I think that Preisendorfer would roll over in his grave if he saw how Ammann, Schmidt and Mann were bastardizing his Rule N.
Here’s what Ammann and Schmidt said at realclimate::
For instance, the data can be normalized to have an average of zero over the whole record, or over a selected sub-interval. The variance of the data is associated with departures from the whatever mean was selected. …
MM05 claim that the reconstruction using only the first 2 PCs with their convention is significantly different to MBH98. Since PC 3,4 and 5 (at least) are also significant they are leaving out good data. It is mathematically wrong to retain the same number of PCs if the convention of standardization is changed. In this case, it causes a loss of information that is very easily demonstrated.
This follows pretty similar hyperventilation by Mann here:
Claims by MM to the contrary are based on their failure to apply standard ‘selection rules’ used to determine how many Principal Component (PC) series should be retained in the analysis. Application of the standard selection rule (Preisendorfer’s "Rule N’") used by MBH98, selects 2 PC series using the MBH98 centering convention, but a larger number (5 PC series) using the MM centering convention. Curiously undisclosed by MM in their criticism is the fact that precisely the same “Åhockey stick’ pattern that appears using the MBH98 convention (as PC series #1) also appears using the MM convention, albeit slightly lower down in rank (PC series #4) (Figure 1). If MM had applied standard selection procedures, they would have retained the first 5 PC series, which includes the important ‘hockey stick’ pattern.
I talked about the issue of centering "conventions" the other day. Preisendorfer, cited here as an authority, said quite explicitly that principal components are about the analysis of variance and are "by definition" centered on the time-average (not on a subset.) The MBH98 method is not a "convention" or even principal components analysis. Preisendorfer:
If Z’ in (2.56) 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 interpretations of S’ and S are quite distinct. PCA, by definition, works with variances i.e. squared anomalies about a mean. (p.27)
The cheekiness never ceases to amaze me. We said quite clearly that the bristlecones were demoted from the PC1 to the PC4. And yet, Mann here has the gall to say that "curiously undisclosed" by MM is this very demotion. I’ve posted on several occasions on whether MBH98 actually used the form of Preisendorfer’s Rule N, claimed in recent realclimate posts. It is impossible to replicate the actual selection in other networks using this rule. However, today, I wish to talk about Preisendorfer’s own justification for Rule N – which I’ve argued to be at most necessary for significance and not sufficient. This seems to be an obvious distinction, but obviously not understood by realclimate. I’ve illustrated this obvious point by showing what happens if you replace bristlecones with dot.com stocks.
Here are some quotes from Preisendorfer on Rule N and other selection rules:
The null hypothesis of a dominant variance selection rule [such as Rule N] says that Z is generated by a random process of some specified form, for example a random process that generates equal eigenvalues of the associated scatter [covariance] matrix S…
One may only view the rejection of a null hypothesis as an attention getter, a ringing bell, that says: you may have a non-random process generating your data set Z. The rejection is a signal to look deeper, to test further. One looks deeper, for example, by drawing on one’s knowledge and experience of how the map of e[i] looks under known real-life synoptic situations or through exhaustive case studies of e[i]’s appearance under carefully controlled artificial data set experiments. There is no royal road to the successful interpretation of selected eigenmaps e[i] or principal time series a[j] for physical meaning or for clues to the type of physical process underlying the data set Z. The learning process of interpreting [eigenvectors] e[i] and principal components a[j] is not unlike that of the intern doctor who eventually learns to diagnose a disease from the appearance of the vital signs of his patient. Rule N in this sense is, for example, analogous to the blood pressure reading in medicine. The doctor, observing a significantly high blood pressure, would be remiss if he stops his diagnosis at this point of his patient’s examination. ….Page 269.
Thus, even if the PC4 is "significant" under Preisendorfer’s Rule N, that doesn’t mean that the PC4 is a temperature proxy. It means that you need to look at it. That’s what we did. But we found out, as Mann had found out before us, that the hockey stick pattern came from bristlecones, which had been identified by specialists as being related to CO2 (or other) fertilization in the 20th century and not due to temperature. There was a "ringing bell" – it should have been paid attention to.
One of the specific criteria set out for significance by Preisendorfer is that the analyst be able to identify a plausible "dynamical origin" for the selection:
In all of this research, one basic question should be kept in mind: is there a dynamical basis for the candidate selection rule? (cf Preisendorfer, 1979a), For practiced meteorologists and oceanographers, such rules should always be checked for the reasonableness of the synoptic patterns in the first few retained eigenvector maps. Both theorists and synopticians should agree that the basis for a selection rule should as far as possible be the dynamical process underlying the data set. (p. 251)
Preisendorfer’s concept of a "dynamical origin" is not fulfilled by ragbag tree ring networks. Presiendorfer’s examples are all PC calculations for data sets consisting of time series of uniform fields (temperature, sea level pressure) over gridded geographical regions. The regularity of the geographical index is not explicitly stated, but is clearly assumed in Preisendorfer. The MBH collection has 16 sites in the gridcell containing Sheep Mountain and 1 site in other gridcells; it has lots of bristlecones. It’s not geographically regular. The need for geographical regularity can be seen in the discussion of "dynamical origins" which requires that the underlying process be representable by differential equations. Preisendorfer has a lengthy discussion of the "asympototic PCA property" for one-dimensional harmonic motion (a spring-linked mass model) and for two-dimensional wave motion on a rectangular domain of an ocean surface. Preisendorfer:
The PCA property (2.96) of Z turns out to have deep connections with various models of physical processes studied in meteorology and oceanography. As we shall see in the next chapter, various data sets generated by solutions of any of a large class of linear ordinary or of linear partial differential equations exhibit the PCA property in the limit of large sample sizes n. When this is the case, the eigenvectors of the data set resemble the theoretical orthogonal spatial eigenmodes of the solutions. In this way, empirical orthogonal functions arise with definite physical meaning… The eigenmotions U of the dynamical system manifest themselves as the eigenvectors of the PCA of the dynamical field’s output field Z. The formal connections of PCA with harmonic analysis exist on quite general levels (p. 87)
“3. Dynamical Origins of PCA: We will give examples of the asymptotic PCA property (2.125) for two systems. This will provide an objective basis for the selection rules stated in Chapter 5.”
Rule N is described by Preisendorfer as a "dominant variance rule" and the chapter formulating Rule N is entitled “5b. Dynamical Origins of the Dominant-Variance Selection Rules. "
MBH98 itself referred to the use of Preisendorfer’s Rule N only in the context of temperature PCs, which do meet Preisendorfer’s conditions. The basis of selection for tree ring PCs was not explicitly stated. The supposed use of Preisendorfer’s Rule N was not mentioned until there was a need to include the bristlecones from a lower PC series. The ragtag and motley collection of tree ring site chronologies obviously do not meet the conditions used by Preisendorfer to justify Rule N. But more importantly, as Preisendorfer clearly says and is obvious, passing a Rule N significance test is merely a "ringing bell". It is not a sufficient test of statistical significance. Roll over, Preisendorfer, and tell Pachauri the news.