250
Ghapter 13: Spatial Patterns: EOFs and GGA
• between ar and ar
are zero for all k =/:- I.
3. The correlation between af and o{ is maximum.
4. The correlation between af and ar is the maximum under the constraints of 2) and 3). The correlations for the higher indexed pairs of
coefficients satisfy similar constraints (namely of being maximum while
being independent with all previously determined coefficients.)
It can be shown (see, for instance, Zorita et al. , 1992) that the adjoint patterns
are the eigenvectors of somewhat complicated looking matrices, namely:
(13.42)
Here ~ X and ~y are the covariance matrices of X and Y. ~ Xy is the crosscovariancematrixofXand Y,Le., ~Xy = E(Xy
T ) ifE(X) = E(Y) = O.
The matrix Ax is a mx x mx matrix and Ay is a my x my matrix. The
two matrices A x and Ay may be written as products B1B2 and B2B1 with
two matrices B1 and B2 • Therefore the two matrices share the same non zero
eigenvalues, and if il is an eigenvector ofAx with an eigenvalue A =/:- 0 then
~y1~~yil is an eigenvector of Ay with the same eigenvalue.
Note that for univariate random variables X = X and Y = Y the two
matrices Ax and Ay in (13.42) reduce to the squared correlations between
X and Y.
The k-adjoint pattern is given by the eigenvector with the k-largest eigenvalue of A. The correlation between ar and ar is given by the k-th largest
nonzero eigenvalue ofAx or Ay.
The covariance between the "Canonical Correlation Coefficients" ar and
the original vector X is given by
(13.43)
... T
k
so that, because of ar = X (ix )A:
-k '["I (-k)
py = ~y Py A
(13.44)
Thus, to determine the "Canonical Correlation Patterns" (CCP) and the
canonical correlation coefficients one has first to calculate the covariance matri ces and cross covariance matrices. From products ofthese matrices (13.42)
the adjoint patterns are derived as eigenvectors. With the adjoint pattern the
CCPs are calculated via (13.44) and the coefficients through (13.41). Because
of the specific form of the matrices, it is advisable to solve the eigenvector
problem for the sm aller one of the two matrices.
Ghapter 13: Spatial Patterns: EOFs and GGA
• between ar and ar
are zero for all k =/:- I.
3. The correlation between af and o{ is maximum.
4. The correlation between af and ar is the maximum under the constraints of 2) and 3). The correlations for the higher indexed pairs of
coefficients satisfy similar constraints (namely of being maximum while
being independent with all previously determined coefficients.)
It can be shown (see, for instance, Zorita et al. , 1992) that the adjoint patterns
are the eigenvectors of somewhat complicated looking matrices, namely:
(13.42)
Here ~ X and ~y are the covariance matrices of X and Y. ~ Xy is the crosscovariancematrixofXand Y,Le., ~Xy = E(Xy
T ) ifE(X) = E(Y) = O.
The matrix Ax is a mx x mx matrix and Ay is a my x my matrix. The
two matrices A x and Ay may be written as products B1B2 and B2B1 with
two matrices B1 and B2 • Therefore the two matrices share the same non zero
eigenvalues, and if il is an eigenvector ofAx with an eigenvalue A =/:- 0 then
~y1~~yil is an eigenvector of Ay with the same eigenvalue.
Note that for univariate random variables X = X and Y = Y the two
matrices Ax and Ay in (13.42) reduce to the squared correlations between
X and Y.
The k-adjoint pattern is given by the eigenvector with the k-largest eigenvalue of A. The correlation between ar and ar is given by the k-th largest
nonzero eigenvalue ofAx or Ay.
The covariance between the "Canonical Correlation Coefficients" ar and
the original vector X is given by
(13.43)
... T
k
so that, because of ar = X (ix )A:
-k '["I (-k)
py = ~y Py A
(13.44)
Thus, to determine the "Canonical Correlation Patterns" (CCP) and the
canonical correlation coefficients one has first to calculate the covariance matri ces and cross covariance matrices. From products ofthese matrices (13.42)
the adjoint patterns are derived as eigenvectors. With the adjoint pattern the
CCPs are calculated via (13.44) and the coefficients through (13.41). Because
of the specific form of the matrices, it is advisable to solve the eigenvector
problem for the sm aller one of the two matrices.