Yesterday’s results connecting eigenvector patterns in the Stahle SWM network to Toeplitz matrices and spatial autocorrelation were obviously pretty interesting. Needless to say, I was interested to test these ideas on out some other networks and see how they held up.
There is a large literature on spatial autocorrelation and there appear to be well-known mathematical approaches (that I’ll take a look at at some point.) In a quick google, I didn’t notice any discussion that was exactly on point to the discussions here (other than my earlier post at CA which ranked first in my search), but the need to discuss the relationship of tree ring chronologies and spatial autocorrelation seems pretty obvious though the connection to eigenvectors could well not have been noticed in the tree ring literature.
Today I’ll report on applying this approach to the Stahle Texas-Oklahoma network also used in MBH98, the combination of the Stahle SWM earlywood network with the TX-OK network (for the purposes of a first-cut analysis, this was more tractable than carrying both earlywood and latewood series) and then the combination when 16 Stahle sites in the NOAMER network were added in. Exactly why the Stahle sites are strewn all over the place in MBH is a question you’d have to ask Mann. The pattern of spatial autocorrelation holds up well in the expanded networks. It’s not quite as strong as in the Stahle SWM series, but is statistically highly significant. As more networks are added in, the region becomes “squarer”: whereas there is a plausible “order” for the Stahle SWM sites, the larger networks are much more rectangular. But the patterns are highly geometric.
Stahle TX-OK Network
This is a network of 14 sites in Texas and Oklahoma with all sites available form AD1700 on. Because the sites don’t go back to AD1400, we’ve not discussed this network much. The locations are shown below.
As with the Stahle SWM network, there is a strong pattern of spatial autocorrelation (see Top left). I’ve arranged the sites more or less from NE to SW and made barplots of the eigenvalues below, which show the distinctive Toeplitz pattern: the PC1 is sort of a weighted average with all coefficients positive; the PC2 is a gradient with one sign change; the PC3 has two sign changes. All of which indicates that the network can be approximated by a Toeplitz network – which is more restrictive than mere spatial autocorrelation.
Again the pattern of eigenvalues can be replicated rather nicely with a simple Toeplitz model as shown below, with the observed eigenvalues being slightly more diffuse than the Toeplitz model. (Mann said that 3 were “significant”).
Combining the Stahle SWM and TX-OK Networks
No reason was given in MBH for not combining the SWM and TX-OK networks. But regardless, one would like to see how stable the eigenvector properties are under simple mathematical operations like combining networks. Intuitively, if the eigenvectors of each network separately represented an important and distinct property, one would presume that the pattern would be recovered from the combined network. Let’s see.
First here is a fresh location map on the left showing the SWM network in blue and the TXOK network in red, yielding a “squarer” overall span. On the right is a plot of intersite correlation versus intersite distance, again showing strong spatial autocorrelation (r-squared of about 0.7).
Next here is graphic showing on the left the barplot of eigenvector coefficients for the PC1 and the contour map. All of the sites are positively correlated and thus the PC1 is a weighted average. There is a concentration of weight at a few interior sites which is predictable in a spatial autocorrelation model on the basis that they are more correlated with sites on the perimeter than the sites on perimeter are to each other, due to shorter distances. Thus, any “accidental” properties of the interior sites in this network will get accentuated at the price of whatever signal may exist.
Next here are corresponding plots for the PC2 and PC3. In these cases, the eigenvector structure no longer has the simple patterns of the Toeplitz matrix though there are elements of gradient seeking. The PC2 has sort of a gradient from west to east, with low values in the south; while the PC3 runs from the SW to the NE with sort of a “valley”. It is my impression that these shapes are strongly controlled by the geometry of site locations combined with spatial autocorrelation; and that it is highly dubious that these eigenvector patterns have the faintest climatic significance. (Even if there was some second order information, it would be over-ridden by the geometrically controlled patterns anyway.)
In case you were wondering what the PC1 of this network looked like, here it is, with warm 1934 highlighted. This area actually has a relatively distinct (negative) correlation of ring widths to temperature. Tamino may be able to discern a HS pattern here, but I’m unable to do so.
Expansion to Stahle NOAMER Sites
Next I experimented with an expansion to include the 16 Stahle sites in the NOAMER network (located in Arkansas and SE United States.) The hiving off the TXOK and SWM networks creates an odd discontinuity in the NOAMER network which is otherwise heavily oriented to SW United States sites. The strong pattern of spatial autocorrelation is preserved in the larger network.
Again because the sites are positively correlated, the PC1 is again a sort of weighted average with interior sites once again being much more heavily weighted. In this case, because of the changed geometry, a different set of interior sites are so weighted.
Next as before, here are the corresponding plots for the PC2 and PC3. The PC2 sort of looks for a West to East gradient in the present geometry, while the PC3 seems to pick out a few sites for highlighting.
When one is developing statistical methodologies, it’s really a bad idea to develop them on data sets where you’re also trying to obtain an important applied result. If Mann et al believed that principal components were a useful method of analyzing tree ring networks, then prior to incorporating output from such calculations into a complicated multiproxy study, the validity of the methodology itself should be demonstrated on relatively uncontroversial data sets. In the above figures, neither Mannian principal components nor bristlecones rear their ugly heads. And yet we can see many issues and questions arising from the simplest sort of technical analysis of principal components applied to uncomplicated tree ring networks.
In these networks, it seems certain that even a centered PC1 is a far from ideal estimator of whatever field is governing this particular network (and establishing a connection between that field and climate fields is a separate topic unto itself.) The PC1 is over-influenced by interior points merely from the geometry. It seems impossible to visualize any plausible meaning to the patterns of lower order eigenvector weighting other than the combination of accidental geometric availability and selection together with spatial autocorrelation.
Having said all that, I can see one useful outcome in terms of statistical testing, which may enable all but the most hard-core Mannians to appreciate the problems inherent in the Graybill chronologies. If tree ring chronologies are governed by spatial autocorrelation, it is highly unlikely, to say the least, that Graybill should uniquely have been able to detect shifts in 20th century mean from historic levels. The failure of Ababneh to replicate Graybill’s Sheep Mountain chronology is a very serious problem. It’s far more likely that Mann (and Tamino) are in possession of a flawed instrument rather than a magic flute.