Here’s an interesting little graphic and analysis of the new satellite data. It’s hard not to scratch your head sometimes at this entire subject matter, when you see the effect of pretty simple alternatives. The satellite data is modelled very nicely as ARMA c(1,0,1), which would imply entirely different conclusions about this data set and completely different projections.
Figure 1. Global Satellite (downloaded August 9, 2005). Black- raw data; red- arima c(1,0,1) fit and projections; blue – "trend". Updated below.
If you simply look at a plot of the satellite data, it looks like data generated from an arima (or even an ARMA) process. Arima models are about 2nd year statistics and not much harder than trend lines and well within the reach of even climate scientists. In very rough terms, an ARMA model is a bit like your bankroll when you’re gambling in a "fair" game. The fit with an ARMA process (red) with only two parameters (the same number as fitting the "trend" line) is obviously pretty spectacular as compared with the trend. For those interested, the log-likelihood of the ARMA model is 250.7, as compared with the logLik for the trend model of only 95.0 (this more or less quantified the better fit of the red than the blue.)
The ARMA coefficients are AR1 of +0.9257 and MA1 of -0.3183. This is a very high AR1 coefficient; I’m going to be posting about AR1 coefficients in some upcoming posts. The projection using the ARMA process is shown: it is a "fair" game and reverts to 0, with a 2-sigma confidence interval of 0.39. deg C. The "trend" is 0.122 per decade. The difference between the two models illustrates a very important statistical point in a simple case: the residuals from the trend are not white noise, and so the "trend" model is mis-specificied in statistical terms. I’ve not shown here the mis-specification in the "trend model" through the trend residuals – I’ll have to look up how that’s done. However, the counter-example with a superb fit with a c(1,0,1) ARMA error structure is really quite remarkable.
The obvious question is: does this data set show any statistically significant trend at all? It’s not the issue that people are discussing. The appropriate comparison is to ARMA models, as shown here. I’m not familiar with satellite literature and perhaps this ARMA modeling is old hat to them, but it’s sure not at the tip of their tongues (both sides) or we would be seeing quite different discussions about "trends". I’m not saying that there isn’t a trend here – only that the statistical methods being used appear to be so naive that no light is being shed on the matter – actually it’s worse than no light. To examine the matter from first principles really needs to be done.
UPDATE (Evening) : I did a new ARMA run adding a time-term for regression. The regression coefficient (with ARMA errors) was 0.0000223329 deg C/decade and the logLikelihood was essentially unchanged at 251.05 (versus 250.71.) Obviously, in statistical terms, you cannot reject the null hypothesis of no trend. I re-emphasize that the usual "trend" calculation looks to be mis-specified. This is a very routine calculation that I’ve done and you would think that it would have been discussed somewhere in the huffing and puffing about satellite versus surface. We shall say. In the mean time, I don’t see anything wrong with the calculation that I’ve done; it "looks" logical as well.