**(understanding the extended analysis)**

A way to understand the results from the extended analysis is through the use of synthetic data sets. There are two issues to be tested here, one is the splitting of the patterns under the window of 5 "lags" of 3 months, and the other is the importance of the VARIMAX rotation of the unrotated patterns. Three synthetic data sets are created with help of the results of an extended rotated EOF (EREOF) analysis of Pacific Ocean SST anomalies from the Hadley Centre. Spatial patterns are drawn from the fields corresponding to the central time of the EREOF analysis while their temporal variations are synthetically created with a sinusoidal function spanning 696 months (the equivalent of the 1945-2002 period) with periods and phases identified from the principal components (i.e. time series) of the EREOF analysis.

The first data set is built using the 1st ENSO spatial pattern with time series of period of 42 months. The second data set is built using the 2nd ENSO pattern with a period of 42 months too but leading by 9 months the time series of the first data set. The third data set uses the Pacific Decadal pattern and a time series with period of 342 months. Several multivariate analyses will be performed on linear combinations of the following basic data sets:

1) Only 1st ENSO (called "ENSO"
now on).

2) Only 2nd ENSO (called "Shifted
ENSO" now on).

3) Only Pacific Decadal.

4) ENSO
plus Shifted ENSO.

5) ENSO
plus Shifted ENSO plus Pacific Decadal.

**Extended Rotated EOF Analysis.**

Extended Rotated EOF analyses will be performed using 5 "lags" shifted in time by 3 months. Knowing a priori what does the data set has before the EREOF analysis it is possible to identify the strengths (and weakness) of the method. The EREOF for the "ENSO" case pick up two main patterns explaining similar variances with a combined variance of 100%:

Correlation and spectral analyses of the associated principal component time series reveal the patterns have a shift of 10.5 months, and that both have maximum single peaks around 42 months.

The EREOF for the "Shifted ENSO" case pick up two main patterns explaining equal variances with a combined variance of 100%:

Correlation and spectral analyses of the associated principal component time series reveals the same shift in time and peaks as for the "ENSO" case.

The EREOF for the "only Pacific
Decadal" case pick up two main patterns too. However, the first is meaningful
while the second can be considered meaningful in the transition months
of the first because the quadrature of the time series, explaining a combined
variance of 100%:

Correlation and spectral analyses of the associated principal component time series reveal the patterns have a shift of at least 72 months, and that both have maximum peaks around 342 months.

The EREOF of the "ENSO plus Shifted ENSO" data set gives similar results to the EREOF of the "ENSO" case. Two main modes are picked up again:

Important features to notice from
the patterns are that the spatial patterns at the central times for both
modes resemble the individual initial anomalies. Correlation and spectral
analyses of the associated principal component time series reveal the patterns
have a shift of 10.5 months, and that both
have maximum single peaks around 42 months.

The EREOF of the "ENSO plus Shifted ENSO plus Pacific Decadal" data set gives similar results to the EREOF of the individual cases of ENSO and Pacific Decadal. ENSO modes explain the largest variance.

1st Mode 2nd Mode 3rd Mode 4th Mode

As for the combined case of "ENSO" and "Shifted ENSO" the patterns of this combined analysis reveals that the spatial patterns at the central times for the three main modes resemble the individual initial anomalies. Correlation and spectral analyses of the associated principal component time series for the ENSO patterns reveal the same shift of 10.5 months and no correlation with the Pacific Decadal pattern, while the Pacific Decadal pattern still has the shift of 72 plus months with respect to the fourth pattern, and that the main patterns still have their maximum single peaks around 42 and 342 months.

Thus, it is clear that the method
is doing well in separating the meaningful spatial and temporal information
from the combined data sets. Now, it will be tested if rotation is necessary
to extract these important information.

**Extended Unrotated EOF Analysis.**

Now, to contrast the good results shown for the rotated case an extended but unrotated EOF (EEOF) analysis will be performed using 5 "lags" shifted in time by 3 months for the combined case of "ENSO plus Shifted ENSO plus Pacific Decadal" data set. That is, the only difference with the last experiment of the previous section is that the rotation is not performed.

