Least-squares function approximation: Difference between revisions

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In [[mathematics]], the idea of '''least squares''' can be applied where is interest into [[function approximation|approximating a given function]] by a weighted sum of other functions. The best approximation can be defined as that which minimises the difference between the original function and the approximation; for a least-squares approach the quality of the approximation is measured in terms of the squared differences the two.
 
====Functional analysis====
 
{{See also|Fourier series|Generalized Fourier series}}
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A generalization to approximation of a data set is the approximation of a function by a sum of other functions, usually an [[orthogonal functions|orthogonal set]]:<ref name=Lanczos>
 
{{cite book |title=Applied analysis |author=Cornelius Lanczos |pages =212–213 |isbn=048665656X |publisher=Dover Publications |year=1988 |edition=Reprint of 1956 Prentice-HallPrentice–Hall |url=http://books.google.com/books?id=6E85hExIqHYC&pg=PA212}}
 
</ref>
 
:<math>f(x) \approx f_n (x) = a_1 \phi _1 (x) + a_2 \phi _2(x) + \cdots + a_n \phi _n (x), \ </math>
 
with the set of functions {<math>\ \phi _j (x) </math>} an [[Orthonormal_set#Real-valued_functions|orthonormal set]] over the interval of interest, {{nowrap|say [a, b]}}: see also [[Fejér's theorem]]. The coefficients {<math>\ a_j </math>} are selected to make the magnitude of the difference ||{{nowrap|''f''''f ''<sub>''n''</sub> }}||<sup>2</sup> as small as possible. For example, the magnitude, or norm, of a function {{nowrap|''g'' (''x'' )}} over the {{nowrap|interval [a, b]}} can be defined by:<ref name=Folland>
 
{{cite book |title=Fourier analysis and its application |page =69 |chapter=Equation 3.14 |author=Gerald B Folland |url=http://books.google.com/books?id=ix2iCQ-o9x4C&pg=PA69 |isbn=0821847902 |publisher=American Mathematical Society Bookstore |year=2009 |edition=Reprint of Wadsworth and Brooks/Cole 1992}}
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{{cite book |title=Statistical methods: the geometric approach |author= David J. Saville, Graham R. Wood |chapter=§2.5 Sum of squares |page=30 |url=http://books.google.com/books?id=8ummgMVRev0C&pg=PA30 |isbn=0387975179 |year=1991 |edition=3rd |publisher=Springer}}
 
</ref>
 
:<math>\|f_n\|^2 = |a_1|^2 + |a_2|^2 + \cdots + |a_n|^2. \, </math>
 
The coefficients {''a<sub>j</sub>''} making {{nowrap begin}}||''f''''f<sub>''n''</sub>''||<sup>2</sup>{{nowrap end}} as small as possible are found to be:<ref name=Lanczos/>
 
:<math>a_j = \int_a^b \phi _j^* (x)f (x) \, dx. </math>