Don’t estimate acceleration by fitting a quadratic…

… if your data do not look like a quadratic!

This is a post about global sea-level rise, but I put that message up front so that you’ve got it even if you don’t read any further.

The reputable climate-statistics blogger Tamino, who is a professional statistician in real life and has published a couple of posts on this topic, puts it bluntly:

Fitting a quadratic to test for change in the rate of sea-level rise is a fool’s errand.

I’d like to explain why, with the help of a simple example. Imagine your rate of sea-level rise changes over 100 years in the following way:

Fig. 1. Rate of sea-level rise as it changes over 100 years. This is a fictitious example chosen for illustrative purposes. It’s simply a polynomial curve, see appended matlab script.

It starts at 1 mm/year, then in the middle of the century it hovers for a while around 2 mm/year (namely between 1.8 and 2.2), and in the end it climbs to 3 mm/year. There can be no question about the fact that the rate of sea-level rise increases overall during those 100 years. It increases by a factor of three, from one to three millimetres per year, although not at a steady rate. You could fit a linear trend to the above curve and also find an increase in the rate – although a linear trend would not be a great description of what is going on, because the increase in rate is clearly not linear. (Note that a linear increase in the rate corresponds to a quadratic sea-level curve.)

You can easily compute the sea-level curve that follows from the above rate by integrating it over time, and it looks like this:

Fig. 2. This sea level curve is the integral of the curve in Fig. 1 and thus contains the same information, but when viewed in this way it is hard to judge by eye whether sea-level rise has accelerated. The better way to answer this question is by looking at the rate curve, i.e. Fig. 1.

Now here it comes: if you fit a quadratic (by the standard least-squares method) to this sea-level curve, the quadratic term (i.e. the acceleration) is negative! So by this diagnostic, sea level rise supposedly has decelerated, i.e. the rate of rise has slowed down! This clearly is nonsense and misleading (we know the truth in this case, it is shown in Fig. 1), and this nonsense results from trying to fit a quadratic curve to data that do not resemble a quadratic. You can call it a misapplication of curve fitting, or the use of a bad model.

Now to real data

Is this just a bizarre, unrealistic example? No! Because the basic shape of this example resembles the observed global sea-level curve from about 1930 to 2000. The red curve below is the rate of rise as diagnosed from the Church&White (2006) global sea-level data set, as shown and described in more detail in Rahmstorf (Science 2007):

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