Section 15.3: Multivariate AR(l) Process
289
period 1j = Jj. The real and imaginary patterns describe together spatially
propagating structures (more detailed description see H. von Storch et al.,
1995). Since they are normally not orthogonal to each other but may have
any spatial phase relationship, the POPs are not restricted to describing
sinusoidal propagating waves.
This points to a limitation of a POP model in describing a pure standing
oscillation. One extreme of the spatial phase relationship between two patterns is the zero-phase relationship. In this limit, the two patterns coincide
and we have a pure standing oscillation with weH defined time period and
fixed spatial structure. The amplitude of the spatial structure can become
larger or smaHer foHowing an oscillation, but the form of the spatial structure
is always the same. Such a standing oscillation can only be identified by one
pattern, in a POP model by a real POP. Since the spectrum of a real POP
is one which has no spectral peak within the frequency interior (15.16), the
oscillation frequency cannot be detected by a POP model. The solution of
this problem is suggested by Bürger (1993).
15.3.4 Eigenstructure of Cross-Spectrum Matrix
at a Frequency Interval: Complex EOFs
The cross spectrum matrix of a multivariate time series can also be defined
as the square of the Fourier transform of the time series:
(15.18)
where the rn-dimensional complex vector F(f) is the Fourier transform of i
at frequency f. One has
1=+00
X(t) = L F(f)ei27rlt
(15.19)
1=-00
Defining the random vector Y by:
Y(t) = 1/2 [X(t) + iX\t)]
(15.20)
where i
h
(t) = Ej:O( -i)F(f)ei27rlt + E~=-oo iF(f)ei27rlt - iF(O) is the
Hilbert transform of X, it can be shown
1=00
Y(t) = E F(f)ei27r/f
(15.21)
1=0
Thus (15.18) is also the cross-spectrum matrix of the random vector Y for
positive frequencies f .
Equation (15.18) is formallyequivalent to (15.4) and satisfies also (15.5).
As discussed in Section 15.2, it is difficult to investigate the cross-spectrum
289
period 1j = Jj. The real and imaginary patterns describe together spatially
propagating structures (more detailed description see H. von Storch et al.,
1995). Since they are normally not orthogonal to each other but may have
any spatial phase relationship, the POPs are not restricted to describing
sinusoidal propagating waves.
This points to a limitation of a POP model in describing a pure standing
oscillation. One extreme of the spatial phase relationship between two patterns is the zero-phase relationship. In this limit, the two patterns coincide
and we have a pure standing oscillation with weH defined time period and
fixed spatial structure. The amplitude of the spatial structure can become
larger or smaHer foHowing an oscillation, but the form of the spatial structure
is always the same. Such a standing oscillation can only be identified by one
pattern, in a POP model by a real POP. Since the spectrum of a real POP
is one which has no spectral peak within the frequency interior (15.16), the
oscillation frequency cannot be detected by a POP model. The solution of
this problem is suggested by Bürger (1993).
15.3.4 Eigenstructure of Cross-Spectrum Matrix
at a Frequency Interval: Complex EOFs
The cross spectrum matrix of a multivariate time series can also be defined
as the square of the Fourier transform of the time series:
(15.18)
where the rn-dimensional complex vector F(f) is the Fourier transform of i
at frequency f. One has
1=+00
X(t) = L F(f)ei27rlt
(15.19)
1=-00
Defining the random vector Y by:
Y(t) = 1/2 [X(t) + iX\t)]
(15.20)
where i
h
(t) = Ej:O( -i)F(f)ei27rlt + E~=-oo iF(f)ei27rlt - iF(O) is the
Hilbert transform of X, it can be shown
1=00
Y(t) = E F(f)ei27r/f
(15.21)
1=0
Thus (15.18) is also the cross-spectrum matrix of the random vector Y for
positive frequencies f .
Equation (15.18) is formallyequivalent to (15.4) and satisfies also (15.5).
As discussed in Section 15.2, it is difficult to investigate the cross-spectrum