An exercise about meaningful numbers: examples from celestial “attribution studies”

A recent paper by Loehle & Scafetta (L&S2011) in a journal known as the ‘Bentham Open Atmospheric Science Journal‘ (also discussed at Skeptical Science) presents some analysis using regression to describe cycles in the global mean temperature, showing us many strange tricks one can do with curves and sinusoids, in something they call “empirical decomposition” (whatever that means).

They fit 20-year and 60-year sinusoids to the early part (1850-1950) of the global mean temperature from the Hadley Centre/Climate Research Unit. I have reproduced their analysis below, although I do not recommend using this for any meaningful purposes than just having fun (source code)

Reproduction of L&S2001 Fig. 2A

It’s typical, however, that geophysical time series, such as the global mean temperature, are not characterised by one or two frequencies. In fact, if we try to fit sinusoids with other frequencies (here only one was used rather than two), we get the following picture (source code):

The blue curves represent a set of different best-fits for sinusoids with different frequencies.

In fact, we can compare the amplitudes of these different fits, and we see that the frequencies of 20 and 60 years are not the most dominant ones (source code):

Amplitudes of the sinusoids, based on the regression coefficients.

Fitting sinusoids with long time scales compared to the time series is dangerous, which can be illustrated through constructing a synthetic time series that is much longer than the one we just looked at. This time series is shown below (source code):

A synthetic series mimicking random 10.000-yr variations

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