The entropic analog of the Schwarzian derivative (ESf) is used to explore the consequences of scaling and partitioning operations on time series that are known to be edge-of-chaos, or random. The nonlinear deterministic series may be fractal trajectories, and if so self-similarity may be a property that characterises their evolution. Examples are generated, and statistical methods that follow from their generation are noted. Some invariance properties of the ESf index that are associated with fractal self-similarity but not with other series appear to be a basis for process identification on relatively short time series. Surrogate and parameter optimisation methods are illustrated in real data. A possible relationship between the difference between an ESf and the ESf on its random surrogates, and the first derivative of the local largest Lyapunov exponent, is noted.