Roughness as a Fractal Property in Univariate Time Series Data

In the analysis of time series data, roughness is sometimes seen as a distinct feature of fractality. This paper seeks to distinguish it from other aspects of that construct (self-affinity and long-range memory processes) and it examines the reliability of the roughness measures currently available, i.e., Gneiting et al.'s (2010) fractal dimension and Marmelat et al.'s (2012) relative roughness. The response of these estimators is evaluated to simulations at varying levels of persistence, as specified by the Hurst exponent, and to the presence or absence of short-range ARMA processes. Four empirical time series datasets are subjected to roughness estimation: the flow of the river Nile, daily recordings of the number of births to teens in the state of Texas, daily school attendance rates at an urban middle school, and unemployment figures provided by the US Department of Labor. Results from the simulation study indicate that persistence levels are faithfully reproduced by both estimation techniques, which also show the (dis)attenuating effects of the short-range dependencies. Analysis of the empirical data indicates that the fractal dimension works best for non-stationary data, while relative roughness is more suitable for stationary data. In the simulations as well as the empirical situation, both estimations reliably identify randomness, and are therefore recommended as goodness of fit measures when time series are analyzed.