Editing B-spline (section)

== Curve fitting ==
Usually in [[curve fitting]], a set of data points is fitted with a curve defined by some mathematical function. For example, common types of curve fitting use a polynomial or a set of [[exponential function]]s. When there is no theoretical basis for choosing a fitting function, the curve may be fitted with a spline function composed of a sum of B-splines, using the method of [[least squares]].<ref>de Boor, Chapter XIV, p. 235.</ref><ref group=note>de Boor gives FORTRAN routines for least-squares fitting of experimental data.</ref> Thus, the [[objective function]] for least-squares minimization is, for a spline function of degree ''k'',
: <math>U = \sum_{\text{all}~x} \left\{ W(x)\left[y(x) - \sum_i \alpha_i B_{i,k,t}(x)\right] \right\}^2,</math>
where ''W''(''x'') is a weight, and ''y''(''x'') is the datum value at ''x''. The coefficients <math>\alpha_i</math> are the parameters to be determined. The knot values may be fixed or treated as parameters.

The main difficulty in applying this process is in determining the number of knots to use and where they should be placed. de Boor suggests various strategies to address this problem. For instance, the spacing between knots is decreased in proportion to the curvature (2nd derivative) of the data.{{Citation needed|date=November 2016}} A few applications have been published. For instance, the use of B-splines for fitting single [[Cauchy distribution|Lorentzian]] and [[Normal distribution|Gaussian]] curves has been investigated. Optimal spline functions of degrees 3–7 inclusive, based on symmetric arrangements of 5, 6, and 7 knots, have been computed and the method was applied for smoothing and differentiation of spectroscopic curves.<ref>{{cite journal |last1=Gans |first1=Peter |last2=Gill |first2=J. Bernard |year=1984 |title=Smoothing and Differentiation of Spectroscopic Curves Using Spline Functions |journal=Applied Spectroscopy |volume=38 |issue=3 |pages=370–376 |doi=10.1366/0003702844555511 |bibcode=1984ApSpe..38..370G |s2cid=96229316 }}</ref> In a comparable study, the two-dimensional version of the [[Savitzky–Golay filter]]ing and the spline method produced better results than [[moving average]] or [[Chebyshev filter]]ing.<ref>{{cite journal |last1=Vicsek |first1=Maria |last2=Neal |first2=Sharon L. |last3=Warner |first3=Isiah M. |year=1986 |title=Time-Domain Filtering of Two-Dimensional Fluorescence Data |journal=Applied Spectroscopy |volume=40 |issue=4 |pages=542–548 |doi=10.1366/0003702864508773 |url=http://www.dtic.mil/get-tr-doc/pdf?AD=ADA164954 |archive-url=https://web.archive.org/web/20170623000852/http://www.dtic.mil/get-tr-doc/pdf?AD=ADA164954 |url-status=dead |archive-date=June 23, 2017 |bibcode=1986ApSpe..40..542V |s2cid=28705788 }}</ref>