454
0.4 ,-----~------_,------~------._------~----_.
C\I
I
,...
0.3
0.2
0.1
u.
0.0
o -0.1
l.U -0.2
-0.3
-0.4
0.3
,.
0.2
('I)
0.1
u.
0.0
o -0.1
l.U -0.2
-0.3
- - EOF1
- - EOF2
-0.4 L-L~~-----'-----'--~~-'--~~--'-----"~---,--~-----'-----,--~~---L-~~---'------.!
o
10
20
30
Rankk
40
50
60
Figure 4: First four EOFs of the SOl time series. The EOFs are grouped into two pairs:
(1,2) and (3,4).
the nonlinear counterpart of a sine-cosine pair in standard Fourier analysis
of linear problems. In the terminology of our Sec. 1 here, such a pair gives
a representation of a ghost limit cycle.
Projecting the time series onto each EOF yields the corresponding principal components (PCs) Ak :
M
Ak(t) = L X(t + j)pkU),
j=l
(8)
Figure 5 shows the variations of the four leading PCs. Again, the pairs
of PCs (1,2) and (3,4) are in quadrature and strongly suggest periodic
variability at two different periods, of about 4 and 2 years, respectively.
We can reconstruct that part of a time series that is associated with a
single EOF or several by combining the associated PCs:
RK.(t) = M1 L I: Ak(t - j)PkU),
t kEIC j=l
(9)
where J( is the set of EOFs on which the reconstruction is based, and
M t is a normalization factor which is M for the central part of the time
series and has slightly different values near its endpoints (Ghil and Vautard
1991, Vautard et al. 1992). The reconstructed components (RCs) have the