164
Chapter 9: Field Intercomparison
Monte Carlo or permutation test is needed to resolve the quest ion of field
significance of the correlation map.
9.2.2 Test Example
In this situation two approaches are viable. Livezey and Chen (1983) first
substituted a random series generated from Gaussian white noise for the
SOl series, recomputed correlations and reperformed the local t-tests, and
counted the number of rejections of the null hypothesis. An alternative approach would have been to use a random reordering (resampling without
replacement; apermutation procedure as described in Section 9.3) of the
SOl instead of the Gaussian noise.
Two hundred proxy SOl series were produced and all of the calculations
repeated for each series. Finally, the experimental result (11.4%) was tested
against the 5% tail of the empirical distribution function of the Monte Carlo
test. This distribution function and the lag-O result are shown in Figure
9.3a. It is dear that the null hypothesis cannot be rejected at the 5% level
for the DJF lag-O result. Note, however, that the lag-1 and -2 results may be
significant at the 5% level, suggesting predictability of the DJF height field
from the SOl, although so far the multiplicity of maps generated (different
seasons, different lags) has not been taken into account.
In summer, it is known that the structural scale ofthe low-frequency height
variability is smaller than in winter. This implies a larger number of effective
tests and a narrower distribution of random outcomes. This is reflected in
Figure 9.3b which also shows that none of the lag correlation maps has field
significant signals.
The comparison of summer" to winter is analogous to sampling a signal
with different grid resolutions; because structures are generally larger in the
winter they are sampled more densely than summer structures.
Instead, a comparison can be made in which spatial scales are comparable
but the grid coverage is reduced. In the context of Figure 9.2 this amounts
to the criterion for field significance moving along the solid curve from right
to left and increasing, in other words a broadening of the empirical distribution function. This is illustrated in Figure 9.3c, which shows the empirical
distribution function and experimental results for the Pacific/North America
(PNA) subdomain. Only the lag-2 results are possibly field significant for
this subdomain. The relative difliculty in detecting signals in the two contrasting situations (winter vs. summer and full vs. PNA domain) illustrated
in Figure 9.3 was first pointed out by Hasselmann (1979).
Incidentally, Livezey and Chen (1983) did not account for the multiplicity and interdependence of seasons and lags considered, but easily could and
should have by use of randomizations of the DJF SOl series. The same randomizations could be used simultaneously with the four preceding seasonal
mean SOls (DJF,MAM,JJA,SON) to consistently produce all ofthe lag correlations for both summer and winter, thereby preserving season to season
Chapter 9: Field Intercomparison
Monte Carlo or permutation test is needed to resolve the quest ion of field
significance of the correlation map.
9.2.2 Test Example
In this situation two approaches are viable. Livezey and Chen (1983) first
substituted a random series generated from Gaussian white noise for the
SOl series, recomputed correlations and reperformed the local t-tests, and
counted the number of rejections of the null hypothesis. An alternative approach would have been to use a random reordering (resampling without
replacement; apermutation procedure as described in Section 9.3) of the
SOl instead of the Gaussian noise.
Two hundred proxy SOl series were produced and all of the calculations
repeated for each series. Finally, the experimental result (11.4%) was tested
against the 5% tail of the empirical distribution function of the Monte Carlo
test. This distribution function and the lag-O result are shown in Figure
9.3a. It is dear that the null hypothesis cannot be rejected at the 5% level
for the DJF lag-O result. Note, however, that the lag-1 and -2 results may be
significant at the 5% level, suggesting predictability of the DJF height field
from the SOl, although so far the multiplicity of maps generated (different
seasons, different lags) has not been taken into account.
In summer, it is known that the structural scale ofthe low-frequency height
variability is smaller than in winter. This implies a larger number of effective
tests and a narrower distribution of random outcomes. This is reflected in
Figure 9.3b which also shows that none of the lag correlation maps has field
significant signals.
The comparison of summer" to winter is analogous to sampling a signal
with different grid resolutions; because structures are generally larger in the
winter they are sampled more densely than summer structures.
Instead, a comparison can be made in which spatial scales are comparable
but the grid coverage is reduced. In the context of Figure 9.2 this amounts
to the criterion for field significance moving along the solid curve from right
to left and increasing, in other words a broadening of the empirical distribution function. This is illustrated in Figure 9.3c, which shows the empirical
distribution function and experimental results for the Pacific/North America
(PNA) subdomain. Only the lag-2 results are possibly field significant for
this subdomain. The relative difliculty in detecting signals in the two contrasting situations (winter vs. summer and full vs. PNA domain) illustrated
in Figure 9.3 was first pointed out by Hasselmann (1979).
Incidentally, Livezey and Chen (1983) did not account for the multiplicity and interdependence of seasons and lags considered, but easily could and
should have by use of randomizations of the DJF SOl series. The same randomizations could be used simultaneously with the four preceding seasonal
mean SOls (DJF,MAM,JJA,SON) to consistently produce all ofthe lag correlations for both summer and winter, thereby preserving season to season