236
13.3
Ghapter 13: Spatial Patterns: EOFs and GGA
Empirieal, Orthogonal Functions
For the sake of simplicity we assurne in this Section that the expectation of
the considered random vector X is zero: i1 = O. Then the covariance matrix
of Xis given by :E = E(XX T ).
The vector X may represent very different sets of numbers, such as
• Observations of different parameters at one location (such as daily mean
temperature, sunshine, wind speed etc.)
• Gridpoint values of a continuous field which was spatially discretized on
a regular grid (as is often the case for horizontal distributions) or on an
irregular grid (such as the vertical discretization in GCMs).
• Observations of the same parameter (such as temperature) at irregularly
distributed stations (see the example of Central European temperature
in Section 13.3.4; also Briffa dealt with this case in Section 5.6.1 when
he considered tree ring data from different sites).
There is some confusion with the terms, since several alternative sets of
expressions are in use for the same object. What is labelled an EOF here
is also named a principal vector or a loading, whereas EOF coefficients are
sometimes principal components (for instance in Chapter 8) or scores. 5
13.3.1 Definition of EOFs
Empirical Orthogonal Functions are defined as that set of K orthogonal vectors (i.e., ple T pi = Clei) which minimize the variance of the residual n in
(13.1).6 Because ofthe enforced orthogonality the coefficients ä are given by
(13.8).
The EOFs are constructed consecutively: In a first step the pattern pl of
unit length (plT pl = 1) is identified which minimizes
(13.16)
After the first EOF pl is determined, the second EOF is derived as the
pattern minimizing
5 The expressions "principal vector" sterns !rom the geoetrical interpretation that these
vectors are the principal vectors of an ellipsoid described by the covariance matrix. The
tenns "loading" and "scores" come from factor analysis, a technique widely used in social
sciences. The tenn "EOF" seems to be in use only in meteorology and oceanography.
6The approach of minimizing the variance of the residual has a mathematical background: the variance of the residual is a measure of the "misfit" og X by L~=l Otk(t)pk.
Other such measures of misfit could be chosen but this quadratic fonn allows for a simple
mathematical solution of the minimization problem.
13.3
Ghapter 13: Spatial Patterns: EOFs and GGA
Empirieal, Orthogonal Functions
For the sake of simplicity we assurne in this Section that the expectation of
the considered random vector X is zero: i1 = O. Then the covariance matrix
of Xis given by :E = E(XX T ).
The vector X may represent very different sets of numbers, such as
• Observations of different parameters at one location (such as daily mean
temperature, sunshine, wind speed etc.)
• Gridpoint values of a continuous field which was spatially discretized on
a regular grid (as is often the case for horizontal distributions) or on an
irregular grid (such as the vertical discretization in GCMs).
• Observations of the same parameter (such as temperature) at irregularly
distributed stations (see the example of Central European temperature
in Section 13.3.4; also Briffa dealt with this case in Section 5.6.1 when
he considered tree ring data from different sites).
There is some confusion with the terms, since several alternative sets of
expressions are in use for the same object. What is labelled an EOF here
is also named a principal vector or a loading, whereas EOF coefficients are
sometimes principal components (for instance in Chapter 8) or scores. 5
13.3.1 Definition of EOFs
Empirical Orthogonal Functions are defined as that set of K orthogonal vectors (i.e., ple T pi = Clei) which minimize the variance of the residual n in
(13.1).6 Because ofthe enforced orthogonality the coefficients ä are given by
(13.8).
The EOFs are constructed consecutively: In a first step the pattern pl of
unit length (plT pl = 1) is identified which minimizes
(13.16)
After the first EOF pl is determined, the second EOF is derived as the
pattern minimizing
5 The expressions "principal vector" sterns !rom the geoetrical interpretation that these
vectors are the principal vectors of an ellipsoid described by the covariance matrix. The
tenns "loading" and "scores" come from factor analysis, a technique widely used in social
sciences. The tenn "EOF" seems to be in use only in meteorology and oceanography.
6The approach of minimizing the variance of the residual has a mathematical background: the variance of the residual is a measure of the "misfit" og X by L~=l Otk(t)pk.
Other such measures of misfit could be chosen but this quadratic fonn allows for a simple
mathematical solution of the minimization problem.