statistics Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...

and signal processing, the method of empirical orthogonal function (EOF) analysis is a decomposition of a signal or data set in terms of

orthogonal In mathematics, orthogonality is the generalization of the geometric notion of ''perpendicularity''. By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in ...

basis function In mathematics, a basis function is an element of a particular basis for a function space. Every function in the function space can be represented as a linear combination of basis functions, just as every vector in a vector space can be represen ...

s which are determined from the data. The term is also interchangeable with the geographically weighted Principal components analysis in geophysics. 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. The method of EOF analysis is similar in spirit to

harmonic analysis Harmonic analysis is a branch of mathematics concerned with the representation of Function (mathematics), functions or signals as the Superposition principle, superposition of basic waves, and the study of and generalization of the notions of Fo ...

, but harmonic analysis typically uses predetermined orthogonal functions, for example, sine and cosine functions at fixed

frequencies Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is eq ...

. 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 In probability theory and statistics, a covariance matrix (also known as auto-covariance matrix, dispersion matrix, variance matrix, or variance–covariance matrix) is a square matrix giving the covariance between each pair of elements of ...

of the data set. A more advanced technique is to form a kernel 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).

See also

References and notes

Further reading