Section 13.3: Empirical Orthogonal Functions
245
Also the above sketched approach does not consider the quality of the
estimation of the patterns; instead the selection rules are concerned with
the eigenvalues only.
I recommend forgetting about the identification of "significant" EOFs
by means of selection rules and resorting to more honest approaches like
North'8 rule-of-thumb outlined in the next paragraph.
• N orth'8 Rule-of- Thumb
Using a scale argument North et a1. (1982) found as the "typical errors"
ß.x"
~
~.x"
(13.38)
ßi"
f'V
ß.x" ~i
p
.xi - A"
(13.39)
with .xi being the eigenvalue closest to .xi and n being the number of
independent sampies. Approximation (13.39) compares patterns, and
not the lengths of vectors, since we are dealing with normalized vectors.
- The first order error t::"i" is of the order of /f. The convergence
to zero is slow.
- "The first order error ßi" is orthogonal to the true EOF p".
- The estimation of the EO F p" is most contaminated by the patterns
of those other EOFs pi which belong to eigenvalues .xi closest to
.x". The contamination will be the more severe the sm aller the
difference Ai - .x" iso
North et a1. (1982) finally formulated the following "rule-of-thumb":
"If the sampling error of a particular eigenvalue dA is comparable or larger than the spacing between . x and a neighboring
eigenvalue, then the sampling error dp of the EOF will be
comparable to the size of the neighboring EOF."
When using this rule-of-thumb one should be careful not to oversee the
condition of independent sampies - in most geophysical data this assumption is not valid.
13.3.4 Example: Central European Temperature
As an example we consider the covariance structure of the winter mean temperature anomalies (Le., deviations from the overall winter mean) at eleven
Central European stations (Werner and H. von Storch, 1993). Thus m = 1l.
Relatively homogeneous time series were available for eighty winters from
1901 to 1980. For both of the 40-year interval before and after 1940 an EOF
analysis was performed. The results of the analysis are very similar in the
245
Also the above sketched approach does not consider the quality of the
estimation of the patterns; instead the selection rules are concerned with
the eigenvalues only.
I recommend forgetting about the identification of "significant" EOFs
by means of selection rules and resorting to more honest approaches like
North'8 rule-of-thumb outlined in the next paragraph.
• N orth'8 Rule-of- Thumb
Using a scale argument North et a1. (1982) found as the "typical errors"
ß.x"
~
~.x"
(13.38)
ßi"
f'V
ß.x" ~i
p
.xi - A"
(13.39)
with .xi being the eigenvalue closest to .xi and n being the number of
independent sampies. Approximation (13.39) compares patterns, and
not the lengths of vectors, since we are dealing with normalized vectors.
- The first order error t::"i" is of the order of /f. The convergence
to zero is slow.
- "The first order error ßi" is orthogonal to the true EOF p".
- The estimation of the EO F p" is most contaminated by the patterns
of those other EOFs pi which belong to eigenvalues .xi closest to
.x". The contamination will be the more severe the sm aller the
difference Ai - .x" iso
North et a1. (1982) finally formulated the following "rule-of-thumb":
"If the sampling error of a particular eigenvalue dA is comparable or larger than the spacing between . x and a neighboring
eigenvalue, then the sampling error dp of the EOF will be
comparable to the size of the neighboring EOF."
When using this rule-of-thumb one should be careful not to oversee the
condition of independent sampies - in most geophysical data this assumption is not valid.
13.3.4 Example: Central European Temperature
As an example we consider the covariance structure of the winter mean temperature anomalies (Le., deviations from the overall winter mean) at eleven
Central European stations (Werner and H. von Storch, 1993). Thus m = 1l.
Relatively homogeneous time series were available for eighty winters from
1901 to 1980. For both of the 40-year interval before and after 1940 an EOF
analysis was performed. The results of the analysis are very similar in the