One of the most ridiculous aspects and most misleading aspects of MBH (and efforts to rehabilitate it) is the assumption that principal components applied to geographically heterogeneous networks necessarily yield time series of climatic interest.
Preisendorfer (and others) state explicitly that principal components should be used as an exploratory method – and disavowed any notion that merely passing a Rule N test imbued a time series with magical properties.
In primitive societies, there is a “correct” order for magical incantations and woe betide anyone who dares to question the authority of the witch doctors. We see this with the recent discussion at Tamino’s of Rule N: total disregard for any need to establish the scientific veracity of a relationship between ring widths and temperature and replacement of such investigation by incantations that certain relatively arbitrary methods are “correct” or “proper”, using the language of ritual or magic rather than science.
There is much that needs to be said on this topic and I will do so next week.
Today, I want to discuss something very interesting that arose in the discussion of the Stahle SWM network – an “uncontroversial” network, but which may have great utility in understanding the properties of principal components applied to geographically heterogeneous tree ring networks.
I’d plotted up barplots of the Stahle SWM network eigenvectors and, at the suggestion of a reader, eigenvector contour maps. These plots had several extremely interesting properties which are connected with known properties of Toeplitz matrices – hence the title – which, in turn, give some considerable insight into the meaning of principal components applied to such networks – a topic conspicuously lacking in all literature to date..
The sites in the Stahle/SWM network span a north-south range in Texas-Mexico (See location map here). The network contains both earlywood and latewood sites.
Prior to making a barplot of the eigenvector weights, I arranged the series by Earlywood-Latewood and then in a N-S order. The PC1 was more or less a type of weighted average; while the PC2 more or less yielded a “gradient” in eigenvector weights in each case, as shown below, first in barplot form and second in contour map format. Note that, for each of the EW and LW sequences, there is one change in sign for the eigenvector coefficients. This is important.
The next PC showed a contrast between earlywood and latewood; the PC4 showed a type of “upside-down quadratic” or “horseshoe” shape. In this case, for both the EW and LW, there are 2 changes in sign with negative weights at the N and S ends of the range and positive weights in the middle.
Recent Mathematical Literature on “Horseshoes”
I ran across two interesting articles that describe eigenvector patterns that are strongly reminiscent of this particular pattern (I’ll explain how I ran across these articles in the conclusion.)
Diaconis et al 2008, HORSESHOES IN MULTIDIMENSIONAL SCALING AND LOCAL KERNEL METHODS, discusses the provenance of “horseshoe” patterns in principal component representations arising out of ecological ordinations, using the lively example of roll call voting patterns in the U.S. House of Representatives. Despite the lively example, the math is not easy.
Another slightly easier survey is by De Leuuw here where De Leuuw observes of Toeplitz matrices A:
Suppose is the eigenvector of A corresponding with eigenvalue the kth largest eigenvalue. We can plot the n elements of against 1; 2;…;n and connect successive points. The zero-crossings of the resulting piecewise linear function are called the nodes of the eigenvector.
Eigenvector has exactly k-1 sign changes. The nodes of successive eigenvectors interlace,
Notice the similarity in pattern to the Stahle SWM patterns shown above: the Earlywood PC2 has exactly one sign change; the next PC has two sign changes and so on,… The eigenvector patterns have the same form as the patterns for this specialized matrix form (Toeplitz matrix.)
If you’re familiar with AR1 autocorrelation matrices, you’re familiar with a type example of a Toeplitz matrix, which is defined as a matrix where the value of an entry A[j;i] is given by the difference in indices i.e. For the autocorrelation matrix, .
Applying to the Stahle SWM Network
Because the eigenvector patterns in the Stahle SWM network are so distinct, one is therefore led to examine the underlying data for evidence of a Toeplitz structure, which in this case would have a very easy interpretation: spatial autocorrelation.
If the site chronologies are spatially autocorrelated with negative exponential (or other monotonic) decorrelation, then this would go a considerable way towards yielding the desired structure. Below is a plot of EW, LW and EWxLW correlations by intersite distance. A spatial autocorrelation model has an extremely good fit to the EW correlations (r2=0.90!) and almost as good to the LW correlations (r2=0.83).
