220
Chapter 12: Teleconnections Patterns
locally significant correlations 2 correlations are visible, pointing to a possible
dynamical link between precipitation in the Sahel and sea level pressure in
those areas.
The teleconnection technique is one of the most elementary techniques
to identify patterns. Other mathematical techniques, like the Empirical Orthogonal Function (EOF) analysis (see Section 13.3) can also reveal patterns.
One of the strong points of the tele connections maps is that there is no a priori assumption on the shape of the patterns to be found. EOFs require by
definition the patterns to be orthogonal to each other, constructing in the
solution their wave-like character to some extent. The orthogonality requirement can be relaxed if one accepts the usage of rotated EOF (see Section
13.2.4), though this option implies some other subtleties.
In a EOF analysis there is no information on the relation between grid
points. The wave-like patterns obtained through EOFs do not necessarily point to a strong coherency between the centers of action (see Section
13.3.2.) . On the other hand, the EOFs provide a measure of the relative
importance of a pattern by yielding the percentage of variance of the fiels
that can be attributed to that pattern. The information provided by EOF
analysis and teleconnections maps is complementary and they should always
be used together to avoid some danger of misinterpretations.
12.2 Singular Value Decomposition
The heterogenous teleconnection maps extend to different physical fields in
the investigation of teleconnections. It would be very desirable to have
a technique similar to EOF analysis, but that could be applied to crosscovariances between different fields. The generalization of EOF analysis to
cross-covariance is based on the Singular Value Decomposition of the crosscovariance. Eigenvalues algorithms can also be used, but they are much more
computational expensive.
The Singular Value Decomposition (SVD) is a powerful algebraic technique
to decompose arbitrary matrices in orthogonal matrices. It is based on a
result from linear operator theory (Smithies, 1970; Gohberg and Krein, 1969)
that proves that the SVD realize a completely orthogonal decomposition for
any matrix A.
The SVD of a matrix A of order mx n (with m colunms and n rows) is
its decomposition in the product of three different matrices, that is
(12.1)
The m x m matrix U and the n x n matrix V obey the orthogonality relations UU T = land VV T = I. The diagonal matrix lj is defined by
lj = diag(ul, U2· .. Umin(n,m) where, and UlJ .. , Umin(n,m) are nonnegative
2 Refer to the discussion in Chapter 9 of the subtleties involved in the assessment of the
over-all significance by plotting distributions of local significance.
Chapter 12: Teleconnections Patterns
locally significant correlations 2 correlations are visible, pointing to a possible
dynamical link between precipitation in the Sahel and sea level pressure in
those areas.
The teleconnection technique is one of the most elementary techniques
to identify patterns. Other mathematical techniques, like the Empirical Orthogonal Function (EOF) analysis (see Section 13.3) can also reveal patterns.
One of the strong points of the tele connections maps is that there is no a priori assumption on the shape of the patterns to be found. EOFs require by
definition the patterns to be orthogonal to each other, constructing in the
solution their wave-like character to some extent. The orthogonality requirement can be relaxed if one accepts the usage of rotated EOF (see Section
13.2.4), though this option implies some other subtleties.
In a EOF analysis there is no information on the relation between grid
points. The wave-like patterns obtained through EOFs do not necessarily point to a strong coherency between the centers of action (see Section
13.3.2.) . On the other hand, the EOFs provide a measure of the relative
importance of a pattern by yielding the percentage of variance of the fiels
that can be attributed to that pattern. The information provided by EOF
analysis and teleconnections maps is complementary and they should always
be used together to avoid some danger of misinterpretations.
12.2 Singular Value Decomposition
The heterogenous teleconnection maps extend to different physical fields in
the investigation of teleconnections. It would be very desirable to have
a technique similar to EOF analysis, but that could be applied to crosscovariances between different fields. The generalization of EOF analysis to
cross-covariance is based on the Singular Value Decomposition of the crosscovariance. Eigenvalues algorithms can also be used, but they are much more
computational expensive.
The Singular Value Decomposition (SVD) is a powerful algebraic technique
to decompose arbitrary matrices in orthogonal matrices. It is based on a
result from linear operator theory (Smithies, 1970; Gohberg and Krein, 1969)
that proves that the SVD realize a completely orthogonal decomposition for
any matrix A.
The SVD of a matrix A of order mx n (with m colunms and n rows) is
its decomposition in the product of three different matrices, that is
(12.1)
The m x m matrix U and the n x n matrix V obey the orthogonality relations UU T = land VV T = I. The diagonal matrix lj is defined by
lj = diag(ul, U2· .. Umin(n,m) where, and UlJ .. , Umin(n,m) are nonnegative
2 Refer to the discussion in Chapter 9 of the subtleties involved in the assessment of the
over-all significance by plotting distributions of local significance.