Section 13.3: Empirical Orthogonal Functions
247
Figure 13.4: EOF coeflicient time series al (t) and a2(t) of the first two EOFs
of winter mean temperature at 11 Central European stations. Note that the
time series have been normalized to one so that the information about the
strength of the variation is carried by the patterns. (From Werner and H.
von Storch, 1993).
1690 1900 1910 U20 1930 1940 1950 1860 1870 1860 1880
I
•
I
•
t
'
I
,
!
!
I
,
!
'
!
,
,
!
•
,
,
•
I
I
I
2nd: EOF
:
2
I
.1. ___ _
I
~'~_4+A~~~~~hr~~~~~~~--To
I
--~---1
I
I
-2
I
I
I
I
I
3 ---~-------~-------~-------~-------~--- -3
I
•
t
I
2
I
I
I
I
O+---~~~~~~~~~~~~~~_1r
-1
-2
1880 1800 1810 1820 1830 \840 1850 1980 1970 1980 1880
two intervals - as a demonstration we show in Figure 13.3 the first two EOFs
for both periods. The representation of the patterns deviates from the definition introduced above: The contribution of the k-th EOF to the fuH signal is
given by ak(t)pk. According to our definitions the variance of the coefficient
is given by the k-th eigenvalue Ak and the vector has unit length. For a better display of the results sometimes a different normalization is convenient,
namely akpk = (akj..;J:k) x (pk..;J:k) = akP'k. In this normalization the coefficient time series has variance one for all indices k and the relative strength
of the signal is in the patterns pi. A typical coefficient is a' = 1 so that the
typical reconstructed signal is pi. In this format the first two EOFs and
their time coefficients obtained in the analysis of Central European winter
temperature are shown in Figures 13.3 and 13.4.
In both time periods the first EOF has a positive sign at all locations,
represents about 90% of the total variance and exhibits "typical anomalies"
of the order of 1 - 2K. The second EOF represents a northeast-southwest
gradient, with typical anomalies of ±0.5K, and accounts for 6% and 7% of
the variance in the two time periods. The remaining 9 EOFs are left to
represent together the variance of mere 5%.
In Figure 13.4 the EOF coefficients are shown. As mentioned above, they
are normalized to variance one. The first coefficient al(t) varies most of the
time between ±1 but exhibits a number of spiky excursions to large negative
247
Figure 13.4: EOF coeflicient time series al (t) and a2(t) of the first two EOFs
of winter mean temperature at 11 Central European stations. Note that the
time series have been normalized to one so that the information about the
strength of the variation is carried by the patterns. (From Werner and H.
von Storch, 1993).
1690 1900 1910 U20 1930 1940 1950 1860 1870 1860 1880
I
•
I
•
t
'
I
,
!
!
I
,
!
'
!
,
,
!
•
,
,
•
I
I
I
2nd: EOF
:
2
I
.1. ___ _
I
~'~_4+A~~~~~hr~~~~~~~--To
I
--~---1
I
I
-2
I
I
I
I
I
3 ---~-------~-------~-------~-------~--- -3
I
•
t
I
2
I
I
I
I
O+---~~~~~~~~~~~~~~_1r
-1
-2
1880 1800 1810 1820 1830 \840 1850 1980 1970 1980 1880
two intervals - as a demonstration we show in Figure 13.3 the first two EOFs
for both periods. The representation of the patterns deviates from the definition introduced above: The contribution of the k-th EOF to the fuH signal is
given by ak(t)pk. According to our definitions the variance of the coefficient
is given by the k-th eigenvalue Ak and the vector has unit length. For a better display of the results sometimes a different normalization is convenient,
namely akpk = (akj..;J:k) x (pk..;J:k) = akP'k. In this normalization the coefficient time series has variance one for all indices k and the relative strength
of the signal is in the patterns pi. A typical coefficient is a' = 1 so that the
typical reconstructed signal is pi. In this format the first two EOFs and
their time coefficients obtained in the analysis of Central European winter
temperature are shown in Figures 13.3 and 13.4.
In both time periods the first EOF has a positive sign at all locations,
represents about 90% of the total variance and exhibits "typical anomalies"
of the order of 1 - 2K. The second EOF represents a northeast-southwest
gradient, with typical anomalies of ±0.5K, and accounts for 6% and 7% of
the variance in the two time periods. The remaining 9 EOFs are left to
represent together the variance of mere 5%.
In Figure 13.4 the EOF coefficients are shown. As mentioned above, they
are normalized to variance one. The first coefficient al(t) varies most of the
time between ±1 but exhibits a number of spiky excursions to large negative