Last year, I did a few posts connecting spatial autocorrelation to something as mundane as the Stahle/SWM tree ring network. In the process, I observed something that I found quite interesting – that principal components applied to geometric shapes with spatially autocorrelated series generated Chladni patterns, familiar from violins and sounds. The Antarctica vortex represents an interesting example of another fairly constrained geometric shape. I’ve alluded a few times to the similarity of the supposedly “physical” Steig eigenvectors to Chladni patterns and today I’ll show why I made this comparison. In the figure below, I show left – the first 3 Steig eigenvectors and right – the first 3 eigenvectors from spatially autocorrelated sites arranged on a disk.
Both the similarities and differences are worthy of notice.
The first eigenvector of an isolated disk weights the interior points more heavily than points around the circumference – a pattern observable in the Steig eigenvector as well. The first Steig eigenvector is displaced somewhat to the east – but you’ll notice that 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.
The 2nd and 3rd Steig eigenvectors also have points in common and points of difference with the simple circular case. Both Steig eigenvectors are “two lobed”. Because the eigenvalues of the first two circular eigenvectors are identical, these two eigenvectors (EOFs are the same thing) are not “separable” in the sense of North et al 1982. Any linear combination is as likely as any other one. So in that sense, for a disk, there is no preferred axis. In the Antarctic case, the Transantarctic Mts look like they provide a reason for orienting the second Steig eigenvector. It would be an interesting exercise to evaluate just how much extra oomph there is in this symmetry breaking. The third Steig eigenvector is not as cleanly perpendicular to the 2nd eigenvector as in the disk example, but I think that one can readily discern this structure.
In the circular case, it’s not as though eigenvectors of lower order are uninterpretable. I’ve posted up a pdf showing the plots for the first 16 eigenvectors, which interesting and pretty patterns (thanks to Roman for showing how to make pdfs straight from R). In the circular case, you can get radially symmetric eigenvectors – a structure that seems like it might be highly relevant if one wants to incorporate seemingly negatively correlated data from islands in the Southern Ocean. I’m not saying that this is a relevant structure, only that this eigenvector might be empirically relevant – or it might not be.
As to how to summarize or truncate – I have no prescriptions. There are lots of schemes – Preisendorfer’s Rule N has been mentioned. But I haven’t seen any head-to-head analyses of Preisendorfer’s Rule N against eigenvectors generated from spatially autocorrelated data – so I don’t know whether it’s a good rule in these circunstances or not.
Let’s return to the description of these three eigenvectors in Steig:
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
No proof or evidence was offered for any of these assertions. You’d think that Nature would require authors to provide evidence, but seemingly not in this case. The observation about the second eigenvector seems untrue on its face: the reported eigenvector has 2 lobes, not 3, which, in my eyes, disqualifies this interpretation.
The plots here provide more evidence (though not “proof”) that the eigenvectors are simply what you’d expect from applying principal components to spatially autocorrelated data on a not quite circular shape than Steig offered up in support of his assertion that these things are “meaningfully related to important dynamical features of high-latitude Southern Hemisphere atmospheric circulation, as defined independently by extrapolar instrumental data.”