Editing Empirical orthogonal functions (section)

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In [[statistics]] and [[signal processing]], the method of '''empirical orthogonal function''' ('''EOF''') analysis is a decomposition of a [[signal processing|signal]] or data set in terms of [[orthogonal]] [[basis function]]s which are determined from the data.  The term is also interchangeable with the geographically weighted [[Principal components analysis]] in [[geophysics]].<ref name=eofa>{{cite web
  | last1 = Stephenson
  | first1 = David B.
  | last2 =  Benestad
  | first2 =  Rasmus E.
  | title = Empirical Orthogonal Function analysis
  | work = Environmental statistics for climate researchers
  | date =2000-09-02
  | url = http://www.uib.no/people/ngbnk/kurs/notes/node87.html
  | access-date = 2013-02-28
  }}</ref>

The ''i'' <sup>th</sup> basis function is chosen to be orthogonal to the basis functions from the first through ''i'' &minus; 1, and to minimize the residual [[variance]]. That is, the basis functions are chosen to be different from each other, and to account for as much variance as possible.

The method of EOF analysis is similar in spirit to [[harmonic analysis]], but harmonic analysis typically uses predetermined orthogonal functions, for example, sine and cosine functions at fixed [[frequency|frequencies]]. In some cases the two methods may yield essentially the same results.

The basis functions are typically found by computing the [[eigenvector]]s of the [[covariance matrix]] of the data set. A more advanced technique is to form a [[kernel (matrix)|kernel]] out of the data, using a fixed [[kernel (statistics)|kernel]]. The basis functions from the eigenvectors of the kernel matrix are thus non-linear in the location of the data (see [[Mercer's theorem]] and the [[kernel trick]] for more information).