Guest Post by Willis Eschenbach
Gavin Schmidt, who I am given to understand is a good computer programmer, is one of the principals at the incongruously named website “RealClimate”. The name is incongruous because they censor anyone who dares to disagree with their revealed wisdom.
I bring this up because I’m on Twitter, @WEschenbach. You’re welcome to join me there, or at my own blog, Skating Under The Ice … but I digress. I always tweet about my new posts, including my most recent post, Changes in the Rate of Sea Level Rise, q.v.
To my surprise, Gavin responded to my tweet, saying:
I responded, saying:
Now, to paraphrase Pierre de Fermat, “I have an explanation of this claim which the margin of this tweet is too small to contain.” So I thought I’d write it up. Let me start with the money graph from my last post:
Figure 1. 31-year trailing trends in the rate of sea level rise.
Is the sea level rise accelerating in this graph? It depends on which section you choose. It decelerated from 1890 to 1930. Then it accelerated from 1930 to 1960, decelerated to 1975, stayed flat until about 2005, and accelerated since then … yikes.
As I mentioned, until we have some explanation of those changes, making predictions about the future sea levels is a most parlous endeavor …
HOWEVER, Gavin wants to look at the overall changes, so let’s do that. Figure 2 shows the entire record shown in Figure 1, along with lines indicating the best linear fit, and the best accelerating (quadratic) fit.
Figure 2. The Church and White sea level record, along with best-fit linear (no acceleration, blue) and quadratic (acceleration, red) lines.
Now, this is what Gavin is talking about … and yes, it certainly appears that the quadratic (accelerating) red line is a better fit. But that’s the wrong question.
The right question is, is that a significantly better fit? When we have two choices, we can only pick one with confidence if it is statistically a significantly better fit than the other option.
The way that we can measure this is to look at what are called the “residuals”, or sometimes the “residual errors”. These are the distances between the actual data points, and the predicted data points from the red or blue fitted line. The line which is a better fit will have, on average, smaller error residuals than the other option.
Now, we can use a measure called the “variance” of the residuals to determine which one has the better fit on average. And as you might expect from looking at Figure 2, the variance of the straight-line residuals (no acceleration) is larger (80.2 mm) than that of the residuals of the red line showing acceleration (53.3 mm) … so the acceleration does indeed give the better fit.
But how much better?
There’s no easy way to answer that, so we have to do it the hard way. The hard way means a “Monte Carlo” analysis of the two sets of residuals. We create “pseudo-data”, random data which has statistical characteristics which are similar to the real residuals. Now, the residuals are not simply random numbers. Instead, they both have a high “Hurst Exponent”, which can be thought of as measuring how much “memory” the data has. If there is a long memory (high Hurst Exponent), then e.g. this decade’s data depends in part on the last decade’s data. And this changes what the pseudo-data looks like
So what I did was to generate a thousand samples of pseudodata which had about the same Hurst Exponent (± 0.05) as each residual, and which on average had the same variance as each residual. Then, I looked at the variance of each individual example of the groups of pseudo-data, to determine how much of a range the individual variances covered. From that, I calculated the “95% confidence interval” (95%CI), the range in which we would expect to find the variance for that exact type of data.
It turns out that the 95% confidence interval of the variance is not symmetrical about the variance. It is larger on the positive side and smaller on the negative side. This is a known characteristic of the uncertainty of the variance, and it is what I found for this data.
So with that as prologue, here is the comparison of the variances of the two options, acceleration and no acceleration, along with their 95% confidence intervals:
Figure 3. Variance and 95%CI for the acceleration and no acceleration situations.
Here’s the thing. The 95% CI for each of the residuals encompasses the variance of the other residual … and this means that there is no statistical difference between the two. It may just be a random fluctuation, or it might be a real phenomenon. We cannot say at this point.
We can understand this ambiguity by noting that from the start to about 1930, the trailing trend line in Figure 1 shows a strong deceleration in the rate of sea level rise. We have no clear idea why this occurred … but it increases the uncertainty in our results. If there were a clear acceleration from the beginning to the end of the dataset, the uncertainty would be much smaller, and we could say confidently that there was acceleration over the entire period … however, that’s not the case. The trends went up and down like a yo-yo … and no one knows why.
Finally, let me caution Gavin and everyone else against extending such a trend into the future. This happens all the time in climate science, and it is a pernicious practice. If we had extended the decelerating trend back in 1930, we would have predicted a large fall in sea levels by the year 2000 … and obviously, that didn’t happen.
My own strong wish in all of this is that climate scientists should declare a twenty-year hiatus in making long-range predictions of any kind, and just focus on trying to understand the past. Why did the rate of sea level rise decelerate in the early part of the Church and White record, and then accelerate so rapidly? Why did we come out of the Little Ice Age? Why are we not currently in a glacial epoch?
Until we can answer such questions, making predictions for the year 2050 and the like is a fool’s errand …
My best to all, including Gavin. Unlike many folks on Twitter, he tweets under his own name, and I applaud him for that. In my experience, anonymity, whether here or on Twitter, leads to abuse. I also invite him to come here and make his objections, rather than trying to cram them into 240 characters on Twitter, but … as my daughter used to say, “In your dreams, Dad” …
As Always—Please quote the exact words that you are discussing, so we can all understand who and what you are referring to.
PS—Note that I have not included all of the uncertainty in these calculations. Remember that each of the Church and White data points has an associated uncertainty. I have only calculated the statistical uncertainty, as I have not included the uncertainty of the individual data points. This can only increase the uncertainty of the variance of both of the conditions, acceleration and no acceleration.
Why didn’t I include the uncertainty of the individual data points? Work and time … the statistical uncertainty alone was large enough to let me know that there is no statistical difference between acceleration and no acceleration, so I forbore doing a bunch more Monte Carlo analyses which would show even larger uncertainty. So many interesting questions … so little time. Clearly, I need minions to give some of this work to … all the evil global overlords in the comic books have minions, where are the minions of Willis The Merciless? Or at least the educational equivalent of minions … graduate students … my regards to everyone, especially the poor overworked graduate students.