Empirical orthogonal functions: Difference between revisions

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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 [[spatial]] patterns. The term is also interchangeable with the geographically weighted [[Principal components analysis|PCAs]] in [[geophysics]] <ref name=eofa>{{cite web

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| last = Stephenson

{{One source|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 [[spatial]] patterns. The term is also interchangeable with the geographically weighted [[Principal components analysis|PCAs]] in [[geophysics]] .<ref name=eofa>{{cite web

| coauthors =

| lastlast1 = Stephenson

| firstfirst1 = David B.

| last2 = Benestad

| first2 = Rasmus E.

| title = Empirical Orthogonal Function analysis

| work = Environmental statistics for climate researchers

| work =

| publisherdate =2000-09-02

| date =

| url = http://www.uib.no/people/ngbnk/kurs/notes/node87.html

| doiaccess-date = 2013-02-28

}}</ref>.

| accessdate = 20 September 2008

}}</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.

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

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 (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==

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<references />

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* Bjornsson Halldor and Silvia A. Venegas [http://www.vedur.is/~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.

==Further reading==

* Bjornsson Halldor and Silvia A. Venegas [http://wwwbrunnur.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. "[httphttps://dx.doi.org/10.1093/biomet/86.4.815 A dimension reduced approach to space-time Kalman filtering]", ''[[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 EmpericalEmpirical Orthogonal Function Analysis of Currents near the Oregon Coast"], "J. Phys. Oceanogr.", 14, 35-4635–46, 1984.

* Christopher K. Wikle and Noel Cressie. "[http://dx.doi.org/10.1093/biomet/86.4.815 A dimension reduced approach to space-time Kalman filtering]", ''[[Biometrika]]'' 86:815-829, 1999.

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

* 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 Emperical Orthogonal Function Analysis of Currents near the Oregon Coast"], "J. Phys. Oceanogr.", 14, 35-46, 1984.

{{DEFAULTSORT:Empirical Orthogonal Functions}}

[[Category:Spatial data analysis]]

[[Category:TimeStatistical seriessignal analysisprocessing]]

[[ja:経験的直交関数]]