In our GRL article, we pointed out that MBH98 had misrepresented their principal components methodology as being "conventional", when it wasn’t.
At realclimate , they argued that it was an alternative centering "convention".
Since they elsewhere rely on Preisendorfer, it’s interesting to see what Preisendorfer has to say about centering.
On page 26 of his opus Principal Component Analysis in Meteorology and Oceanography, Preisendorfer says:
t-centering the data set
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… Using these t-centered values z[t,x], we form a new n x p matrix.
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 properties of S’ and S [the covariance matrix] are quite distinct. PCA, by definition, works with variances i.e. squared anomalies about a mean.
According to Preisendorfer, centering is not a "convention"; it is integral to PCA.
I doubt that many paleoclimatologists have read Preisendorfer’s book. It’s not easy. I don’t know his background, but, from reading this book, Preisendorfer seems like a very capable mathematician who happened to end up in meteorology. For example, he has a section on abstract PCA and is attentive to dual spaces, a fundamental concept in abstract linear algebra, but not one that you see a whole lot in paleoclimate. It has lots of interesting digressions – for example, there’s an interesting digression on the distribution of eigenvalues in random matrices that leads to very difficult mathematics. (Persis Diaconis has written on these problems and has a survey which is internet accessible.)