Chapter 6 Fluctuation Analyses: Global Scaling

“If you have not found the 1/f spectrum, it is because you have not waited long enough. You have not looked at low enough frequencies.”
- Machlup (1981)

As Machlup (1981) noted, in order to find long range temporal correlations, you need to be able to observe them, that is, observe the process for a sufficiently long time. In general, variables measured from Complex Adaptive Systems will display long-range correlations Olthof, Hasselman, Wijnants, & Lichtwarck-Aschoff (2020). What the presence of such correlations, and specifically the pattern known as 1/f noise (pink noise), signifies is a matter of debate. This chapter lists some of the methods available in package casnet to quantify the presence of temporal patterns that can be associated to the different colours of noise discussed in the previous chapter. Most analysis outcomes can be converted to an estimate of the Fractal Dimension of the time series (see Hasselman2012a? for rationale and conversion formula’s).

If you haven’t done so already, it is recommended you study the materials listed in the paragraph Fractals of the previous chapter.

The presence of power-law scaling means that the observed time series (and potentially the data generating process) cannot be described by referring to a characteristic scale. This can be a scale of fluctuation (e.g. a population variance) or a central tendency (e.g. a population mean). However, we do not know anything about the mechanism responsible for the emergence of the powerlaw. This is explained in the video linked below. Other useful resources are:

Fractals and Scaling: What do power laws mean?

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