Yesterday, I showed an interesting comparison between the 3 Steig eigenvectors and “Chladni patterns” generated by performing principal components on a grid of spatially autocorrelated sites on a disk. Today I’ll show a similar analysis, but this time using a random sample of points from actual Antarctica. The results are pretty interesting, to say the least.
Key points for the disk included:
1) the first disk eigenvector and the first Steig eigenvector weight interior points more heavily than points around the circumference, but the first Steig eigenvector is displaced somewhat to the east. I speculated as follows:
Antarctica is by no means perfectly circular and the “center of gravity” is displaced to the east as well. My guess is that the same sort of graphic done on the actual Antarctic shape will displace to the east as well. I’ll check that some time.
2) the 2nd and 3rd Steig eigenvectors and disk eigenvectors were both “two lobed”. In the disk, any axis orientation is as likely as any other, while the axis of the Steig eigenvectors could perhaps be construed as being related to the peninsula and the Transantarcti Mts.
The Steig AVHRR grid information contains 5509 gridcells with lat-longs. I took a random sample of 300 cells (by taking a random sample from 1:5509 and taking the corresponding gridcells). I calculated the distances between gridcells (a network with 90000 from-to pairs) and the correlations assuming an exponential decorrelation of exp(-distance/1200) – this is sort of consistent with what we see, but I’m mainly just experimenting right now. I converted this into a 300×300 correlation matrix and took principal components. I then used the akima program to make this into a contour map (changing everything into x-y coordinates, using Roman’s pretty extraction of the Antarctic contour from mapproj to overlay the continent onto the contour map.) I need to white out some of the ocean areas, but that’s a little fiddly and not germane to the plots shown below.
On the left as before are the Steig eigenvectors; on the right are eigenvectors from the above procedure (with the order of the 2 and 3 eigenvectors reversed for a reason that will be obvious). Using preferred Team terminology, I submit that the patterns are “remarkably similar”.
As before, let’s return to Steig’s assertions about these three eigenvectors (for which no evidence was provided):
The first three principal components are statistically separable and can be meaningfully related to important dynamical features of high-latitude Southern Hemisphere atmospheric circulation, as defined independently by extrapolar instrumental data. The first principal component is significantly correlated with the SAM index (the first principal component of sea-level-pressure or 500-hPa geopotential heights for 20S–90S), and the second principal component reflects the zonal wave-3 pattern, which contributes to the Antarctic dipole pattern of sea-ice anomalies in the Ross Sea and Weddell Sea sectors 4,8
Now consider some of the following as possible “confirmation” that the form of these eigenvectors result from nothing more than principal components on spatially autocorrelated series on a figure with an Antarctic shape.
In addition to the high interior weighting of the 1st eigenvector, it is displaced towards the east as I predicted.
The orientations of the 2nd and 3rd eigenvectors now match the 3rd and 2nd spatially autocorrelated eigenvectors. So the axis orientations seem to be derived merely from the shape of the continent. There is a little extra oomph in the eigenvector with a NW-SE axis relative to the eigenvector with a perpendicular axis.
I’m not saying that this model explains everything in the Steig eigenvectors, but it sure accounts for most of the major features.
Note: we still haven’t seen any actual AVHRR data, only the rank 3 AVHRR version. As Jean S observed, it appears increasingly likely that the rank 3 data was what Steig, Mann et al used in their RegEM process. Keep an eye on this story.