Section 15.3: Estimation of POPs and Interpretation Problems
295
is r( + f) =I r( - f) and a spectral peak is found around /1 for POP1 (solid
line in Figure 15.2a) and around h for POP2 (dashed line in Figure 15.2a).
Around the frequency range on which POP1 and POP2 oscillate, two modes
are not related (Figure 15.2b).
This example demonstrates that the fuH spectral and spatial features of
ENSO and QBO can be described by two POPs and their corresponding
eigenvalues. The spectral relationship between the two modes is described by
the cross-spectrum between the POP coefficients. Thus, we have considered
here not only the relationship between one stage of the ENSO mode and one
stage ofthe QBO mode, which is normaHy done by applying a cross-spectrum
analysis to, say, the Southern Oscillation Index and the stratospheric wind at
one level, but the relationship between two evolutions. One is characterized
by an eastward propagation of surface wind anomalies and strengthening of
SST in the central and eastern Pacific and the other by a down ward propagation of stratospheric wind anomalies.
Since both ENSO and QBO modes display only one frequency peak, their
spatial evolutions can certainly be also derived from a complex EOF analysis
of filtered data. However, if the above combined time series is considered,
the obtained modes have to satisfy the mathematical constraints discussed
in Section 15.3.4. It is impossible to study the relationship between them,
because by construction there is none.
15.4 Estimation of POPs and Interpretation
Problems
In practical situations, when only a finite time series x(t) is available, A
is estimated by first deriving the sampie lag-1 covariance matrix :E(1) =
Ltx(t + l)xT(t) and the sampie covariance matrix 1:(0) = LtX(t)XT(t)
and then forming.A = :E(1)1:(0)-1. The eigenvectors of.A are the estimated
POPs. The eigenvalues of this matrix satisfy I.AI < 1 for a stationary time
series.
It can be shown that A = 1:(1)I:(0)-1 is the solution of the minimization
problem:
n
2
L (Xj,t - AXj,t-1) = minimum
j=l
(15.25)
An estimation of multivariate spectral features of x(t) is given by the eigenvalues of A.
Let us assume that one (dynamical) process generates one POP and there
are n (dynamical) processes which generate an m-dimensional time series.
From the m x m system matrix, m POPs can be estimated. In the case
m < n, a POP analysis of the rn-dimensional time series can identify only
rn of n modes. One has to extend the spatial dimension in order to get
295
is r( + f) =I r( - f) and a spectral peak is found around /1 for POP1 (solid
line in Figure 15.2a) and around h for POP2 (dashed line in Figure 15.2a).
Around the frequency range on which POP1 and POP2 oscillate, two modes
are not related (Figure 15.2b).
This example demonstrates that the fuH spectral and spatial features of
ENSO and QBO can be described by two POPs and their corresponding
eigenvalues. The spectral relationship between the two modes is described by
the cross-spectrum between the POP coefficients. Thus, we have considered
here not only the relationship between one stage of the ENSO mode and one
stage ofthe QBO mode, which is normaHy done by applying a cross-spectrum
analysis to, say, the Southern Oscillation Index and the stratospheric wind at
one level, but the relationship between two evolutions. One is characterized
by an eastward propagation of surface wind anomalies and strengthening of
SST in the central and eastern Pacific and the other by a down ward propagation of stratospheric wind anomalies.
Since both ENSO and QBO modes display only one frequency peak, their
spatial evolutions can certainly be also derived from a complex EOF analysis
of filtered data. However, if the above combined time series is considered,
the obtained modes have to satisfy the mathematical constraints discussed
in Section 15.3.4. It is impossible to study the relationship between them,
because by construction there is none.
15.4 Estimation of POPs and Interpretation
Problems
In practical situations, when only a finite time series x(t) is available, A
is estimated by first deriving the sampie lag-1 covariance matrix :E(1) =
Ltx(t + l)xT(t) and the sampie covariance matrix 1:(0) = LtX(t)XT(t)
and then forming.A = :E(1)1:(0)-1. The eigenvectors of.A are the estimated
POPs. The eigenvalues of this matrix satisfy I.AI < 1 for a stationary time
series.
It can be shown that A = 1:(1)I:(0)-1 is the solution of the minimization
problem:
n
2
L (Xj,t - AXj,t-1) = minimum
j=l
(15.25)
An estimation of multivariate spectral features of x(t) is given by the eigenvalues of A.
Let us assume that one (dynamical) process generates one POP and there
are n (dynamical) processes which generate an m-dimensional time series.
From the m x m system matrix, m POPs can be estimated. In the case
m < n, a POP analysis of the rn-dimensional time series can identify only
rn of n modes. One has to extend the spatial dimension in order to get