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Curve fitting

Fitting of a noisy curve by an asymmetrical peak model, with an iterative process (Gauss-Newton algorithm with variable damping factor α).
Top: raw data and model.
Bottom: evolution of the normalised sum of the squares of the errors.

Curve fitting is constructing a mathematical function which best fits a set of data points.[1][2][3]

Curve fitting may involve either interpolation[4] or smoothing.[5] Using interpolation requires an exact fit to the data. With smoothing, a "smooth" function is constructed, that fit the data approximately. A related topic is regression analysis,[6][7] which focuses more on questions of statistical inference such as how much uncertainty is present in a curve that is fit to data observed with random errors.

Fitted curves can be used to help data visualization,[8][9] to guess values of a function where no data is available,[10] and to summarize the relationships among two or more variables.[11] Extrapolation refers to the use of a fitted curve beyond the range of the observed data.[12] This is subject to a degree of uncertainty,[13] since it may reflect the method used to construct the curve as much as it reflects the observed data.

Curve Fitting Media

  • Polynomial curves fitting points generated with a sine function. The black dotted line is the "true" data, the red line is a first degree polynomial, the green line is second degree, the orange line is third degree and the blue line is fourth degree.

  • Relation between wheat yield and soil salinity

  • Circle fitting with the Coope method, the points describing a circle arc, centre (1 ; 1), radius 4.

  • different models of ellipse fitting

  • Ellipse fitting minimising the algebraic distance (Fitzgibbon method).

References

  1. ↑ 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.)
  2. ↑ The Signal and the Noise: why so many predictions fail - but some don't. Nate Silver
  3. ↑ Data Preparation for Data Mining: Text. Dorian Pyle.
  4. ↑ Numerical Methods in Engineering with Python 3. Jaan Kiusalaas. p21, 24.
  5. ↑ Numerical methods of curve fitting. Philip George Guest. Page 349.
  6. ↑ Fitting Models to Biological Data Using Linear and Nonlinear Regression. Harvey Motulsky, Arthur Christopoulos.
  7. ↑ Regression analysis. Rudolf J. Freund, William J. Wilson, Ping Sa. p 269.
  8. ↑ Visual Informatics. Halimah Badioze Zaman et al. Page 689.
  9. ↑ Numerical Methods for Nonlinear Engineering Models. By John R. Hauser. Page 227.
  10. ↑ Methods of Experimental Physics: Spectroscopy, vol 13, Part 1. Claire Marton. p 150.
  11. ↑ Encyclopedia of research design, vol 1. Edited by Neil J. Salkind. p 266.
  12. ↑ Community analysis and planning techniques. Richard E. Klosterman. p 1.
  13. ↑ An introduction to risk and uncertainty in the evaluation of environmental investments. DIANE Publishing. pg 69
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