ARMA (1,1) Coefficients for CRU Gridcells

I did an ARMA (1,1) on the CRU monthly (as opposed to the usual annual) global data set to see what it looked like. The ARMA (1,1) coefficients for the GL data for the second half of the data set were similar to those observed in the satellite data (AR1 +0.93; MA1 -0.46), but were different in the first half (AR1 +0.71; MA1 -0.27). The "look" of the persistence in the 19th century was quite a bit different than in the 20th century. I don’t see why the ARMA(1,1) structure should change with increased temperatures or increased CO2 and I wonder how much of this is due to measurement issues. It may be an interesting quality control cross-classification.

Here’s another interesting calculation – the AR1 and MA1 coefficients for all CRU gridcells by gridcell (this is from the July 2003 CRU dataset). There seems to be a strong ARMA (1,1) structure to the underlying temperature data(whatever the basis) – it seems plausible that ARMA (1,1) outliers would be an interesting form of quality control on the CRU data set: I wonder what causes the outliers, which are obviously more prevalent in higher latitudes. I can see weakening of the AMRA(1,1) structure away from the tropics, but negative AR1 coefficients are surprising.

Figure 1. ARMA (1,1) Model for CRU Gridcell Temperature Data By Latitude. Top: AR1 Coefficient; Bottom – MA1 Coefficient.


  1. Peter Hartley
    Posted Aug 13, 2005 at 12:30 PM | Permalink

    An interesting research exercise might be to take the satellite temperatures for different latitude bands (if not CRU grid cells), estimate separate ARMA models for them and then compare them to the corresponding ARMA processes for the CRU data as you heve presented them here. Do you know whether it is easy to get the satellite data by latitude and longitude grid cells?

    Steve: Yes. It retrieves quite easily. The cell size is 2.5 x 2.5 instead of 5×5 so a little fiddling is needed to make them compatible. There’s lots of interesting exercises with the temperature data set. I wish that I’d left it alone.

  2. Ross McKitrick
    Posted Aug 13, 2005 at 4:20 PM | Permalink

    This graphic reminds me of a similar one in Karner’s latest paper (Central European Jnl of Physics, see ffor pdf) where he graphs the Hurst coefficient H by latitude band. People have been looking for evidence of polar greenhouse effect amplification by looking for increased trend coefficients near the poles. But increased trend coefficients is consistent with other hypotheses as well so it’s a confounded test. It seems to me a more precise test is decreased speed of mean-reversion, which is to say that, given a + greenhouse shock there is a slow, and possibly zero-speed reversion to the previous mean (ie nonstationarity). If greenhouse warming is amplified and more dominant in the polar regions then persistency should increase, or in this case the AR1 coefficient should approach 1, though it’s not measuring quite the same thing as H. A lower AR coefficient near the poles implies stronger mean-reversion on a short time scale; the negative AR coefficient just implies an oscillation rather than a decay back to the mean. If polar amplification is really so strong at the surface it seems to me the graphs should be U-shaped.

  3. JerryB
    Posted Aug 13, 2005 at 5:50 PM | Permalink

    > There’s lots of interesting exercises with the temperature data set.


    > I wish that I’d left it alone.

    Best punchline of the month!

  4. Sandy Taylor
    Posted Aug 13, 2005 at 9:19 PM | Permalink

    You wanted one of the 500,000, well I’m about 200 of those.
    Thanks for your efforts to keep the PC world and their true believers honest. I know how hard it can be to be the one eyed man in the world of the blind.

    I’ve been reading this blog for a few months. I’m naturally skeptical and found this site through at general search on the topic of global warming. It’s one of the few I come back to and check at least weekly. I don’t trust the “global warming is obvious believers” about how obvious it is, but I am still willing to read their sites. No matter how many times I email them articles to read, they never do (move into the light, Carol Anne). I did have my faith shaken by reading Roy Spencer, a big U Alabama climate scientist, come down on the ID side of the evolution debate, but I’m still skeptical.
    Keep on keeping them honest, if you don’t I don’t know who will.

  5. Steve McIntyre
    Posted Aug 14, 2005 at 1:24 PM | Permalink

    ###CALCULATION OF ARMA COEFFICIENTS load("c:/climate/data/jones/") stat<-array(rep(NA,3*2592),dim=c(2592,3)) count<-apply(!,2,sum) #this calculaiton had to be re-started on a couple of occasions due to arima calculation problems; I just determined where it stopped ansd started at the next number for (i in 1:2592) { if ( count[i]>50) stat[i,]<- coef( arima(v[,i],order=c(1,0,1)) ) } dummy<-array(c(1:2592,jonesinv(1:2592)),dim=c(2592,3)) stat<-cbind(dummy,stat) trend<-rep(NA,2592) #this is OLS trend year<-(1:1769)/12 for (i in 1:2592) { if (count[i]>50) { fm<-lm(v[,i]~ year) trend[i]<-coef(fm)[2] } } stat<-cbind(stat,trend) arma.coefficients<-stat save(arma.coefficients,file="c:/climate/data/jones/")##PLOT BY LATITUDE nf <- layout(array(1:2,dim=c(2,1)),heights=c(1.1,1.3)) par(mar=c(0,4,2,2)) plot(stat[,1],stat[,3],xlab="",ylab="AR1 Coef",axes=FALSE) axis(side=2) axis(side=1,at=seq(-60,+60,30),labels=FALSE) box() par(mar=c(4,4,0,2)) plot(stat[,1],stat[,4],xlab="",ylab="MA1 Coef",axes=FALSE) axis(side=2) axis(side=1,at=seq(-60,+60,30)) box()

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