Section 15.3: MuItivariate AR(1) Process
291
In practice, it is a delicate task to choose the width f. of the frequency
interval. On the one hand, one wishes to get a stable es tim at ion of the crossspectrum matrix. Therefore, the frequency interval should be not too narrow.
The resulting matrix describes the second moment information averaged over
fo ± f.. On the other hand, one wishes to focus on a specific frequency so that
the frequency interval should be as sma11 as possible. In the limit of f. -+ 0,
the contributions of the time series at frequency 10 are obtained. In the limit
of f. -+ 00, (15.23) reduces to cross-covariance matrix of the unfiltered time
series. The complex EOF analysis merges to the conventional EOF analysis
which is discussed in Chapter 13. In this case, there would be no separation
between contributions from different frequencies.
Several "nice" mathematical features can be derived for complex EOFs.
Since the matrices in (15.23) are Hermitian, the complex EOFs form an
orthonormal basis and the eigenvalues are real. With P being the eigenvector
matrix of r;,,(f) or ~y,(O), it holds p.Tp = I. The orthogonality implies
that differently to a complex POP, a complex EOF describes only sinusoidal
spatial structures. In analogy to the conventional EOFs (Section 13.3.1),
~I
the complex EOFs maximize the variance of Y with the first eigenvector
explaining the maximum amount of total variance, the second one explaining
the maximum amount of the residual variance and so forth. If the original
.... /
.... /
....
time series Y is transformed into an eigenvector basis via Y = PZ with
~I
T ~I
Z = p. Y being the complex EOF coefficients, it can be shown that the
~I
cross-spectrum matrix of Z satisfies:
(15.24)
where ~Y' (0) = P Ap·T is used with A being the corresponding eigenvalue
matrix.
The cross-spectrum matrix of complex EOF coefficient in a frequency interval around 10 is a real diagonal matrix. The j-th diagonal component
indicate variances contributed by the j-th mode in this frequency interval.
The zero off-diagonal components indicate that cross-spectrum between any
two different complex EOF coefficients is zero in this frequency interval.
Both the frequency and the time domain approach can be used to identify
modes within a filtered time series. However, the modes describe only spectral
features in the chosen frequency interval. Furthermore, they must satisfy a11
features required for a Hermitian matrix. In practice, it is a-priori not dear
why two mo des should explain different amounts of total variance, and be
orthogonal in space and independent in time with zero co- and quadrature
spectrum between their coefficients. In some cases, it is just the goal of
dimate research to investigate wh ether two processes are related to each
other.
291
In practice, it is a delicate task to choose the width f. of the frequency
interval. On the one hand, one wishes to get a stable es tim at ion of the crossspectrum matrix. Therefore, the frequency interval should be not too narrow.
The resulting matrix describes the second moment information averaged over
fo ± f.. On the other hand, one wishes to focus on a specific frequency so that
the frequency interval should be as sma11 as possible. In the limit of f. -+ 0,
the contributions of the time series at frequency 10 are obtained. In the limit
of f. -+ 00, (15.23) reduces to cross-covariance matrix of the unfiltered time
series. The complex EOF analysis merges to the conventional EOF analysis
which is discussed in Chapter 13. In this case, there would be no separation
between contributions from different frequencies.
Several "nice" mathematical features can be derived for complex EOFs.
Since the matrices in (15.23) are Hermitian, the complex EOFs form an
orthonormal basis and the eigenvalues are real. With P being the eigenvector
matrix of r;,,(f) or ~y,(O), it holds p.Tp = I. The orthogonality implies
that differently to a complex POP, a complex EOF describes only sinusoidal
spatial structures. In analogy to the conventional EOFs (Section 13.3.1),
~I
the complex EOFs maximize the variance of Y with the first eigenvector
explaining the maximum amount of total variance, the second one explaining
the maximum amount of the residual variance and so forth. If the original
.... /
.... /
....
time series Y is transformed into an eigenvector basis via Y = PZ with
~I
T ~I
Z = p. Y being the complex EOF coefficients, it can be shown that the
~I
cross-spectrum matrix of Z satisfies:
(15.24)
where ~Y' (0) = P Ap·T is used with A being the corresponding eigenvalue
matrix.
The cross-spectrum matrix of complex EOF coefficient in a frequency interval around 10 is a real diagonal matrix. The j-th diagonal component
indicate variances contributed by the j-th mode in this frequency interval.
The zero off-diagonal components indicate that cross-spectrum between any
two different complex EOF coefficients is zero in this frequency interval.
Both the frequency and the time domain approach can be used to identify
modes within a filtered time series. However, the modes describe only spectral
features in the chosen frequency interval. Furthermore, they must satisfy a11
features required for a Hermitian matrix. In practice, it is a-priori not dear
why two mo des should explain different amounts of total variance, and be
orthogonal in space and independent in time with zero co- and quadrature
spectrum between their coefficients. In some cases, it is just the goal of
dimate research to investigate wh ether two processes are related to each
other.