In statistics and signal processing, the method of empirical orthogonal functions is a decomposition of a signal or data set in terms of orthogonalbasis functions which are determined from the data. The ith 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. Thus this method has much in common with the method of kriging in geostatistics, and Gaussian process models.
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 frequencies. In some cases the two methods may yield essentially the same results.
The basis functions are typically found by computing the eigenvectors 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. 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).
Christopher K. Wikle and Noel Cressie. "A dimension reduced approach to space-time Kalman filtering", Biometrika 86:815-829, 1999. (Also at CiteSeer: [1] (http://citeseer.nj.nec.com/wikle99dimensionreduction.html))
David B. Stephenson and Rasmus E. Benestad. "Environmental statistics for climate researchers" (http://www.gfi.uib.no/~nilsg/kurs/notes/). (See: "Empirical Orthogonal Function analysis" (http://www.gfi.uib.no/~nilsg/kurs/notes/node87.html))
EOFs are a transform of the data; the original set of numbers is transformed into a different set with some desirable properties.
Hence the term `empirical'; we still have an orthogonal basis, like the Fourier or Legendre bases, but whose members are not chosen based on analytic considerations, but based on maximization of the projection of the data on them.
The 3rd, while clearly structured (unlike the case with the 2 orthogonal cosines), is a combination of the generating patterns, not an individual pattern.
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The eigenfunctions of the covariance matrix of the various state parameters (temperature, pressure, sea ice, precipitation, etcetera) are termed "empiricalorthogonalfunctions" or "EOFs" and the set of corresponding eigenvalues are the "EOFvariance spectra".
EOFs and their variance spectra are of particular interest, because they represent basic information on the parent probability distribution of the climate.
Empirical estimates of these are based on a finite number, say n, realizations of the the instantaneous state of a geophysical field during a limited time period, and the field is sampled at a finite number of points, say p, covering a limited spatial domain.
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