Juckes and the Euro Team spent a lot of time on the topic of MM normalization, stating as follows (continuing the academic check kiting initiated by claims made in Wahl and Ammann (Clim Chq 2006) using the rejected Ammann and Wahl (GRL 2006)):
Wahl and Ammann (2006) ascribe the difference between MM2005 and MBH1998 to another apparent error by McIntyre and McKitrick: the omission of the normalisation of proxies prior to the calculation of proxy principal components.
Juckes, in his usual style, raised this issue again on the blog as follows (in the same message as his claim about failing to “disclose” “machine-specific” code):
re #29: Using your code, I can show that the sensitivity you describe only exists when you use your own “arbitrary” normalisation for the calculation of the proxy PCs (ranging from a standard deviation of 0.0432 for wy023x to 0.581 for nm025). Why do you use this normalisation? Is the effective elimination of much of the data intentional?
Remarkably enough, this very issue with the specific series wy023x has already been discussed in print (Huybers Comment and Reply to Huybers). My published corpus of work is not large – indeed, I receive many criticisms for it not being larger than it is. Given that we’re talking about only 5 papers and given Juckes’ preoccupation with matters MM, you’d think that Juckes would have familiarized himself with prior discussion of this issue before making various attacks, both here and in his submission to Climate of the Past.
The discussion of the density series wy023x was initiated by Huybers as follows:
NOAMER records are standardized chronologies [Cook and Kairiukstis, 1990], reported as fractional changes from mean tree ring width or maximum ring density after correcting for the effects of increasing tree age. The variance of the chronology is a function of both environmental variability and the trees’ sensitivity to the environment. Sensitivity depends on factors such as species, soil, local topography, tree age, location within a forest, and what quantity is being measured [Fritts, 1976]. The most striking example of varying sensitivity is that the two NOAMER chronologies indicating changes in tree ring density (co509x and wy023x) have variances roughly thirty times smaller than the other chronologies indicating changes in tree ring width.
From this, Huybers concluded that the most informative method of applying PCs to tree ring networks would be dividing the chronologies as standardized in the ITRDB data bank by their standard deviations, while elsewhere noting that it might make the most sense simply to take the mean of the ring widths. Note Huybers’ recognition that ITRDB chronologies are already standardized – although not converted to unit standard deviation. ITRDB chronologies are in dimensionless units with a common mean of 1.
In our Reply to Huybers, we made it clear that we were not advocating covariance PCs (or PCs at all) as a method of extracting information from the ragbag North American tree ring chronologies, but had simply (and logically) applied covariance PCs as the most logical implementation of the stated MBH98 method (“conventional” principal components).
We re-emphasize that our comparison between the MBH98 method and a covariance PC1 was not presented as an attempt to “Å”Åremove the bias in MBH98’s method”, and that we take no position on the relative merits of using a mean, a covariance PC1, or even using PC analysis at all, in paleoclimate work. The onus for demonstrating validity of a statistic as a temperature proxy rests entirely with its advocate. Any valid climate reconstruction should not depend on whether a correlation matrix or covariance matrix is used in tree ring PC analysis. No variation on PC methodology can overcome the problems of using bristlecones as a temperature proxy.
In our Reply to Huybers (and elsewhere), we took the view (see citations in Reply to Huybers: Rencher 1992, 1995; Overland and Preisendorfer 1982; also North et al 1982; quotations here) that tree ring networks were already in common dimensionless units and that, under such circumstances, statistical authorities recommended the use of principal components on a covariance matrix. Jean S, who is knowledgeable in such matters, has endorsed this. Neither Juckes nor any other Team member have ever given a single countervailing statistical authority; they merely assert that use of a correlation matrix is the “correct” methodology.
However, regardless of whether correlation or covariance PCs are the “correct” methodology, in our Reply to Huybers, we provided compelling reasons why the decision in the case of the AD1400 North American network should not be based on the standard deviation of wy023x. In this network, there were only 2 density series and 68 ring width series. Both density series were also represented by ring width series at the same site (and in the case of co509x, it was additionallyhad quadruplicate use in the SWM network). We stated:
One of Huybers’  principal justifications for proposing a correlation PC1 is his observation that the covariance PC1 underweights density series, which have lower variances. But in the MBH98 network, only 2 of 70 series are density series, and both are from sites also represented in the same network with a ring width series. Indeed, the Spruce Canyon site (density series co509x and ring width series co509w) also occurs in 4 series in the MBH98 Stahle/SWM network. Accommodating these 2 density series should not be at the expense of the most appropriate treatment for the other 68.
