Estimating Fractional Dependencies and Scale Invariance in Univariate Time Series Data: A Primer

The estimation of fractal patterns in time series data is a small but important specialty in the expansion of research methods that are specifically attuned to dealing with nonlinear dynamics and complex processes. This paper provides a brief methodological overview of the detection and confirmation of fractal patterns in time series data, focusing on two approaches, fractional differencing, a regression-based approach that estimates the relative contribution of a fractal parameter to the overall variability in the series, and spectral density analysis, which decides whether a Fourier-transformed series yields a linear relationship between the log relative frequencies and the log amplitude in the power spectrum. It is demonstrated how finding such a relationship points to a fractal pattern in the data (self-affinity). Three existing datasets are analyzed for illustrative purposes: annual recordings of the flow of the River Nile between 622 and 1285AD, monthly recordings of US unemployment figures from 1948 to 2020, and weekly survey responses concerning self-reported left-right political orientation in the Netherlands. It is shown how the two methods are able to detect fractals in the political orientation and River Nile data, but not in the unemployment data.