Section 13.3: Empirical Orthogonal Functions
243
(
X1(1) x1(2)
x2(1) x2(2)
X=
.
.
·
.
·
.
xm (1) x m (2)
(13.34)
The n columns of the n X m data matrix X are the sampie vectors
x(j), j = 1, ... n; the rows mark the m coordinates in the original space.
The matrix product X X T is a quadratic m x m matrix even if X itself
is not quadratic. The product X T X, on the other hand, is a n x nmatrix. The above mentioned trick teIls us that one should calculate
the eigenvalues and eigenvectors of the smaIler of the two matrices X T X
and X X T . In practical situations we often have the number of sampies
n being much smaIler than the number of components m.
A byproduct of the "trick" is the finding that we can estimate only the
first n EOFs (or n - 1 if we have subtracted the overall mean to get
anomalies) of the m EOFs of the m-variate random variable.
• Numerically, the EOF analysis of a finite set of observed vectors may by
done by a Singular Value Decomposition (SVD, see Chapter 14).
(
a1(1) a2(1)
a1(2) a2(2)
X=
.
.
·
.
·
.
a1(n) a2(n)
. .. an(1))
... an ( n )
( ~
~ ) T
'" -11 I-m
.
v p ... p
an(n)
(13.35)
with a rectangular n x m matrix V with zero elements outside the diagonal and positive elements on the diagonal: dij = SiOij ~ O. The
quadratic n x n and m x m matrices to the right and left of V are
orthogonal.
The eigenvalues of the estimated covariance matrix are ~j = s1. The
coefficients of the estimated EOFs are given by aj = SjO:j. Again, there
is a maximumofmin(n, m) nonzero sj-values so that at most minen, m)
useful EOFs can be determined.
• The choice of the numerical algorithm is irrelevant for the mathematical
character of the product - EOFs are the eigenvectors of the estimated
covariance matrix independently if the number crunching has been done
via the eigenvector problem or via SVD.
As always, when estimating parameters of a random variable from a finite
sampie of observations, one may ask how accurate the estimation probably
lS:
• Biases
If Ak is an estimate ofthe true eigenvalue Ak and ak the EOF coefficient
243
(
X1(1) x1(2)
x2(1) x2(2)
X=
.
.
·
.
·
.
xm (1) x m (2)
(13.34)
The n columns of the n X m data matrix X are the sampie vectors
x(j), j = 1, ... n; the rows mark the m coordinates in the original space.
The matrix product X X T is a quadratic m x m matrix even if X itself
is not quadratic. The product X T X, on the other hand, is a n x nmatrix. The above mentioned trick teIls us that one should calculate
the eigenvalues and eigenvectors of the smaIler of the two matrices X T X
and X X T . In practical situations we often have the number of sampies
n being much smaIler than the number of components m.
A byproduct of the "trick" is the finding that we can estimate only the
first n EOFs (or n - 1 if we have subtracted the overall mean to get
anomalies) of the m EOFs of the m-variate random variable.
• Numerically, the EOF analysis of a finite set of observed vectors may by
done by a Singular Value Decomposition (SVD, see Chapter 14).
(
a1(1) a2(1)
a1(2) a2(2)
X=
.
.
·
.
·
.
a1(n) a2(n)
. .. an(1))
... an ( n )
( ~
~ ) T
'" -11 I-m
.
v p ... p
an(n)
(13.35)
with a rectangular n x m matrix V with zero elements outside the diagonal and positive elements on the diagonal: dij = SiOij ~ O. The
quadratic n x n and m x m matrices to the right and left of V are
orthogonal.
The eigenvalues of the estimated covariance matrix are ~j = s1. The
coefficients of the estimated EOFs are given by aj = SjO:j. Again, there
is a maximumofmin(n, m) nonzero sj-values so that at most minen, m)
useful EOFs can be determined.
• The choice of the numerical algorithm is irrelevant for the mathematical
character of the product - EOFs are the eigenvectors of the estimated
covariance matrix independently if the number crunching has been done
via the eigenvector problem or via SVD.
As always, when estimating parameters of a random variable from a finite
sampie of observations, one may ask how accurate the estimation probably
lS:
• Biases
If Ak is an estimate ofthe true eigenvalue Ak and ak the EOF coefficient