So I would submit that Juckes’ question is asked and answered and that this was an issue that Juckes should have been familiar with before submitting his article to Climate of the Past (which does not cite either the Huybers Comment or our Reply, although both papers deal at considerable length with issues of “normalization” of the North American network).
In addition, this very issue was specifically visited by the NAS panel (also not cited by Juckes et al), which stated, referring to Huybers 2005.
Huybers (2005), commenting on McIntyre and McKitrick (2005a), points out that normalization also affects results, a point that is reinforced by McIntyre and McKitrick (2005b) in their response to Huybers. Principal components calculations are often carried out on a correlation matrix obtained by normalizing each variable by its sample standard deviation. Variables in different physical units clearly require some kind of normalization to bring them to a common scale, but even variables that are physically equivalent or normalized to a common scale may have widely different variances. Huybers comments on tree ring densities, which have much lower variances than widths, even after conversion to dimensionless “standardized” form. In this case, an argument can be made for using the variables without further normalization. However, the higher-variance variables tend to make correspondingly higher contributions to the principal components, so the decision whether to equalize variances or not should be based on the scientific considerations of the climate information represented in each of the proxies.
In our Reply to Huybers, we had stated that the “onus for demonstrating validity of a statistic as a temperature proxy rests entirely with its advocate” – a position that is surely reflected in the conclusion of the NAS panel. For Juckes et al to assert that a calculation demonstrating the impact of a covariance matrix is an “error” is pretty insolent.
Another point – it’s pretty annoying to listen to Juckes (following Wahl and Ammann) falsely accuse us of “omitting” consideration of the impact of dividing North American chronologies by their standard deviation. In MM05(EE) we stated:
If the data are transformed as in MBH98, but the principal components are calculated on the covariance matrix, rather than directly on the de-centered data, the results move about halfway from MBH to MM. If the data are not transformed (MM), but the principal components are calculated on the correlation matrix rather than the covariance matrix, the results move part way from MM to MBH, with bristlecone pine data moving up from the PC4 to influence the PC2. In no case other than MBH98 do the bristlecone series influence PC1, ruling out their interpretation as the “dominant component of variance” [Mann et al, 2004b]
Now these sentences are expressed in terms of correlation and covariance matrices, but PC with a correlation matrix is identical to PC using a covariance matrix on series standardized by dividing by their standard deviation. PC using the covariance matrix of the Mannian network is relatively close to PC on the correlation matrix (up to the difference between the standard deviation over 1400-1980 and 1902-1980). Huybers acknowledged the first point acknowlelged. Juckes et al (and Wahl and Ammann) should have been familiar with both points. In retrospect, I’d rephrase the last sentence quoted above a little – the bristlecones do not “dominate” the correlation PC1, but they do “influence” it, a point that was discussed in the Reply to Huybers. In our Reply to Huybers, we illustrated the correlation PC1 and various other permutations and combinations as follows:
This figure clearly shows not just the covariance PC1, but the correlation PC1 – indeed, even a correlation PC1 standardized with an autocorrelation-consistent standard deviation. Our Reply to Huybers contains a detailed analysis of issues pertaining to correlation and covariance PCs. For Huybers (or Wahl and Ammann) to assert that we “omitted” a discussion of “normalized” chronologies can only mean that they are somehow unaware that correlation PCs are the same as PCs from chronologies divided by their standard deviation. Having failed to cite our prior discussion of correlation PCs, they then have the gall to allege that we committed an “error” by omitting a discussion equivalent to discussing correlation PCs. They make it worse by omitting to consider the NAS panel’s specific consideration of the topic.
Now going back to Juckes’ question – the way that he framed the question is very odd. Juckes accuses me of using
your own “arbitrary” normalisation for the calculation of the proxy PCs (ranging from a standard deviation of 0.0432 for wy023x to 0.581 for nm025). Why do you use this normalisation? Is the effective elimination of much of the data intentional?
The way that this question is expressed suggests, among other things, that Juckes is not using a principal components algorithm to do principal components, but uses svd on data matrices (an observation confirmed by inspecting the archived series). As we stated clearly and illustrated, the results follow directly from a conventional principal components analysis on a covariance matrix.
