Today’s post, which has a forbidding title, contains an interesting mathematical and statistical point which illuminates the controversy over how many PCs to retain. In my re-visiting the totally unknown corner of Mannian methodology – regression weights and their determination – I re-parsed the source code, finding something new and unexpected in Mannian methodology and resolving a puzzling issue in my linear algebra.
The key to the idea is very simple. After creating a network of proxies, Mann does what are in effect two weighted regressions: first, a calibration regression in which the proxies are regressed against the temperature PCs weighted by their eigenvalues; second , an estimation regression in which the reconstructed temperature PCs are estimated by a weighted regression in which different weights are assigned to each proxy. These weights ranged from 0.33 to 1.5.
Now there’s an interesting aspect to Mann’s implementation of the weights using SVD, which has taken me a long time to figure out. After a new push at parsing the source code, I’ve determined that the code actually implements weighting by the square of the weights. I’m going to do a very boring post on the transliteration of these weighting calculations from Mannian cuneiform into intelligible methodology in a day or two.
The effective weighting thus varies from 0.11 to 2.25 (a range of over 20) – with the largest weight being assigned to the Yakutia series. This is amusing, because Yakutia is an alter ego for the Moberg Indigirka series which has a very elevated MWP – we discussed this in connection with Juckes. Use of the updated long Yakutia-Indigirka series will have a dramatic impact on MBH99 as well as Juckes – something that I’d not noticed before.
In the AD1400 network, the sum of squares of all 22 weights is 12.357, of which the NOAMER network accounts for about 16% of the total weight.
How were these weights determined? No one knows. Actually I challenge Tamino or anyone else to locate these weights. Weighted regression is not mentioned in MBH98 itself. Indeed, MBH98 contains an equation for this step which clearly fails to show that weighting was done. They are not discussed in the Corrigendum but weights are given in the Corrigendum SI in files http://www.nature.com/nature/journal/v430/n6995/extref/PROXY/data1400.txt. In the source code provided to the House Energy and Commerce Committee, the weights are referred to in the code by the variable weightprx.
Weights are mentioned in passing in Wahl and Ammann, who stated incorrectly that:
A second simplification eliminated the differential weights assigned to individual proxy series in MBH, after testing showed that the results are insensitive to such linear scale factors.
Additionally, the reconstruction is robust in relation to two significant methodological simplifications-the calculation of instrumental PC series on the annual instrumental data, and the omission of weights imputed to the proxy series. Thus, this new version of the MBH reconstruction algorithm can be confidently employed in tests of the method to various sensitivities.
The argument below shows exactly how incorrect these claims are and how theses claims are inconsistent with Wahl and Ammann’s own confirmation of results using 2 and 5 PCs.
Weights and PC Retention
To illustrate the connection between weights and PC retention, showing how PC retention can be viewed as a special form of regression weighting, let’s construct a network using centered PCs with 5 PCs included. Perhaps through Rule N (even if it wasn’t used in MBH98, it’s not an implausible rule.)
Here’s how special choices of weights yield the two cases that have been at issue so far. If you give weights of (1,1,0.001,0.001,0.001) to the 5 PCs, then you’ll get a result that is virtually identical to the retention of only 2 PCs, while if you give equal weights to the 5 PCs, then you’ll get a result that looks like the Stick – heavy weighting on the bristlecones. Wahl and Ammann agree that you get different results using 2 and 5 covariance PCs. So obviously the selection of weights can “matter” – their own calculations confirm this. They happened to show one calculation where the two choices didn’t matter much, but that hardly rises to the standard of mathematical proof. It’s amazing how low the standards of reasoning are in this field.
If you assign equal unit weights to all 5 PCs in the network now represented by 5 PCs instead of 2 PCs, you also increase the proportion of weighting assigned to the NOAMER network relative to all weights. With two PCs, the NOAMER network had about 16% of the weights, but, if 5 PCs are each assigned equal weight, then the proportion of weight increases from 2/12.357 to 5/15.357 or nearly doubles.
Let’s suppose for a moment that there was a reason why Mann allocated 16% of the total weights to the NOAMER network. Perhaps it wasn’t clearly articulated but one can’t exclude the possibility that there was a reason. Why would the proportion assigned to this network increase because of the decision to increase the number of PCs used to represent the network? Mann has changed this aspect of his methodology. On the basis that this network should represent 16% of the total weight, then the weight allocated to all 5 PCs should still amount to 16% of the total weight. Instead of equal unit weights, the 5 PCs should be assigned lower weights.
There’s another weighting issue which reflects the reduced proportion of total variance accounted for by a PC4 (and is a pleasing and rational way of dealing with the frustration that a PC4 should dominate the subsequent regression.) In Mann’s calibration, he weighted the temperature PCs by their eigenvalues. Now this procedure is undone by the subsequent rescaling (hey, this is Mann we’re talking about), but the idea of weights by eigenvalues is not a bad one.
So a plausible implementation for the tree ring network would be to weight each tree ring PC by its eigenvalue so that the more important PCs were weighted more heavily, requiring also, as in the prior paragraph, that the total weights be apportioned so that the network still accounts for 16% of total weight. If this quite plausible procedure is done, then the weights assigned to the Graybill bristlecones are reduced substantially – so that the resulting recon is only marginally different from the reconstruction using 2 PCs as shown below:
I absolutely don’t want readers to get the impression that any of these squiggles mean anything. They are just squiggles. The MBH squiggle gets a HS shape from higher weighting of Graybill bristlecone chronologies. As I’ve said over and over, the Graybill Sheep Mt chronology was not replicated by Ababneh and all calculations involving the Graybill Sheep Mt chronology (including MBH, Mann and Jones 2003, Mann et al 2007, Juckes, Hegerl etc) should be put in limbo pending a resolution of the Sheep Mt fiasco.
Today I’m just making an appealing mathematical point – differences between the number of retained PCs can and should impact the weights in the weighted regression. The statement by Wahl and Ammann that these weights don’t matter – as a mathematical point – is, in Tamino’s words, “just plain wrong”. Purporting to salvage MBH by increasing the number of retained PCs without reflecting this in the overall weight assigned to the network increases the proportion of weight assigned to the network – in this case dramatically.
If the overall weight of the network is left unchanged and the PCs additionally weighted in proportion to their eigenvalues, the net result does not yield a HS using AD1400 proxies.