Next I did a one-parameter model of the EW correlation, estimating the coefficient for decorrelation by distance. I then created a matrix (Toeplitz) with the modeled intersite correlations and calculated the eigenvalues, comparing these eigenvalues to the observed eigenvalues, as shown below. The match is fantastically good, compared to anything that I’ve ever seen in dendro or proxy studies. I’m sure that there will be some statistic that measures this correspondence, but for now, the visual match is obviously almost exact.
There is one conclusion that can be drawn from this: everything here turns on spatial autocorrelation and the geometry of the sites (i.e. the intersite distances.) If you have another geometric arrangement of sites, then you’ll get different eigenvalue patterns.
How does this tie back to Toeplitz matrices? In this particular example, it seems to be possible to assign a linear order to the sites (mostly N-S, but the Texas sites are ordered a little “north” of all the Mexican sites, slightly against the latitude by itself). Using this linear order, the correlation matrix can be approximated by a Toeplitz matrix.
The r squared for a fit of EW correlation against abs(i-j) where i and j are simply the matrix index is 0.70! So the approximation by a Toeplitz matrix (a one-dimensional ordering) is itself very good. Because the approximation is as good as it is, the various properties of the eigenvector coefficients deriving from Toeplitz matrices can be observed in the eigenvector patterns of the Stahle SWM network.
Instead of this network containing a huge amount of “information”, what we have is a network that is approximated closely by a spatially autocorrelated sites arranged in a straight line.
I take a couple of thoughts away from this exercise – aside from it being interesting in itself.
First, the spatial autocorrelation in the site chronologies is exceptionally strong.
Second, the eigenvectors (and eigenvalues) in such situations will be affected by geographic inhomogeneity and it should be possible to say something about this. In the Stahle SWM case, the sites are not random but there is a clump of sites in the north and a clump in the south. If one is seeking a measurement of the “field” over the geographic space, then you can’t get there through PC analysis on its own. The PC1 gets overweighted by the clump of NW sites, where the three nearby sites are spatially autocorrelated and then overweighted in the PC analysis relative to their geographic significance. To make a proper estimate over the geographic region would require kriging or some other utilization of the geographic information. (This is one of many noticeable problems with Mannian methods – geographic information is never used and it doesn’t even “matter” if it is wrong – Mann stubbornly refused to correct geographic errors in Mann et al 2007.)
Third, Rule N, uninformed by the geography and information on spatial autocorrelation, is not very helpful. Supposedly using Rule N (though this is probably untrue), Mann deduced that there were 9 significant PCs in the Stahle SWM network. It’s hard to believe that there is more than 1. And it seems clear that even the PC1, as an effective index of whatever “climate field” is at work, is biased by geographical inhomogeneity and that a more useful index could be constructed by kriging.
Interpretation of the SWM network is simplified by its quasi-linearity in geographic structure. The 2-D situation in larger networks complicates the interpretation, but the salient aspects should carry over. Quite a bit is known about “block Toeplitz” matrices arising out of 2-D spatial autocorrelation, which can perhaps be usefully applied if one feels determined to derive something from tree ring networks.
The very strong spatial autocorrelation observed here seems relevant for assessing data quality in other situations. For example, Graybill’s Niwot Ridge limber pine chronology looks nothing like Woodhouse’s nearby Niwot Ridge chronology. Yeah, yeah, there are microclimate differences, but the inconsistency points to something going well beyond climatic differences. But even if the differences are climatic, this needs to be ironed out. These sites are a half hour drive from UCAR world headquarters and a reconciliation is long overdue.
How did I encounter these interesting articles? In his effort to rehabilitate Mann, while evading the explicit statement of Mann’s citation (Preisendorfer) that PC methods were necessarily centered because they were an analysis of variance, Tamino found some literature mentions of uncentered SVD of data matrices (loosely called “uncentered PC analysis” on some occasions), which he claimed as some sort of justification for Mannian short-centering – again without confronting the problematic qualities of Mannian short-centered in time series analysis. The references by Tamino and his readers were primarily to literature in ecological ordination, where ter Braak has used uncentered PC analysis.
Exactly how usage in ecological ordination can be levered into justification of a quite different use in time series analysis mystifies me, but it sounds good for people that believe in magic.
Anyway, ter Braak and ecological ordination literature are cited by Diaconis et al and de Leuuw. I noticed these papers through googling ter braak and other combinations.