Juckes’ use of the terminology “effective elimination of much of the data” is simply realclimate rhetoric. All that happens is downweighting of bristlcones – for which realclimate and Juckes use the code word “much of the data” – they never use the word “birstlecone” in this context. In an unguarded moment, Wahl and Ammann 2006 admitted that the downweighting from use of covariance PCs is precisely equivalent to bristlecone downweighting – I’ve got a pretty graphic in my Stockholm/Holland presentation illustrating this.
At the time of our 2003 paper where the differences first emerged, we had no idea that it was the bristlecones that caused the problem. We used covariance PCs because that is “conventional” PC methodology in a network denominated in common units and that’s what Mann said that he did. Obviously he did something different than what he said he did, but that’s a long and different story. We only became aware of the role of bristlecones by tracking what Mann’s biased methodology did – gradually realizing that all the series in what Mann described as the “dominant component of variance” came from Graybill and Idso’s strip-bark bristlecone and foxtail network. Later we realized that Mann’s CENSORED directory contained a sensitivity study of what happened without the bristlecones and foxtails – so he realized what was going on long before we did.
A covariance PC emphasizes different data than the bristlecones emphasized by Mann’s data mining method. We do not argue that this index is a plausible temperature indicator – the obligation to do so rests with the proponents of PC analyses as a means of extracting temperature from tree ring networks. We can categorically say that we did not select a PC methodology with a view to downweighting bristlecones; it was only through patient detective work that we learned of the effect of bristlecones.
Update: For convenience, here are relevant quotations from statistical authorities on the selection of PC methodologies (keep in mind that tree ring networks are already in common dimensionless units through standardization):
Preisendorfer is cited in MBH98 as an authority for principal components. In Overland and Preisendorfer , Presiendorfer stated:
In representing the variance of large data sets, the covariance matrix is preferred” (p.4)
I have not found a comparably explicit statement in Preisendorfer . However, this text nearly always talks in terms of “covariance matrices”, rather than “correlation matrices”, such as the following quote:
The first step in the PCA of [data set] Z is to center the values z[ t,x] on their averages over the t series… If Z… is not rendered into t-centered form, then the result is analogous to non-centered covariance matrices and is denoted by S’. The statistical, physical and geometric properties of S’ and S [the covariance matrix] are quite distinct. PCA, by definition, works with variances i.e. squared anomalies about a mean. (p. 26)
Preisendorfer  never suggested a change from the position of Overland and Preisendorfer , which is cited approvingly on several occasions in Preisendorfer . The standard examples in Preisendorfer  are data sets (e.g. sea surface temperature, sea level pressure defined over regions), which are not scaled to unit variance. While Preisendorfer explicitly calls for centering over the time average, he NEVER calls for scaling to unit variance on these data sets which are in common units but have differing variances. Preisendorfer discusses the covariance matrix on virtually every page, but only mentions a correlation matrix on a couple of occasions. The only occasion in which Preisendorfer calls for scaling to unit variance is when a data set is a composite of two data sets denominated in different units, e.g. sea surface temperature in one region and sea level pressure in another — which is not the case in the dataset under discussion here as discussed above.
Rencher  is another prominent statistical authority (cited by Huybers), who also stands as authority for use of covariance matrices as follows:
Generally extracting principal components from S [covariance] rather than R [correlation] remains closer to the spirit and intent of principal component analysis, especially if the components are to be used in further computations. (p. 430)
This follows almost verbatim a similar comment in Rencher [American Statistician 1992, 46, p 221], which said:
“For many applications, it is more in keeping with the spirit and intent of this procedure to extract principal components from the covariance matrix S rather than the correlation matrix R, especially if they are destined for use as input to other analyses. However, we may wish to use R in cases where the measurement units are not commensurate or the variances otherwise differ widely.”
The only exception contemplated by Rencher  is a circumstance in which a few series measured in different units have much larger variance than the bulk of the data, because it would dominate the analysis:
“When one variable has a much larger variance than the other variables, this variable will dominate the first component”.
This is the exact opposite situation to MBH98, where the exception is two density series that have much smaller variances, and moreover are from sites already represented by width series in the major partition of the dataset.
I have been unable to locate any third-party statistical authority that recommends use of correlation matrices in networks denominated in common units. I’m not saying that such a reference doesn’t exist anywhere, but the challenge to produce one has been outstanding for some time and so far no one has located one.