Ritson at realclimate did not thank me for helpful discussions on autocorrelation despite lengthy correspondence on my part with him. I thought that the histogram that I posted up earlier today looked familiar. So I looked back at my correspondence with Ritson (who posted up on autocorrelation at realclimate and sure enough, I’d sent the identical histogram to him in November 2004. So Ritson had seen correctly calculated autocorrelation coefficients a long time ago. The letter was interesting to re-read in the present context.
I tend to be most interested in empirical points and from reading your paper, I see one very obvious bit of empirical information which we could helpfully report (and I think that the absence of this may have been frustrating you) – the AR1 coefficients of the North American tree ring network. In the AD1400 network used by Mann, there are 70 sites with AR1 coefficients ranging from 0 to 0.79. Below is a histogram. Obviously the AR1 persistence in these tree ring site index series is greater than in temperature series and I think that this turns out to be very important both in principal component analysis (especially as done by Mann) and in the downstream regressions. Curiously, the AR1 coefficients seem to me to be more strongly correlated to the author than to any other variable. Series done by Stahle have little autocorrelation (and Durbin Watson statistics around 2), while series done by Jacoby and Graybill have very high autocorrelation and Durbin Watson statistics sometimes under 1. This is inadequately discussed in the literature.
Secondly, the AR1 coefficients underestimate the actual persistence in many sites. For example, the Sheep Mountain site has an AR1 coefficient of 0.76 under an ARMA (1,0) model, but its actual persistence is much greater. The ACF is shown below, with the red line showing the iterated AR1 coefficient. I’ve gotten interested in fractional processes to deal with sort of situation, following Mandelbrot (who actually calculated Hurst parameters for some tree ring series). I suspect that what I’ve described as a fractional process (using Whitcher’s algorithm following Hosking (1981)) would be recognisable to you as your 1/f process. Interestingly, Hurst of the Hurst parameter in 1/f processes was a hydrologist, who studied fluctuations of the Nile, a climatic series.
Thirdly, here is a graphic showing the relationship between the weighting of a site chronology under Mann’s PC method and its AR1 coefficient. I think that you will agree that it is a very strange scatter plot. The 14 series on the right account for over 99% of the variance in the PC1. As you see, there is a strong association between the AR1 coefficient and the EOF1 weighting – which is not an effect that one would normally seek and surely points to some problem with the method. The over-printed sites are sites which Mann excluded in an unreported sensitivity study – which gave him very different results than he reported. As I’ve mentioned to you before, I am very struck by this omission, which would not be legal to omit in a securities offering. When you look at the sites marked with the overprint, they are all bristlecone pine sites, mostly from Donald Graybill and reported in Graybill and Idso (1993). There are many curious features about bristlecone pines and the growth index has never been shown to be a climate proxy. The extreme right hand site is the one shown in the above ACF function.
I find it incredible that Ritson can seriously propose AR1=0.15 as a model for a series with an autocorrelation function looking like Sheep Mountain in the second figure. Especially when the issue had been specifically brought to his attention.
As a passing comment, the above letter also illustrates how constructive I was in correspondence. I probably have been a little unguarded in this respect as, for example, I sent Bürger a considerable amount of detailed information, but did not receive any acknowledgement in Bürger et al 2006.