Empirical orthogonal functions: Difference between revisions

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In [[statistics]] and [[signal processing]],

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the method of '''empirical orthogonal functions''' is a decomposition of a [[signal]] or data set in terms of [[orthogonal]] [[basis function]]s which are determined from the data.

{{One source|date=February 2022}}

The ''i''th basis function is chosen to be orthogonal to the basis functions from the first through ''i'' − 1,

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and to minimize the residual [[variance]].

In [[statistics]] and [[signal processing]], the method of '''empirical orthogonal functionsfunction''' ('''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

That is, the basis functions are chosen to be different from each other,

| last1 = Stephenson

and to account for as much variance as possible.

| first1 = David B.

Thus this method has much in common with the method of [[kriging]] in [[geostatistics]], and [[Gaussian process]] models.

| 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'' ^th 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 empirical orthogonal functions 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 method of empiricalEOF orthogonal functionsanalysis 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. This is the same as performing [[principal components analysis]] on the data. A more advanced technique is to form a [[kernel matrix]] out of the data, using a fixed [[kernel (mathematics)|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).

The basis functions are typically found by computing the [[eigenvector]]s of the [[covariance matrix]] of the data set. This is the same as performing [[principal components analysis]] on the data. A more advanced technique is to form a [[kernel (matrix)|kernel]] out of the data, using a fixed [[kernel (mathematicsstatistics)|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).

==See also==

* [[Blind signal separation]]

* [[Multilinear PCA]]

* [[Multilinear subspace learning]]

* [[Nonlinear dimensionality reduction]]

* [[Orthogonal matrix]]

* [[SourceSignal separation]]

* [[Singular spectrum analysis]]

* [[Transform coding]]

* [[Varimax rotation]]

==References and notes==

*[[Source separation]]

<references />

*[[Blind signal separation]]

*[[Nonlinear dimensionality reduction]]

*[[Transform coding]]

== References ==

* Christopher K. Wikle and Noel Cressie. "{{citeseer|A dimension reduced approach to space-time Kalman filtering|wikle99dimensionreduction}}", ''[[Biometrika]]'' 86:815-829, 1999.

==Further reading==

* 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. "{{citeseer|[https://dx.doi.org/10.1093/biomet/86.4.815 A dimension reduced approach to space-time Kalman filtering|wikle99dimensionreduction}}]", ''[[Biometrika]]'' 86:815-829, 1999.

* 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, 35–46, 1984.

* David M. Kaplan [https://web.archive.org/web/20200701033210/https://websites.pmc.ucsc.edu/~dmk/notes/EOFs/EOFs.html] "Notes on EOF Analysis"

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[[Category:StatisticsSpatial analysis]]

In [[statistics]]Category:Statistical and [[signal processing]],