Fitting the Generic Multi-Parameter Crossover Model: Towards Realistic Scaling Estimates

The primary concern of fractal metrology is providing a means of reliable estimation of scaling exponents such as fractal dimension, in order to prove the null hypothesis that a particular object can be regarded as fractal. In the particular context to be discussed in this contribution, the central question is what is the minimum extent of the scaling range needed to give any meaning to the object's description as a fractal. Preceded by a short review of the motivations for the generic transition model, we present the straightforward manner of extending it over more free parameters. The price paid for fitting such a multi-parametric model is, however, not only computational expense but the danger of obtaining far from optimal fits. Deterministic cross-sections through the parameter space of the model are demonstrated to show insight into the sensitivity of the fitting procedure to parameter variations. Realistic confidence intervals obtained are demonstrated to allow for testing the fractality hypothesis on the base of globally uniform scaling (fractal dimension). This is demonstrated in both examples of non-linear fit to measurements on decreasingly lowered generation level deterministic pre-fractals and genuine human writing samples.