints.[4][5] Curve fitting can involve either interpolation,[6][7] where an exact fit to the data is required, or smoothing,[8][9] in which a "smooth" function is constructed that approximately fits the data. A related topic is regression analysis,[10][11] 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,[12][13] to infer values of a function where no data are available,[14] and to summarize the relationships among two or more variables.[15] Extrapolation refers to the use of a fitted curve beyond the range of the observed data,[16] and is subject to a degree of uncertainty[17] 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.[18][19][20]

  1. ^ Sandra Lach Arlinghaus, PHB Practical Handbook of Curve Fitting. CRC Press, 1994.
  2. ^ William M. Kolb. Curve Fitting for Programmable Calculators. Syntec, Incorporated, 1984.
  3. ^ 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.)
  4. ^ The Signal and the Noise: Why So Many Predictions Fail-but Some Don't. By Nate Silver
  5. ^ Data Preparation for Data Mining: Text. By Dorian Pyle.
  6. ^ Numerical Methods in Engineering with MATLAB®. By Jaan Kiusalaas. Page 24.
  7. ^ Numerical Methods in Engineering with Python 3. By Jaan Kiusalaas. Page 21.
  8. ^ Numerical Methods of Curve Fitting. By P. G. Guest, Philip George Guest. Page 349.
  9. ^ See also: Mollifier
  10. ^ Fitting Models to Biological Data Using Linear and Nonlinear Regression. By Harvey Motulsky, Arthur Christopoulos.
  11. ^ Regression Analysis By Rudolf J. Freund, William J. Wilson, Ping Sa. Page 269.
  12. ^ Visual Informatics. Edited by Halimah Badioze Zaman, Peter Robinson, Maria Petrou, Patrick Olivier, Heiko Schröder. Page 689.
  13. ^ Numerical Methods for Nonlinear Engineering Models. By John R. Hauser. Page 227.
  14. ^ Methods of Experimental Physics: Spectroscopy, Volume 13, Part 1. By Claire Marton. Page 150.
  15. ^ Encyclopedia of Research Design, Volume 1. Edited by Neil J. Salkind. Page 266.
  16. ^ Community Analysis and Planning Techniques. By Richard E. Klosterman. Page 1.
  17. ^ An Introduction to Risk and Uncertainty in the Evaluation of Environmental Investments. DIANE Publishing. Pg 69
  18. ^ Ahn, Sung-Joon (December 2008), "Geometric Fitting of Parametric Curves and Surfaces" (PDF), Journal of Information Processing Systems, 4 (4): 153–158, doi:10.3745/JIPS.2008.4.4.153, archived from the original (PDF) on 2014-03-13
  19. ^ Chernov, N.; Ma, H. (2011), "Least squares fitting of quadratic curves and surfaces", in Yoshida, Sota R. (ed.), Computer Vision, Nova Science Publishers, pp. 285–302, ISBN 9781612093994
  20. ^ Liu, Yang; Wang, Wenping (2008), "A Revisit to Least Squares Orthogonal Distance Fitting of Parametric Curves and Surfaces", in Chen, F.; Juttler, B. (eds.), Advances in Geometric Modeling and Processing, Lecture Notes in Computer Science, vol. 4975, pp. 384–397, CiteSeerX 10.1.1.306.6085, doi:10.1007/978-3-540-79246-8_29, ISBN 978-3-540-79245-1

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