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

Is the number 2.14159 (here rounded off to 5 decimal points) a fundamentally meaningful one? Add one, and you get

π = 3.14159 = 2.14159 + 1.

Of course, π is a fundamentally meaningful number, but you can split up this number in infinite ways, as in the example above, and most of the different terms have no fundamental meaning. They are just numbers.

But what does this have to do with climate? My interpretation of Daniel Bedford’s paper in Journal of Geography, is that such demonstrations may provide a useful teaching tool for climate science. He uses the phrase ‘agnotology’, which is “the study of how and why we do not know things”.

Furthermore, many descriptions of our climate are presented in terms of series of numbers (referred to as ‘time series‘), and when shown graphically, they are known as curves. It is possible to split curves into pieces in an analogous way to how π may be split into random numbers.

All curves (finite time series) can be represented as a sum of sine and cosine curves (sinusoids), describing cycles with different frequency. This is demonstrated in the figure below (source code for the figure):

The random time series, here represented as the bold curve on the top, may be physically meaningful, but the components made up of cosine and sine may not all have a physical interpretation (especially if the time series is from a chaotic or complex system).

However, cosine and sine curves represent only one special case, and there may be other curves that equally well can make up a time series. A technique called ‘singular spectrum analysis‘ (SSA), for instance, is designed to find curves with other shapes than sinusoids.

The process of representing a series of numbers as a sum of sinusoids (cycles) with different frequencies (or wave lengths) is known as a Fourier transform (FT). It is also possible to go the other way, from the information about the frequencies, and reconstruct the original curve – this is known as the inverse Fourier transform.

Fourier transforms are closely related to spectral analysis, but these concepts are not exactly the same. The reason is that all measurements hold a finite number of observations, and provide just a taste – a sample – of the process. The FT makes the assumption that the curve that is analysed repeats itself exactly for infinity, something which clearly is not the case for real noisy or chaotic data.

One of the gravest mistakes in the attribution of cycles is trying to fit sinusoids with long time scales to short time series. We will see some examples of this below.

In the meanwhile, it may be useful to note that spectral analysis tries to account for mathematical artifacts, such as ‘spectral leakage‘, probabilities that some frequencies are spurious, and the significance of the results. Anyway, there is a number of different spectral analysis techniques, and some are more suitable for certain types of data. Sometimes, one can also use regression to find the best-fit combination of sinusoids for a time series.

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