In
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
Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing ''signals'', such as audio signal processing, sound, image processing, images, and scientific measurements. Signal processing techniq ...
, the method of empirical orthogonal function (EOF) analysis is a decomposition of a
signal
In signal processing, a signal is a function that conveys information about a phenomenon. Any quantity that can vary over space or time can be used as a signal to share messages between observers. The ''IEEE Transactions on Signal Processing'' ...
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
Principal component analysis (PCA) is a popular technique for analyzing large datasets containing a high number of dimensions/features per observation, increasing the interpretability of data while preserving the maximum amount of information, and ...
in
geophysics
Geophysics () is a subject of natural science concerned with the physical processes and physical properties of the Earth and its surrounding space environment, and the use of quantitative methods for their analysis. The term ''geophysics'' som ...
.
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
In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbers ...
. 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
eigenvector
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
s 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
Kernel may refer to:
Computing
* Kernel (operating system), the central component of most operating systems
* Kernel (image processing), a matrix used for image convolution
* Compute kernel, in GPGPU programming
* Kernel method, in machine learnin ...
out of the data, using a fixed
kernel
Kernel may refer to:
Computing
* Kernel (operating system), the central component of most operating systems
* Kernel (image processing), a matrix used for image convolution
* Compute kernel, in GPGPU programming
* Kernel method, in machine learnin ...
. The basis functions from the eigenvectors of the kernel matrix are thus non-linear in the location of the data (see
Mercer's theorem In mathematics, specifically functional analysis, Mercer's theorem is a representation of a symmetric positive-definite function on a square as a sum of a convergent sequence of product functions. This theorem, presented in , is one of the most not ...
and the
kernel trick
In machine learning, kernel machines are a class of algorithms for pattern analysis, whose best known member is the support-vector machine (SVM). The general task of pattern analysis is to find and study general types of relations (for example ...
for more information).
See also
*
Blind signal separation
Blind may refer to:
* The state of blindness, being unable to see
* A window blind, a covering for a window
Blind may also refer to:
Arts, entertainment, and media Films
* ''Blind'' (2007 film), a Dutch drama by Tamar van den Dop
* ''Blind' ...
*
Multilinear PCA Within statistics, Multilinear principal component analysis (MPCA) is a multilinear extension of principal component analysis (PCA). MPCA is employed in the analysis of M-way arrays, i.e. a cube or hyper-cube of numbers, also informally referred ...
*
Multilinear subspace learning
Multilinear subspace learning is an approach to dimensionality reduction.M. A. O. Vasilescu, D. Terzopoulos (2003"Multilinear Subspace Analysis of Image Ensembles" "Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVP ...
*
Nonlinear dimensionality reduction
Nonlinear dimensionality reduction, also known as manifold learning, refers to various related techniques that aim to project high-dimensional data onto lower-dimensional latent manifolds, with the goal of either visualizing the data in the low-d ...
*
Orthogonal matrix
In linear algebra, an orthogonal matrix, or orthonormal matrix, is a real square matrix whose columns and rows are orthonormal vectors.
One way to express this is
Q^\mathrm Q = Q Q^\mathrm = I,
where is the transpose of and is the identity ma ...
*
Signal separation
Source separation, blind signal separation (BSS) or blind source separation, is the separation of a set of source signals from a set of mixed signals, without the aid of information (or with very little information) about the source signals or t ...
*
Singular spectrum analysis
*
Transform coding
Transform coding is a type of data compression for "natural" data like audio signals or photographic images. The transformation is typically lossless (perfectly reversible) on its own but is used to enable better (more targeted) quantization, whi ...
*
Varimax rotation In statistics, a varimax rotation is used to simplify the expression of a particular sub-space in terms of just a few major items each. The actual coordinate system is unchanged, it is the orthogonal basis that is being rotated to align with those ...
References and notes
Further reading
* Bjornsson Halldor and Silvia A. Venega
"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
"Environmental statistics for climate researchers" ''(See
''
* Christopher K. Wikle and Noel Cressie.
A dimension reduced approach to space-time Kalman filtering, ''
Biometrika
''Biometrika'' is a peer-reviewed scientific journal published by Oxford University Press for thBiometrika Trust The editor-in-chief is Paul Fearnhead (Lancaster University). The principal focus of this journal is theoretical statistics. It was es ...
'' 86:815-829, 1999.
* Donald W. Denbo and John S. Allen
"Rotary Empirical Orthogonal Function Analysis of Currents near the Oregon Coast" "J. Phys. Oceanogr.", 14, 35-46, 1984.
* David M. Kapla
"Notes on EOF Analysis"
{{DEFAULTSORT:Empirical Orthogonal Functions
Spatial analysis
Statistical signal processing