More on global temperature spectra and trends.

In my previous post, I looked at ARIMA models and their effect on uncertainty of temperature trend, using autocorrelation functions. I was somewhat surprised by the strength of apparent periodicity with a period of 3-4 years.

It is plausible to connect this with ENSO, so in this post I want to probe a bit with FFT's, removing ENSO and other forcings in the spirit of Foster and Rahmstorf. But first I'd like to say something about stat significance and what we should be looking for.

Statistical significance is to do with sampling. If you derive a statistic from a sample of a population, how much would it vary if you re-sampled. How this applies to time series statistics isn't always clear.

If you measure the average weight of people on a boat, and you are interested in whether that boat will float, then yo don't need statistical significance. You only need to measure accurately. But if you want to make inferences about a class of people on boats, then you do. And you need to define your population carefully, and decide to worry about variables like age and sex.

A trend is just a statistic;  is a measure of growth. You can quote a trend (for a fixed period) for something that is growing exponentially, not of course expecting it to be constant over other times. Trend is a weighted average, and the same issue of statistical significance applies. You can quote a trend for 20 Cen temperature anomaly without worrying about its properties as a sample, or that it was not very linear. But if you want to make inferences about climate, you do need to treat it as a sample, and form some model of randomness. That involves the study of residuals, or deviation from linearity, and so apparently non-random deviation from linearity also becomes important, because you have to distinguish the two.

The autocorrelation functions of the previous post illustrate (and help to solve) the issue. Short term lags form the basis of the stochastic models we use (ARIMA). But once we get beyond lag ten or so, there is behaviour that doesn't look like those models. Instead it is due to periodicities, and quadratic terms etc.

Now in that case, the (lag stochastic/secular) regions were not well separated, and it is likely that the secular effects are messing with the ARIMA models.

So, coming back to what "sampling" means here. It's a notion of, what if we could rerun the 20th cen? Of course, with GCM's we can. But would the secular effects repeat too? ENSO is a case in point. It has aspects that can be characterised - periods etc. But they aren't exactly predictable. Phase certainly isn't. Should they have some kind of stochastic representation?

We can defer that by using what we know of that instance of ENSO forcing (and volcanic and solar) to try to focus more on the ARIMA part of the acf's. I'll do that here by looking at the Foster and Rahmstorf residuals.

Power spectra

But first I'll revisit the previous analysis and its apparent periodicity. Here are the acf's:

HADCRUT GISS NOAA UAH MSU-RSS

And here are the corresponding power spectra, which are the discrete Fourier transforms of the acf's

HADCRUT GISS NOAA UAH MSU-RSS

You can see the prominent peak at about 0.22 yr^-1 (45 months) which Is likely ENSO related.

Removal of forcings

Foster and Rahmstorf (2011) used known forcings (ENSO, volcanic and solar) to regress against temperature, and showed that the residuals seemed to have a much more even trend. I did my own blog version here. But for this exercise I used the F&R residuals described by Tamino.

First, here are the power spectra. The marked frequency at about 0.22 yr^-1 (or its harmonic at 0.44 yr^-1) is still often dominant, but its magnitude is reduced considerably:

HADCRUT GISS NOAA UAH MSU-RSS

And here are the acf's. The critical thing is that the oscillation near zero is reduced, and intrudes less on the peak at zero, which is smaller, and should be freer of secular influences.

HADCRUT GISS NOAA UAH MSU-RSS

Here are the acf's with models fitted. It is no longer the case that ARMA(1,1) is clearly better than AR(1), and both taper close to zero before the acf goes negative. For the satellite indices, there is little difference between the models. In all cases, the central peaks of the acf's are much less spread, indicating much tighter CIs.

HADCRUT GISS NOAA UAH MSU-RSS

Finally, here is the table of model fit results:

Result	HADCRUT4	GISS	NOAA	UAH	MSU-RSS
OLS trend β	1.7020	1.7092	1.7500	1.4086	1.5676
OLS s.e. β	0.0463	0.0651	0.0514	0.0665	0.0641
AR(1) trend β	1.6984	1.7084	1.7449	1.4161	1.5678
AR(1) s.e. β	0.0785	0.0936	0.0765	0.1101	0.0978
AR(1) Quen s.e.	0.0781	0.0933	0.0762	0.1096	0.0974
AR(1) ρ	0.4802	0.3445	0.3735	0.4613	0.3948
ARMA(1,1) trend β	1.6953	1.7025	1.7423	1.4188	1.5687
ARMA(1,1) s.e. β	0.0946	0.1132	0.0936	0.1189	0.1027
ARMA(1,1) Quen s.e.	0.0940	0.1123	0.0930	0.1182	0.1022
ARMA(1,1) ρ	0.6901	0.6433	0.6672	0.5696	0.4849
ARMA(1,1) MA coef	-0.2744	-0.3378	-0.3447	-0.1353	-0.1058

As Foster and Rahmstorf found, the trends are not only quite large, but have much reduced standard errors, whatever model is chosen.

Conclusion

Subtracting out the regression contributions of the forcings (ENSO, volcanic and solar) does reduce the long-lag oscillatory contributions to the acf, though they are not eliminated. This enables a better separation between those and the stochastic effects.