University of Washington - Department of Statistics
Fractal behavior and long-range dependence have been described in an astonishing number of physical, biological, geological, and socio-economic systems. Time series, profiles, and surfaces have been characterized by their fractal dimension, a measure of roughness, and by the Hurst coefficient, a measure of long-memory dependence. Either phenomenon has been modeled and explained by self-similar random functions, such as fractional Gaussian noise and fractional Brownian motion. The assumption of statistical self-similarity implies a linear relationship between fractal dimension and Hurst coefficient and thereby links the two phenomena. In this talk, I will introduce stochastic models that allow for any combination of fractal dimension and Hurst coefficient, thereby demonstrating that the two phenomena are independent of each other. I will review the historical development of the aforementioned notions, and will point to current research opportunities. A relevant Technical Report (joint with Martin Schlather) is available (in PDF format).