A few days ago, I showed some plots showing distribution of weights arising from principal components carried out on data from a region arranged as a line segment (think Chile). Today I’ve done a similar analysis for a square shaped region again assuming spatial autocorrelation governed by distance. In this case, I made a regular 10×10 grid (I was prepared to do a finer grid, but the results seem clear enough without going finer) and then did a SVD decomposition of the spatial autocorrelation matrix. I then made contour maps of the coefficients for each PC (called “eigenvectors”), which showed pleasing and interesting geometric symmetries, something that one doesn’t see discussed when PC analysis is applied abstractly to tree ring networks.
Update: Hans Erren observed below that the patterns here are the well-known Chladni figures, known as eigenfunctions of the wave equation for a square plate. This seems to be a related but different derivation: since the wave equation and uniform spatial autocorrelation both generate Chladni figures, they must be doing the same thing at some level and references to such an explanation would be welcome. End update.
Here are 8 plots showing weights for the first 8 PCs, with the first PC being highly symmetric. In the line segment case, the PC weights appear almost certainly generated by sin(kx) functions. My guess is that the functions in the square case are also sine functions of some sort – the first one looks like it has radial symmetry, but I don’t know what formulas would generate the lower ones.
In addition, here is a barplot of the eigenvalues from the above decomposition: the network has 100 sites. The left panel shows all 100 eigenvalues with lower order patterns obviously being very small. The middle panel shows the first 10 eigenvalues; the PC2 and PC3 which are symmetry-breaking have indistinguishable eigenvalues. I’m pretty sure that, in an empirical situation where one has a spatial autocorrelation and a rectangular grid, the empirical PC2 and PC3 would readily get mixed; the “plane” resulting from the PC2 and PC3 would be fairly distinct, but little meaning could be attached to the symmetry-breaking weights.
The right panel shows the square of the first ten eigenvalues (normalized), showing a remarkable concentration of weight on the PC1. Given that the network is constructed with spatial autocorrelation, it makes sense that the relevant “signal” is in the PC1. By the time that you get to the PC4, you are dealing with an inconsequential proportion of weight.
Maybe my rho is higher than realistic; I’m experimenting still. But if the rho is plausible and we re-apply this back to the NOAMER network, which I’ll try to get to some time, this means that, under the model here, one would not expect a PC4 to contribute much to the weighting of the network. The weight of the NOAMER PC4 is significantly higher than the PC4 in this model. So in Preisendorfer’s terms, this should be a “ringing bell” for the examination of exactly why the PC4 is over-contributing to the network?
In this case, we know that the PC4 is made up almost entirely (all but one) of Graybill chronologies? As we’ve discussed elsewhere, this raises questions about both 1) aside from bristlecones, are there are problems in what Graybill did e,g, Ababneh’s failure to replicate at Sheep Mt? 2) are Graybill’s bristlecones actually recovering a signal unavailable to other trees in the network? 3) is that signal relevant to temperature?
All of these questions have been asked before. The only fresh context here is an illustration of what PC weights look like given geometric uniformity. The methodology is also perhaps a simple and interesting way of generating some interesting looking eigenfunctions.