Tamino’s guest post at RC deals with global mean temperature and AR(1) processes. AR(1) is actually mentioned very often in climate science literature, see for example its use in the Mann corpus (refs (1,2,3,4). Almost as often something goes wrong (5,6,7). But this time we have something very special, as Tamino agrees at realclimate that AR(1) is an incorrect model:
The conclusion is inescapable, that global temperature cannot be adequately modeled as a linear trend plus AR(1) process.
This conclusion would be no surprise to Cohn and Lins. But if their view is that global temperature cannot be adequately modeled as a linear trend plus AR1 noise, what are we to make of IPCC AR4, where the caption to Table 3.2 says
Annual averages, with estimates of uncertainties for CRU and HadSST2, were used to estimate. Trends with 5 to 95% confidence intervals and levels of significance (bold: less than 1%; italic, 1 – 5 %) were estimated by Restricted Maximum Likelihood (REML; see Appendix 3.A), which allows for serial correlation (first order autoregression AR1) in the residuals of the data about the linear trend.
This was mentioned here earlier (9). Thus, according to Tamino the time series in question is too complex to be modeled as AR(1)+ linear trend, but IPCC can use that model when computing confidence intervals for the trend!
But there is something else here, how did Tamino reach that conclusion? It was based on this figure:
It would be interesting to try to obtain a similar figure, here’s what I managed to do :
First, download (10) and arrange data to one 1530X1 vector. Divide by 100 to get changes in degrees C. Then some Matlab commands,
n1=detrend(data); % Remove a linear trend, center
[Xn,Rn]=corrmtx(n1,12*12); % Autocorrelation matrix
% note that signal processing guys sometimes
% use odd definition for correlation vs. covariance
% (not center vs. center)
cRn=Rn(:,1)/Rn(1,1); % divide by std
fi=find(cRn<0); % find first negative
thn=-(1:fi(1)-2)’./(log(cRn(2:fi(1)-1,1))); % -dt / ln
Black line is quite close:
Next step is trickier, how to make those AR(1) simulations? For a monthly data, tau=5 years= 60 months. Should we use one-lag autocorrelation p=0.985 then? 5 realizations in red color:
Not a good idea. How about AR(1) with p=0.985 + some white noise, again 5 realizations in red color:
Much better. No wait, I was trying to replicate Tamino’s simulations, not the GISS data result. Any suggestions?
UPDATE(SM): Lubo has an interesting post on this , also linking to a paper coauthored by Grant Foster and realclimate
2 Mann, M.E., Lees. J., Robust Estimation of Background Noise and
Signal Detection in Climatic Time Series , Climatic Change, 33,
3 Mann, M.E., Rutherford, S., Wahl, E., Ammann, C., Robustness of
Proxy-Based Climate Field Reconstruction Methods, J. Geophys. Res.,
4 Trenberth, K.E., P.D. Jones, P. Ambenje, R. Bojariu, D.
Easterling, A. Klein Tank, D. Parker, F. Rahimzadeh, J.A. Renwick,
M. Rusticucci, B. Soden and P. Zhai, 2007: Observations: Surface and
Atmospheric Climate Change. In: Climate Change 2007: The Physical
Science Basis. Contribution of Working Group I to the Fourth
Assessment Report of the Intergovernmental Panel on Climate Change
[Solomon, S., D. Qin, M. Manning, Z. Chen, M. Marquis, K.B. Averyt,
M. Tignor and H.L. Miller (eds.)]. Cambridge University Press,
Cambridge, United Kingdom and New York, NY, USA.