A Pretty Graphic of ARMA Coefficients

Here is a plot of ARMA (1,1) coefficients for the CRU data set plotted up against a world map. I ‘ve not seen anything like this plotted up before. I think that the patterns are really quite pretty.

I’ve mentioned a couple of other studies that have examined relationships between temperature and ARMA processes, but these have been (to my knowledge) at a global level. I’ve not seen gridcell detail as shown in the graphics here. I also don’t think that there has been as specific a focus on ARMA(1,1) processes; the tendency has been to use AR(k) models, but some nice clean information is obviously coming out of the ARMA(1,1) models at gridcell detail. The tropical oceans show very clearly in extremely AR1 coefficients. Continental AR1 coefficients are markedly lower (shown even more clearly in the square root coloring scheme in the figure at the bottom of the page, which accentuates this.) MA1 coefficients in the oceanic Southern Hemisphere are consistently stronger than in the more continental NH. Some nice detail features can be observed: e.g. the El Nino tongue has lower MA1 coefficients. DETAIL

Figure 1. ARMA(1,1) Coefficients for Gridcell CRU Data. Top – AR1 Coefficient; Bottom – MA1 – Coefficient

Coefficients are calculated here if there is a count of at least 100 measurements using the na.pass option. This means that coefficients are calculated for a lot of series with very sparse measurements. I suspect that the cacophony of colors in the Southern Ocean results from calculations on sparse information, rather than actual discontinuities in ARMA coefficient behavior. I get the impression that CRU has grabbed measurements in sparse areas and homogeneity is a hope rather than something that has been demonstrated – the evidence of these coefficients certainly suggests a great deal of non-homogeneity in the data in this area. If you squint, you can see some suspicious entries elsewhere: e.g. northeast Brazil has a gridcell with a highly positive MA1 coefficient; the far northeast Siberian AR1 coefficients look suspicious.

Figure 2. ARMA(1,1) Coefficients for Gridcell CRU Data. Top – AR1 Coefficient; Bottom – MA1 – Coefficient. Both on sqaure root scale.


  1. Armand MacMurray
    Posted Aug 14, 2005 at 10:53 AM | Permalink

    Very nice images (and neat idea!). Might I suggest making these and similar plots 2x as wide and tall, pixel-wise, to make it easier to spot the fine details?

  2. Steve McIntyre
    Posted Aug 14, 2005 at 1:08 PM | Permalink

    HEre’s a script to produce this graphic (it requires prior calculation of coefficients). It’s crazy how simple it is to produce fancy stuff with modern methods.


    # postscript(file.path(“c:/climate/data/jones/arma.coefficients.eps”), width = 8, height = 6,
    # horizontal = FALSE, onefile = TRUE, paper = “special”,
    # family = “Helvetica”,bg=”white”,pointsize=8)
    # this is used to create larger image


  3. Steve McIntyre
    Posted Aug 14, 2005 at 1:12 PM | Permalink

    Armand, try http://www.climateaudit.org/wp-content/arma.coefficients.eps

  4. Posted Oct 16, 2005 at 5:58 PM | Permalink


    sorry for the late reply.
    As far as I understand (KISS since I am no expert), the first AR1-coefficient tells us how good the system can remember its previous state whereas the second MA1-coefficient tells us how good random influences will remember (or compensate for) their previous influence. Is that basically correct?

    Given that I don’t get the point looking at the varying MA1-coefficients in the polar regions, especially the antarctic region. Altough gridcell aereas are much smaller there, some show a tendency to compensate for previous influences while their gridcell neighbours behave just the opposite. Any idea wether that’s due to sparse data or zoom-in effects?

    Very interesting indeed – keep up the good work.

  5. Steve McIntyre
    Posted Sep 21, 2007 at 11:53 AM | Permalink


  6. Jean S
    Posted Sep 21, 2007 at 12:11 PM | Permalink

    Nice! This is from times before even I used to visit this site ;=) Think about it, only two real comments in a post!

    How about an update: the same plot but with detrended series? 🙂

  7. Mike B
    Posted Sep 21, 2007 at 1:45 PM | Permalink

    Thanks for bumping these Steve. Goes to show how much I missed on this site before I first visited.

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