In general the no rotation of the extended patterns is not able to separate clearly both spatial and temporal features of the original data sets. As before three main patterns rise from the analysis, with mixed features evident in both patterns and time series: one resembling the ENSO mode but with a decadal touch, another looking like the Shifted ENSO mode, and another trying to look like the Pacific Decadal mode but with interannual variability; in addition a fourth mode appears and can be considered as meaningful in the transition months of the Pacific Decadal mode.

Important features to notice from the patterns are that the spatial patterns at the central times for both modes do not look like the individual initial anomalies. From the time series is worth to mention that 1) correlation between the 1st ENSO time series and the second ENSO time series is still maximum at a shift of 10.5 months while correlation with the Pacific Decadal related time series can be considered high now, at a shift of 20 months; 2) correlation between the 2nd ENSO time series and the Pacific Decadal related time series has increased now and is maximum at a shift of 10.5 months. Spectral analysis of the associated time series indicates the contamination of the 1st ENSO and Pacific Decadal modes.

Thus, the previous results highlight
the necessity to rotate the modes to separate spatial and temporal features
in a given data set. Those results are now contrasted with the most common
regular (unextended) EOF analysis.

**EOF Analysis.**

Regular EOF analysis can only represent temporal variations that are in phase or out of phase in a data set. However, propagating structures, like those resulting from the interaction between traveling waves of different spatial and temporal scales cannot be well represented in this conventional EOF analysis. In spite of those limitations the regular EOF analysis is widely used to extract recurrent patterns (assuming the existence of only in/out phase relationships in the data) and to generate indices.

The following results show what it is extracted from the conventional unrotated EOF analysis using the combined "ENSO plus Shifted ENSO plus Pacific Decadal" data set. In general it is not possible to recognize the two ENSO patterns as different of each other and are merged as one ENSO pattern. The second pattern brings a not exact representation of the Pacific Decadal pattern while the third pattern is something else that seems to be closely correlated to the first pattern because its time evolution. Spectral analysis of their time series show the "contamination" of the ENSO and Pacific Decadal variability with decadal and interannual variability, respectively.

In the case of the extended analyses
rotation was successful to separate the different signals contained in
the data set. However, in this case of conventional rotated analysis the
extraction of the individual signals is not possible at all. With some
changes a merged ENSO
pattern and a mixed Pacific
Decadal pattern are still extracted, with the additional
pattern still there. The ENSO pattern is purely interannual now, and
although the Pacific Decadal pattern is closer to the original anomalies,
its time series is still "contaminated" with interannual variability; the
decadal signal of the unrotated ENSO pattern goes to the rotated third
pattern, as seen from the spectral
analysis of the time series.

**Effect of phase propagation in
conventional EOF Analysis.**

The previous results are not very
satisfying in terms of the role played by the rotation of the unrotated
eigenvectors. A possible factor contributing to the bad results of the
rotated (conventional) EOF analysis may be the presence of the phase propagation
due to the "Shifted ENSO" anomalies in the combined data set. In order
to see if this is the reason, the conventional EOF analysis is repeated
but for the combined data set of "ENSO plus Pacific Decadal" (i.e. omitting
the "Shifted ENSO" anomalies). In the unrotated case there are two main
patterns, one ENSO
related with interannual and decadal variability, and another Pacific
Decadal related with decadal and interannual variability. As before,
the associated time series are correlated,
and spectral analysis show the "contamination"
of those modes. However, when rotation is performed, the ENSO
and Pacific
Decadal modes are extracted with their original features with not
correlation at all between them and with their distinctive unique
spectral peaks.

**Final note.**

Thus, those results show some important
things to bear in mind when dealing with EOF analyses in their conventional
or extended forms. One thing is that rotation is needed for a "clean" extraction
of modes. Phase propagation implicit in a data set is best extracted
through extended rotated EOF analysis. Conventional rotated EOF analysis
is not able to sort of things and the phase propagation can mix the modes.
However, if no phase propagation is present in the data set conventional
rotated EOF analysis can do extract the standing modes.... but how to know
this a priori?