Stochastic models that separate fractal dimension and the Hurst effect

Fractal behavior and long-range dependence have been observed in an astonishing number of physical, biological, geological, and socio-economic systems. Time series, pro_les, and sur- faces have been characterized by their fractal dimension, a measure of roughness, and by the Hurst coe_cient, a measure of long-memory dependence. Either phenomenon has been modeled and explained by self-a_ne random functions, such as fractional Gaussian noise and fractional Brownian motion. The assumption of statistical self-a_nity implies a linear relationship be- tween fractal dimension and Hurst coe_cient and thereby links the two phenomena. This article introduces stochastic models that allow for any combination of fractal dimension and Hurst coe_cient. Associated software for the synthesis of images with arbitrary, pre-speci_ed fractal properties and power-law correlations is available. The new models suggest a test for self-a_nity that assesses coupling and decoupling of local and global behavior