Least-squares fit of a quadratic to smoothings of HADCRUT3

Vaughan Pratt
Stanford University
February 24, 2011

Figures 1-4 show the HADCRUT3 global land-sea temperature record smoothed with an n-year running average for n = 12, 55, 62.5, and 75 years (red curve), along with the least-squares fit of a quadratic (green curve) to the smoothed curve.

The fitted quadratics in each case are not intended as a model of temperature. One problem with a quadratic is that the slope is implausible in the 18th and earlier centuries, having arbitrarily negative slope going sufficiently far back in time. Our interest in a quadratic fit here is purely as a measure of smoothness having three degrees of freedom, namely the three coefficients in the quadratic am2 + bm + c where m is the month, taking m = 0 to be June 1930 (the middle of 1850-2011).

Figure 1. 12-year smoothing

Figure 1 shows the general shape of the HADCRUT3 record after "ironing out" fast-moving phenomena such as solar cycles and ENSO/ESSO events and episodes via a 12-year smoothing window. (This is the same curve as at Paul Jones' Wood For Trees site.) The green curve is obviously a very weak model of the record, in fact it's implausible even in the 19th century.

Figure 2. 55-year smoothing

55-year smoothing irons out the strange bumps in the left hand half of Figure 1. The quadratic fit is noticeably better.

Figure 3. 62.5-year smoothing

Increasing the smoothing window to 62.5 years flattens the bumps even more, as one would expect when the window is increased.

Figure 4. 75-year smoothing

Strangely, further widening worsens the fit.

A natural question then would be, what is the quality of the fit as a function of smoothing?

Figure 5. 1/(1-r2) as a function of smoothing window width

When fitting, r2 is a number in the range 0 to 1 indicating the quality of the fit, with 1 being a perfect fit. Obviously none of these smoothed (red) curves are perfect quadratics, so 1-r2 will be a positive number, and 1/(1-r2) tends to infinity as r2 tends to 1, thereby accentuating goodness of fit.

What Figure 5 shows is that the quality of fit spikes dramatically when the smoothing window is 62.5 years, and falls off rapidly on both sides, contrary to the intuition that it should continue to rise with increasing width of smoothing window. The peak of 1054 at 62.5 years corresponds to r2 = 0.99905. This indicates an excellent fit, as one might guess based on Figure 3.

The mystery is why such a good fit exactly there and nowhere else. I have a theory, which however I'll sit on for the time being as I may well be overlooking a better explanation and would hate to prejudice people's thinking before they'd had a chance to come up with their own theory. You can find my email at the top left of my home page. And feel free to play around with the C program that generated Figure 5.