Much of what we discussed—but not all—is in this picture. Right click and open it in a new window so that you can follow along.
We have two temperature series, A and B. A is incomplete and overlaps B only about 30% of the time. A and B officially stop at year “80”. We want to know one main thing: what will be the temperatures at A and B for the years 81 – 100?
Official homogenization of A commences by modeling A’s values as a function of B’s. There are no auto-correlations to worry about in this data, because there are none by design: this is entirely simulated data. A and B were generated by a multivariate normal distribution with a fixed covariance matrix. In plain English, this means the two series are correlated with each other, but not with themselves through time. Plus, any trends are entirely spurious and coincidental.
Thus, a linear model of A predicted by B is adequate and correct. That is, we do not have to worry, as we do in real life, that the model A = f(B) is misspecified. In real life, as I mentioned in earlier posts, there is additional uncertainty due to us not knowing the real relationship between A and B.
We also are lucky that the dates are fixed and non-arbitrary. Like I said earlier, picking the starting and stopping dates in an ad hoc manner should add additional uncertainty. Most people ignore this, so we will, too. We’re team players, here! (Though doing this puts us on the losing team. But never mind.)
Step 1 is to model A as a function of B to predict the “missing” values of A, the period from year 1 – 49. The result is the (hard-to-read) dashed red line. But even somebody slapped upside the head with a hockey stick knows that these predictions are not 100% certain. There should be some kind of plus or minus bounds. The dark red shaded area are the classical 95% parametric error bounds, spit right out of the linear model. These parametric bounds are the ones (always?) found in reporting of homogenizations (technically: these are the classical predictive bounds, which I call “parametric”, because the classical method is entirely concerned with making statements about non-observable parameters; why this is so is a long story).
Problem is, like I have been saying, they are too narrow. Those black dots in the years 1 – 49 are the actual values of A. If those parametric error bounds were doing their job, then about 95% of the black dots would be inside the dark red polygon. This is not the case.
I repeat: this is not the case. I emphasize: it is never the case. Using parametric confidence bounds when you are making predictions of real observables is sending a mewling boy to do a man’s job. Incidentally, climatologists are not the only ones making this mistake: it is rampant in statistics, a probabilistic pandemic.
The predictive error bounds, also calculated from the same A = f(B), are the pinkish bounds (technically: these are the posterior-predictive credible intervals). These are doing a much better job, as you can see, and aren’t we happy. The only problem is that in real life we will never know those missing values of A. They are, after all, missing. This is another way of stating that we do not really know the best model f(B). And since, in real life, we do not know the model, we should realize our error bounds should be wider still.
Our homogenization of A is complete with this model, however, by design. Just know that if we had missing data in station B, or changes in location of A or B, or “corrections” in urbanization at either, or measurement error, all our error bounds would be larger. Read the other four parts of this series for why this is so. We will be ignoring—like the climatologists working with actual data should not—all these niceties.
Next step is to assess the “trend” at B, which I have already told you is entirely spurious. That’s the overlaid black line. This is estimated from the simple—and again correct by design—model B = f(year). Our refrain: in real life, we would not know the actual model, the f(year), and etc., etc. The guess is 1.8oC per century. Baby, it’s getting hot outside! Send for Greenpeace!
Now we want to know what B will be in the years 81 – 100. We continue to apply our model B = f(year), and then mistakenly—like everybody else—apply the dark-blue parametric error bounds. Too narrow once more! They are narrow enough to induce check-writing behavior in Copenhagen bureaucrats.
The accurate, calming, light-blue predictive error bounds are vastly superior, and tell us not to panic, we just aren’t that sure of ourselves.
How about the values of A in the years 81 – 100? The mistake would be to use the observed values of A from years 50 – 80 augmented by the homogenized values in the years 1 – 49 as a function of year. Since everybody makes this error, we will too. The correct way would be to build a model using just the information we know—but where’s the fun in that?
Anyway, it’s the same story as with B, except that the predictive error bounds are even larger (percentage-wise) than with B, because I have taken into account the error in estimating A in the years 1 – 49.
Using the wrong method tells us that the trend at A is about 1.0oC per century, a worrisome number. The parametric error bounds are also tight enough to convince some that new laws are needed to restrict people’s behavior. But the predictive bounds are like the cop in the old movie: Move along; nothing to see here.
This example, while entirely realistic, doesn’t hit all of the possible uncertainties. Like those bizarre, increasing step-function corrections at Darwin, Australia.
What needs to be done is a reassessment of all climate data using the statistical principles outlined in this series. There’s more than enough money in the system to pay the existing worker bees to do this.
Our conclusion: people are way too certain of themselves!