Once again, I am nattering on about volcanoes and GMST (global mean surface temperature).
Yes, volcano eruptions affect GMST. Does anyone really doubt this anymore? Though no one doubts volcano eruptions affect GMST, it appears some wish to estimate the variability of GMST (weather noise) since 2001 as if a volcano eruption the size of Pinatubo “might” have occurred sometime in the last 8 years.
In today’s post, I’ll actually make a statistical argument showing that the variability of “weather noise” during periods with major eruptions like Pinatubo and El Chicon is greater than during periods when the volcanoes aren’t erupting. This means when doing hypothesis tests on GMST data collected during periods with no major volcano eruptions, we should not use the ARMA(1,1) parameters Tamino suggested. Rather, we should compute them based on periods without eruptions the size of Pinatubo, or El Chicon. 🙂
As usual, the main result is a picture at the end. Feel free to s_c_r_o_l_l down!
Background
Some of you recall Tamino recently suggested that the earth’s “weather noise”– i.e. the variability in global mean surface temperature — GMST is reasonably well described by an ARMA(1,1) process, which I endorse as better than AR(1) only. (Obviously, I would not object to ARMA(1,1): I was already using AR(1)+weather, which Tamino agrees is equivalent to ARMA(1,1).)
However, after deciding to use ARMA(1,1), Tamino decided to estimate parameters for the ARMA(1,1) process using observations of GMST from a period of time affected by three major volcanic eruptions: by major, I mean eruptions that hurled significant amounts of aerosols into the stratosphere. Pinatubo, and El Chicon both erupted during the period used to compute parameters, and the stratosphere was just clearing from the Fuego eruption.
Because major eruptions of this sort are believed to cause the GMST to first fall, and then recover, I object to estimating the variability of “weather” after 2000 using that data from that time periods. After all: the stratosphere have been entirely clear of volcanic aerosols using weather data from a time when the aerosols were rapidly varying.
After all, the recent flat temperature trend is not due to Pinatubo. So, why estimate the variability of weather during years with no eruptions using a period with a huge dip due to Pinatubo and another one due to El Chicon? (And don’t forget the small Fuego recovery.)
But maybe phenomenological arguments aren’t enough.
Why not do a statistical test?
I think it’s always worthwhile to test my phenomenological arguments using statistics. Consequently, I decided I would compare the residuals to an ordinary least squares fit to GMST vs time, during the period of time chosen by Tamino to a period of time when aerosol loadings varied less. Residuals are a first order measure of “weather noise”, so that seemed like a nice easy test.
By focusing my test on residuals, Tamino’s time period, and the “volcano free” period I’ve discussed in earlier blog posts, my question about variability becomes,
“Are the residuals to an ordinary least squares fit during the 403 months since 1975 consistent with the ARMA(1,1) that fits the weather from Nov. 1913- Dec 1943.”
The first period of time is the period Tamino selected to show that weather is highly variable, which would tend to make it difficult to test hypothesis. The second period of time is a period when volcanic aerosols were relatively light. The choice of that periods is discussed in a previous blog post.
If the answer to the question is, “No, the residuals to the OLS fit since 1975 are inconsistent with the ARMA(1,1) weather noise during the “no-volcano” period.” that would strongly support my contention that we can not use the ARMA(1,1) fit since 1975 to estimate “weather noise” for the period since 2001. This is because we know there have been no volcanic eruptions. S
If the answer is, “Yes, the residuals from Tamino’s time are consistent with the earlier time period, then either a) I am wrong and the volcanoes don’t increase variability of GMST, or ) the volcanoes do increase variability in GMST, but there is insufficient data to demonstrate any statistically significant effect.
Nuts and bolts
Anyway, I did the following:
- Selected GISS Temp Land/Ocean. (I can repeat for others on request. I haven’t done them, the answer might be different.)
- Performed OLS fits to the data during the two periods. In particular, I obtained the standard error of the residuals for the fit for both eras.
- Obtained the ARMA(1,1) parameters for the “no volcano” period as described here. So, this is “no volcano” ARMA(1,1) for GISSTemp, rather than Hadley discussed in the earlier blog post.
- Wrote a script to create 403 months of “simulated weather” using the ARMA(1,1) obtained above. Repeated 10,000 times.
- Performed an ordinary least squares (OLS) fit to each of the 403 months of “simulated weather”, and computed the residuals to the fit. (These are “sy” if you use LINEST” in EXCEL. ) This gave me 10,000 sets of residuals that might occur if the “weather noise” had properties similar to those in 1913.
