There has been considerable recent blog discussion of Rahmstorf smoothing and centering, with attention gradually being increasingly directed towards Rahm-sea level.
Last year, these topics were discussed online by Tom Moriarty here in a posting at his blog which unfortunately did not receive as much attention as it deserved. Following recent discussion at CA, Tom posted a follow-up article here. Tom and I got in touch and, from that correspondence, I learned that Rahmstorf had commendably sent code for his sea level article to Tom, commenting that he was “the first outside person to test this code”.
As we’ve learned, such testing is not a prerequisite for publication in the most eminent science journals (Science in this case) nor for use in “important” reviews, such as the Copenhagen Synthesis.
I’ll demonstrate that I’ve got the right data and have emulated things correctly. I’ll conclude this post with an interesting plot of the actual data – something that was omitted in Rahmstorf 2007.
Ramhstorf first rahm-smooths the GISS temperature and sea level ( as usual, using ssatrend Rahmstorf “15-dimensional embedding”, which I’ve shown elsewhere to be an alter ego for a 29-point triangular filter). Then he calculated the sea level rise. Then he binned both into 5-year intervals and did (in effect) a linear regression. R07:
A highly significant correlation of global temperature and the rate of sea-level rise is found (r = 0.88, P = 1.6 × 10−8) (Fig. 2) with a slope of a = 3.4 mm/year per °C.
These results are obtained through a simple linear regression of binned and smoothed rise against binned and smoothed temperature. No allowance is made for the reduced degrees of freedom for autocorrelation. (After making such an allowance, needless to say, there are virtually no degrees of freedom left.) R07 illustrated this regression through his Figure 2, emulated below. This “relationship” is used throughout the article.
Next here is an emulation of the top panel of R07 Figure 3, showing the smoothed rate of sea level rise against the fit from the smoothed GISS temperatures. (Actually and this is something that may surprise readers – the red curve has been rahm-smoothed twice!) I’ll show some further information on this “relationship” below.
Figure 2. Emulation of R07 Figure 3 top panel.
Next here is an emulation of R07 Figure 3 bottom panel, showing the fit of the semi-empirical relationship – this is carried forward into the Figure 4 projections which follow/
Figure 3. Emulation of R07 Figure 3 bottom panel.
Rahmstorf then takes various IPCC projections and uses the R07 “relationship” between temperature and sea level to project sea level.
Figure 4. Emulation of R07 Figure 4.
Something that isn’t actually shown is R07 is a plot of the data, an omission which is remedied below. To me, these curves look like they have very little relationship. But if you’re a Copenhagen Synthesizer or a Science reader, this relationship is, I guess, 99.99999% significant.
What happens when the Rahmstorf relationship is adjusted for autocorrelation, along the lines of the Steig corrigendum (or the Santer, Schmidt 2008 critique of Douglass et al 2007).
There are only 24 bins. The AR1 autocorrelation of residuals is 0.75, resulting in N_eff of 3,36 (using N_eff=N*(1-r)/(1+r) ).
Whereas the OLS standard error was 0.039, the AR1 adjusted standard error is 0.156 (using:
se.obs= sqrt((N-2)/(neff-2))* summary(fm)$coef[2,"Std. Error"];se.obs
With the reduced degrees of freedom, the benchmark t-statistic is closer to 3 than to 2:
t0= qt(.975,neff) # 2.997517
Instead of a rahm-significant relationship as claimed, the confidence intervals are: 0.34 0.47 (not significant at all).
(ci=t0*se.obs) # 0.4659963
Having said that, it makes sense to me that higher temperatures would result in higher sea levels. I think that a heuristic diagram comparing the total sea level rise to the total increase in temperature in the historical period would probably make some sense. It’s Rahmstorf’s effort to dress a heuristic relationship in the language of statistics that fails so miserably. With the recent Steig precedent on the need to issue a corrigendum for failing to allow for autocorrelation, Rahmstorf really needs to do a similar corrigendum.
It’s pretty hard to keep up with Team corrigenda.