Empirical orthogonal functions: Difference between revisions

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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. 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.

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

the method of '''empirical orthogonal functions''' 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 ''i''th 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.

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 empirical orthogonal functionsEOF 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).

==See also==

*[[Source separation]]

*[[Blind signal separation]]

*[[Nonlinear dimensionality reduction]]

*[[Principal components analysis]]

*[[Source separation]]

*[[Transform coding]]

== References ==

* 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.

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

* 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|A dimension reduced approach to space-time Kalman filtering|wikle99dimensionreduction}}", ''[[Biometrika]]'' 86:815-829, 1999.

[[Category:Statistics]]