I never quite got to presenting an attempt to replicate Moberg before. Here’s a try. I’m still a long way off from being able to replicate his results. It is so infuriating to have to try to do such an amount of detective work prior to even atempting any analysis. I presume that has been Hockey Team strategy all along and, only the most intrepid and persistent, can even come close. I’ve provided a collation of Moberg’s data here together with a script showing my present emulation. The script is not very pretty and I’ll try to tidy it at some point when I return to this. I’ve also done a comparison of the archived Moberg reconstruction to actual CRU data with interesting results.
You may recall that I was on my way to trying to replicate Moberg when I ran into lack of data problems with a couple of series. Moberg wouldn’t provide the data – there also seem to be some question about the Lauritzen verion used. Rather than stew about it, I filed a Materials Complaint with Nature and to their credit, Nature has dealt with it. It seems that Moberg used data without permission, hence the problem. He’s going to have to publish a Corrigendum, after which he is supposedly going to provide the data. In the mean time, I’ve plodded along trying to replicate his results with the data versions that I have (although the discrepancies in the Lauritzen versions may be material.) Once again, it’s ridiculous that there’s no source code, making it a bit of a guessing game. Here are some preliminary results.
Figure 1 shows my emulation of Moberg as compared with the archived version, then the residuals. In this case, I’ve done the emulation using a discrete wavelet transform (DWT) rather than a continuous wavelet transform (CWT) which Moberg used, since (1) I’m used to working with the DWT; (2) I figure that any valid result should not be sensitive to the difference between the DWT and CWT; (3) Moberg did not justify a use of CWT in preference to DWT. This might account for the differences but you never know. I’ve posted up my code for this reconstruction. Any bright ideas would be welcome. The maximum difference between my emulation and Moberg’s archived version is 1.15 and the 95% confidence interval has a width of 1.18. By contrast, Moberg stated that the jack-knife confidence interval for this reconstruction was 0.23 standardized units. His confidence interval calculation is really hokey.
Figure 1. Top -emulation of Moberg; middle – archived Moberg; bottom – residuals.
Next is a figure showing the Moberg reconstruction against the CRU series (top panel) and residuals (bottom panel). Moberg is a coauthor of one of the Jones’ CRU temperature collations (Moberg and Jones .) The red lines in the bottom panel show the supposed "confidence intervals" of Moberg (the CI of the reconstruction in standardized units scaled to temperature using Moberg’s factor), which do not appear to bear any relationship whatever to residuals to actual CRU data. Moberg’s SI includes a description of how the confidence intervals are calculated. I wish that – just for once – the Hockey Team would involve an actual statistician in their calculation of confidence intervals rather than their typical concoctions. I’ll spend a little time on this, but, unfortunately, it always takes a while to decode Hockey Team shenanigans.
Below I’ve done a simple plot of the archived Moberg reconstruction against the actual CRU dataset being modeled. You’d think that Moberg and his crowd would have done something as simple as plotting the residuals against the confidence interval. Look at the autocorrelation of the residuals.
Figure 2: top – Moberg reconstruction vs CRU; bottom; residuals with "confidence intervals" in red.
Moberg does not present elementary statistics on the residuals. For example, the Durbin-Watson statistic is 0.6 – a value in the spurious regression range. It is unacceptable for the hockey Team to simply ignore this. See my discussion of Granger and Newbold  who said over 30 years ago:
It is very common to see reported in applied econometric literature time series regression equations with an apparently high degree of fit, as measured by the coefficient of multiple correlation R2 or the corrected coefficient R2, but with an extremely low value for the Durbin-Watson statistic. We find it very curious that whereas virtually every textbook on econometric methodology contains explicit warnings of the dangers of autocorrelated errors, this phenomenon crops up so frequently in well-respected applied work. Numerous examples could be cited, but doubtless the reader has met sufficient cases to accept our point. It would, for example, be easy to quote published equations for which R2 = 0.997 and the Durbin-Watson statistic (d) is 0.53.
I’ve re-read Moberg’s discussion of Confidence Intervals and it makes me ill. And to think that Dano slags me as an "amateur".
Update: Doug Hoyt inquired as to the residuals from my emulation of Moberg. Here’s something curious: my emulation has better performance than Moberg’s original result in the various statistics that I’ve examined so far (and I’ve not been exhaustive). The igure below shows the same as Figure 2 for my emulation.
Figure 2: top – Emulation of Moberg reconstruction vs CRU; bottom; residuals with "confidence intervals" in red.
The correlation of the emulation to CRU is 0.42 ( Moberg – 0.30), and the match is obviously better if you compare the two graphs. The Durbin-Watson is still in the red zone at 1.36 (below 1.5 is in the red zone), but is less bad than the DW for the archived version which is a ghastly 0.6. So riddle me this: why is Moberg’s reconstruction using a CWT "right" as opposed to a reconstruction using a DWT?