University of Oklahoma/Cooperative Institute for Mesoscale Meteorological Studies, School of Meteorology, 1310 Sarkeys Energy Center, 100 E. Boyd ST, Norman, OK 73019 - USA

• Goals
• Data
• Methodology
• Results
• Conclusions

Goals

To determine if eigenpatterns of MSU channel 4 data are robust and physically meaningful. Standard statistical eigenanalysis is applied to the T-mode and S-mode

of the correlation matrices. The T-mode analysis will be highlighted and compared against several solutions for veracity.

Data

Temperature anomalies for the Southern Hemisphere were analyzed for the period January1979 - December 1997.  The data correspond to the MSU channel 4,

temperature anomalies for lower stratosphere.  This channel has its peak weighting at 70hPa and provides a good measure of lower stratospheric deep-layer temperature.  The data are placed in a 5128 (gridpoint) by 228 (month) matrix.

Methodology

The data were decomposed via principal component analysis in the T- and S-mode. The T-mode results will be presented. A standard approach used by the majority of investigators was taken, namely begin by correlating the data into a 228 by 228 matrix and then diagonalizing that matrix into 228 eigenvectors and corresponding eigenvalues. Since the principal component model was used, the eigenvectors were scaled by the square root of the corresponding eigenvalues to arrive at principal component loadings (time series) which are in the same units as the parent similarity matrix (i.e., correlations) and portray the time variability of the data. Principal component scores were then calculated and plotted on a polar stereographic map centered on Antarctica. The resulting spatial patterns are thought to be dominant modes of the relative variability of the MSU channel 4 data. The question remains: Are they? By truncating the number of eigenmodes from 228 to 6 and application of various targets and rotations to these data, a fuller picture of lower stratospheric temperature anomalies emerges. The rotations applied include two orthogonal transformations (Varimax, Quartimax), one oblique transformation (Covarimin), and two Procrustes transformations (targeted trends of temperature and a series of random targets).

Results

• The dimensionality of the analysis can not be accurately gauged by looking at the eigenvalues. There are more interpretable eigenvectors than would be found by application of typical eigenvalue magnitude and spacing techniques, such as the Scree test, LEV test or North et al. test. This suggests that those eigenvectors with variability indistinguishable from white noise have physical signal embedded within them. Developing new tests, based on the spatial structure of the variability, rather than the magnitude of the variability, should be given priority.
• Random targets can be applied by Procrustes with relatively poor fits (e.g., correlations between the target and the fit were typically 0.20), suggesting that the MSU channel 4 temporal fluctuations are not random. However, the PC scores based on the random fits present a shocking picture, that of robust appearing dipoles, mainly wave number 1 and occasionally wave number 2, centered around Antarctica. Undoubtedly, if such patterns were presented to analysts, they would interpret them as physically relevant. However, the Procrustes analysis has a safeguard built into the technique and one can quickly determine that 95 percent of the target is not fit. Therefore, the perceptive analyst can ignore such spatial modes.
• The unrotated PC scores exhibit wave number 1 to wave number 4 patterns with most variability in the middle and high latitudes, although two patterns (PCs 4 and5) had some tropical signal. The higher wave number patterns are associated with about one-half the variance as the larger scales. The PC loadings might be used to track the temporal behavior of the PC scores if the patterns are relatively simple. As the wave number increases, the probability of obtaining a high spatial match to the PC scores decreases.

However, the loadings of PC 3 had a decreasing trend with time and those for PC 5 had an increasing trend.

• The orthogonal rotations (Varimax and Quartimax) had virtually identical PC loadings.

This suggests that the MSU data can be simplified by PC loading number/column (Varimax) or by time/row (Quartimax) with little difference. The PC scores for the two rotations are nearly indistinguishable. It also suggests, for these data, that the solution is insensitive to the choice of orthogonal rotation ? the temporal configuration of temperature anomalies can be well simplified. The PC scores suggested that wave numbers ranged primarily from 1 to 3 with the exception of a high wave number on Varimax/Quartimax PC 6.

• The oblique rotation used (Covarimin) indicated a fairly high correspondence to the orthogonal ones in both loadings and scores, particularly for PC 2, 4, 5 and 6. There were some phase shifts in PCs1 and 3. The average absolute correlation among PC scores (maps) was less than 0.09 whereas the highest correlation was less than 0.22, indicating the underlying modes are close to uncorrelated in space.
• When an increasing trend was fit to the time series using Procrustes analysis, the best fit was a 0.34 correlation to a linear target and slightly lower (0.30) for a monotonic increasing sigmoidal trend. The scores of the Procrustes trend analysis indicated low temperature anomalies from Antarctica into southern Argentina and high anomalies south of Australia.

Conclusions

One must be careful when analyzing data filtered by a PCA or EOF technique. Due to the post-hoc analysis applied to the interpretation of the PC spatial and temporal patterns, one is tempted to over-analyze the results. Moreover, following a standard approach based on inspection of eigenvalue magnitudes or spacing will likely lead to underestimating the information embedded in the data (or similarity matrix used). Reconciling these two points is not an easy task. This research suggests that careful analysis of a range of solutions (unrotated, simple structure rotation) may provide insight into the variability. That said, the use of Procrustes should be encouraged where prior theory exists. Such theory can be incorporated directly into the elements of the target and fit to the data. The degree of fit can then be assessed to determine how well the theory fits the data. In the examples presented herein, the data do not fit a random theory or that of a linear or sigmoidal trend very well. The capability to reject such hypotheses, provided by Procrustes, underscores the advantage of this tool.