An Example of MBH "Robustness"

In the MBH source code, they apply steps that purport to weight the temperature PCs in their regression calculations proportional to their eigenvalues. Comments on their code say:

c set specified weights on data
…
c weights on PCs are proportional to their singular values

This is one of two weighting procedures in MBH for the regression, the other being the weighting of the proxy series. Wahl and Ammann tout one of the key results of their study as showing that the MBH algorithm is “robust” against “important” simplifications and modifications. They claimed:

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.

This claim is repeated in slightly different terms on several occasions:

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.

…

Our results show that the MBH climate reconstruction method applied to the original proxy data is not only reproducible, but also proves robust against important simplifications and modifications.

…

Indeed, our analyses act as an overall indication of the robustness of the MBH reconstruction to a variety of issues raised concerning its methods of assimilating proxy data, and also to two significant simplifications of the MBH method that we have introduced.

I’ll discuss this claim in respect to weights on proxies in detail on another occasion, but today will note that it can trivially be seen to be untrue, merely through considering the acknowledged impact of differing weights on the NOAMER PC4. Obviously results are not “insensitive” to the “linear scale factors” (weight) of the PC4. Low weights remove the HS-ness of the result and yield a low RE statistic. So this particular claim is patently false.

However, it is true that the MBH result is “robust” to the use or non-use of weight factors on the temperature PCs. This can be seen through some trivial (though long-winded) linear algebra as follows.

As noted in the previous post, the matrix of unrescaled reconstructed RPCs can be expressed as follows (using notation from prior post):

Mann re-scaled each such series so that its standard deviation (in the calibration period) individually matched the standard deviation of the corresponding temperature PC . Denoting the matrix of rescaled RPCs by , we then have:

for i=1,…,k

Or stacking:

where applied to a matrix is the square root of the diagonal of – which, when divided by yields the standard deviation and, for these ratio calculations, equivalent results.

The expression for can be used to expand as follows:

We obtain the following long (but easily calculable expression) for by substituting for :

where and

In the 1-dimensional case (one reconstructed temperature PC), these long matrix expressions reduce to a scalar and a very simple expression – as I discussed a long time ago. Even in the multiple PC cases, because the normalization is being done one by one and L is a diagonal matrix, one can apply the simple identity:

where is the diagonal matrix from the square root of the diagonal.

Thus all uses of the weight matrix L cancel out in the above expression yielding:

)

Because the matrix L cancels out because of the underlying linear algebra, one would expect that any Wahl and Ammann “experiments” with varying this procedure would be “robust” to this particular variation.