154
Chapter 8: Statistical Analysis of GCM Output
Figure 8.5: Distribution of observed temperature profiles used in the analysis
for June 1984. (From Reverdin et al., 1991)
10 N
"
...
June 84
..
- .
10
0
10
20 S
60W
50
40
30
20
10
0
10
20E
culated, so that the non-Iocality of the interpolation procedure is accounted
for in the error field. Although details can not be trusted, I}D is treated in
the analysis as a true covariance matrix.
The observed and simulated thermocline depth variations are shown in
Figure 8.6. The agreement is characterized by the misfit
(8.16)
which is approximately distributed as Hotelling's T 2 with m (the dimension
of the intercomparison space) and n (the degrees of freedom of t M + I}D)
degrees of freedom, if the null hypothesis Ho that there is no model error
holds. As m is large, a strong data compression is done. Using Common
Principal Component Analysis 6 , an orthonormal base is first defined where
the main features of simulations and observations are weIl represented. The
first four common Empirical Orthogonal Functions (EOFs) ac count for more
than 80% ofthe observed and simulated fields (Figure 8.7, left). Observations
and simulations are represented in this subspace by the time series in Figure 8.7 (right), where the 95% confidence intervals already suggest that the
observational and forcing uncertainties cannot explain all the model-reality
discrepancies. A time compression is then done by an EOF analysis of the
differences between observed and model time series, so with no further truncation, the dimension of the reduced space is 4 x 4 = 16. Due to the large
estimated values of the degrees of freedom n, T 2 behaves approximately as
6 Generalization of principal component analysis (Chapter 13) that applies simultaneously to two or more fields. The EOFs are common to the different fields but the eigenvalues
and principal components differ, because of sampling variability and(or true differences.
The transformation can be viewed as a rotation yielding variables that are as uncorrelated
as possible simultaneously in k groups (see Flury, 1989).
Chapter 8: Statistical Analysis of GCM Output
Figure 8.5: Distribution of observed temperature profiles used in the analysis
for June 1984. (From Reverdin et al., 1991)
10 N
"
...
June 84
..
- .
10
0
10
20 S
60W
50
40
30
20
10
0
10
20E
culated, so that the non-Iocality of the interpolation procedure is accounted
for in the error field. Although details can not be trusted, I}D is treated in
the analysis as a true covariance matrix.
The observed and simulated thermocline depth variations are shown in
Figure 8.6. The agreement is characterized by the misfit
(8.16)
which is approximately distributed as Hotelling's T 2 with m (the dimension
of the intercomparison space) and n (the degrees of freedom of t M + I}D)
degrees of freedom, if the null hypothesis Ho that there is no model error
holds. As m is large, a strong data compression is done. Using Common
Principal Component Analysis 6 , an orthonormal base is first defined where
the main features of simulations and observations are weIl represented. The
first four common Empirical Orthogonal Functions (EOFs) ac count for more
than 80% ofthe observed and simulated fields (Figure 8.7, left). Observations
and simulations are represented in this subspace by the time series in Figure 8.7 (right), where the 95% confidence intervals already suggest that the
observational and forcing uncertainties cannot explain all the model-reality
discrepancies. A time compression is then done by an EOF analysis of the
differences between observed and model time series, so with no further truncation, the dimension of the reduced space is 4 x 4 = 16. Due to the large
estimated values of the degrees of freedom n, T 2 behaves approximately as
6 Generalization of principal component analysis (Chapter 13) that applies simultaneously to two or more fields. The EOFs are common to the different fields but the eigenvalues
and principal components differ, because of sampling variability and(or true differences.
The transformation can be viewed as a rotation yielding variables that are as uncorrelated
as possible simultaneously in k groups (see Flury, 1989).