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{{Technical|date=February 2022}}
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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. It is the same as performing a [[principal components analysis]] on the data, except that the EOF method finds both [[time series]] and [[Three-dimensional space|spatial]] patterns. The term is also interchangeable with the geographically weighted [[Principal components analysis|PCAs]] in [[geophysics]].<ref name=eofa>{{cite web▼
}}
▲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
| 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
|
}}</ref>
The ''i'' <sup>th</sup> basis function is chosen to be orthogonal to the basis functions from the first through ''i'' − 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).
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* [[Nonlinear dimensionality reduction]]
* [[Orthogonal matrix]]
* [[
* [[Singular spectrum analysis]]
* [[Transform coding]]
* [[Varimax rotation]]
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* Bjornsson Halldor and Silvia A. Venegas [http://brunnur.vedur.is/pub/halldor/TEXT/eofsvd.html "A manual for EOF and SVD analyses of climate data"], McGill University, CCGCR Report No. 97-1, Montréal, Québec, 52pp., 1997.
* David B. Stephenson and Rasmus E. Benestad. [http://www.gfi.uib.no/~nilsg/kurs/notes/ "Environmental statistics for climate researchers"]. ''(See: [http://www.gfi.uib.no/~nilsg/kurs/notes/node87.html "Empirical Orthogonal Function analysis"])''
* Christopher K. Wikle and Noel Cressie. "[
* Donald W. Denbo and John S. Allen. [http://journals.ametsoc.org/doi/pdf/10.1175/1520-0485(1984)014%3C0035%3AREOFAO%3E2.0.CO%3B2 "Rotary Empirical Orthogonal Function Analysis of Currents near the Oregon Coast"], "J. Phys. Oceanogr.", 14,
* David M. Kaplan [https://web.archive.org/web/20200701033210/https://websites.pmc.ucsc.edu/~dmk/notes/EOFs/EOFs.html] "Notes on EOF Analysis"
{{DEFAULTSORT:Empirical Orthogonal Functions}}
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