I only recently found out that Albert Parker/Alberto Boretti and C.D. Ollier published a “Discussion” of my paper with Patrick Brown about the analysis of sea level time series. You can get your own copy of their paper here.
What, you may wonder, is Parker/Boretti and Ollier’s method to look for acceleration of sea level? It’s contained in equations (1) and (2) of their paper. The first is this:
This is just the OLS (ordinary least squares) estimate of the slope from fitting a straight line to data points j through k. It always worries me when scientific papers give this equation, because it suggests that the authors are so unfamiliar with something as simple as least squares regression that they feel the need to “explain” it to the reader.
But at least it’s correct. What Parker/Boretti and Ollier say about it is not:
If j=1 (the oldest record) and k is variable, then
SLR1,k is the estimation of the relative rate of rise at the time xk.
Let’s find out, shall we?
Consider some artificial data, defined by the simple equation , for 101 t values from 0 to 1. We already know what the slope is at each moment () and what the acceleration is (=2 for all times), and we won’t add any noise so the method should diagnose the slope without error.
What does Parker/Boretti and Ollier’s method say the slope is at each moment of time? This:
The solid line is their method’s slope estimate, the dashed lines are the 95% confidence limits according to OLS. Now let me add a red line showing the true slope at each moment of time:
Wha-wha-wha-what??? The Parker/Boretti and Ollier method got it wrong. At every moment of time. In spite of the fact that the data are noise-free.
That’s because their method does not give “the estimation of the relative rate of rise at the time xk” (remember they’re using x to represent the time, while I tend to use t). It gives an estimate of the average rate of rise over the entire time interval, not “at the time xk.” If we had to assign this to a moment of time rather than an interval, the only logical choice would be the mid-point of the time interval (for evenly spaced data), not the end (the time xk).
If we use their method to estimate the acceleration, we get the same value at all times — as we should — but it’s the wrong value. Their method estimates the acceleration is equal to 1, when we already know it’s equal to 2.
How bad is it, when their method of estimating slope and acceleration can’t get it right even for noise-free data?
Parker/Boretti and Ollier don’t actually have a lot to say about my and Brown’s paper. My guess is: the math was over their heads. Way over.
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