- Compared the sample standard deviation of residuals to the OLS fit to 403 months of data from 1975-now to the distribution standard deviation of residuals that would be produced if the “weather noise” is described as ARMA(1,1), but the parameters are those from the “volcano free” period described above.
The result?
A picture is better than words:
(click for larger)But words are also good!
The standard deviation of residuals from 1975-now was sGMST=0.1477C. This exceeds 98.9% of standard deviation of residuals that would occur if the “weather noise” was ARMA(1,1) with properties similar to those seen during the “volcano – free” period described above. So, this result
- Shows the residuals to the GISS temp fit to data for the period from 1975-now are inconsistent with the ARMA(1,1) process fit to data during the closest thing we have to a “volcano-eruption-free” period to a confidence of level of 95%. This is diagnosed by noticing that the 1975-now residuals exceeded 98.9% of all residuals that expected based on the ARMA(1,1) for the “volcano-eruption-free”. The specific cut off for the 95% criterion is falling outside the 2.5% and the 97.5% range. So, since 97.5%<98.9% we can conclude the residuals during the volcano period are larger than during the “volcano free” period and the result is statistically significant..
- Stated qualitatively: supports the hypothesis that volcano eruptions result in increased variability of “weather noise”. This suggests that if we estimate variability of “weather noise” during periods with eruptions like Pinatubo and El Chico, we will overestimate the variabilty during periods when the stratosphere is clear.
So, generally speaking: If we use the ARMA(1,1) fit from 1975 to now to estimate variability in a hypothesis tests for data collected during periods with no volcanic eruptions, our our hypothesis tests of 2C/century may fail to falsify because we have over inflated our uncertainty intervals by pretending we should expect the sort of variability that happens only when volcanoes erupt.
We should, instead, estimate our uncertainty intervals based on periods with no major volcanic eruptions.
Are you wondering if there is a trick?
Are you wondering why I fit the ARMA(1,1) to the volcano free period, and then tested the residuals for the 1975- now fit instead of the other way around. The main reason was that running 403 months simulated weather gives a more powerful test than running 365 months of weather. But, naturally, I was curious, so I repeated but doing the “reverse” test as it were.
Well, it turns out the residuals for the “volcano free” period are inconsistent with the ARMA(1,1) fit during the volcanic period also. So, either way, I get the same result! Only, if I do it the other way around, the result is the residuals for 1913 are less than all but 0.9% of those consistent with the ARMA(1,1) computed from 1975-now. That’s less than 2.5% so, it does indeed appear the “weather noise” is more variable during periods with volcano eruptions.
But we already knew that, didn’t we?
Um. If what we really want to know if we should use one ARMA(1,1) noise model over another, shouldn’t we ask if the two models are inconsistant with each other?
Raphael–
We should absolutely ask that. (ARMA(1,1) is a broader class of models. Those prone to analogies might say:
AR(1) + white noise is to ARMA(1,1) as dogs are to mammals.
All dogs are mammals. All AR(1)+white noise are ARMA(1,1).
However, not all ARMA(1,1) are AR(1)+ white noise. Similarly, not all mammals are dogs.
However, the ARMA(1,1) that fits the data happens to fall in the subclass that is AR(1)+white noise. Since people will, I will likely have to post the algebra to show that AR(1)+white noise always corresponds to a specific ARMA(1,1) model. The algebra is a pain in the neck to post.
But, of course, you don’t have to believe me. When Tamino learned my use of AR(1) plus noise predated post explaining that ARMA(1,1) appears to describe GMST data, he was gracious enough to tell his readers the model is also an ARMA(1,1). He said so in this gracious language:
As I preveiously mentioned: Yes. I knew this. This is one of the reasons my reaction to his getting around to using this model was to say that I liked his new approach– except for his use of a period with very large volcanic eruptions to estimate the parameters which he later applied to a period when the stratosphere is clear.
As for difference in numerical results we are getting right now: difference between the HadCrut AR(1)+white noise and the ARMA(1,1) model are small, and likely due to a combination of rounding and also the fact that even if we write the parameters to four significant figures, the imprecision is larger. (The difficulty with some statistical analyses is once you are doing the math you want to avoid rounding for complicated reasons.)
Anyway, there is a reason I prefer to express it as AR(1)+white noise. It has to do with comparing to estimates of measurement uncertainty.
Lucia,
It wasn’t the lack of acceptance that all dogs are mammals that prompted the question. I simply failed to assume that the OLS mentioned in the nuts and bolts was OLS corrected for AR(1) + white noise you prefer.