CAREER: Self-Similarity: Roadblock or Breakthrough?

University Of Washington, Seattle WA

Investigators

Abstract

Following Mandelbrot's seminal work, fractal based analyses of time series, profiles, and natural or man-made surfaces have found extensive applications in almost all scientific disciplines. The fractal dimension, D, of an object or data set is a roughness measure and routinely estimated in scientific studies. Long-memory dependence in time series or spatial data plays key roles in the discussion of issues such as global warming. It is associated with power-law correlations and quantified by the Hurst coefficient, H. In principle, fractal dimension, D, and Hurst coefficient, H, are independent of each other: fractal dimension is a local property, and long-memory dependence is a global characteristic. Nevertheless, the two notions have been linked through the celebrated relationship that indicates that the fractal dimension added to the Hurst coefficient equals n+1, for a self-similar surface in n-dimensional space. This relationship is based on the assumption of self-similarity, which has hardly ever been put to test. Recently developed stochastic models allow for any combination of fractal dimension, D, and Hurst coefficient, H, and therefore challenge the relationship and the role of self-similarity. Has the prevalence of self-similar model been a roadblock, keeping scientists from exploring and understanding a wide range of local and global behavior in complex systems? Or is it indeed a breakthrough, which lets scientists focus on models for which the above relationship holds, while other combinations of D and H are physically meaningless? This research develops tests for self-similarity and thereby addresses these problems. Along the way, new insight into the behavior of statistical estimators of fractal dimension and Hurst coefficient, software for the fast and exact simulation of self-similar and related classes of random processes, an analysis of topographic data for Oregon and Washington state, and new theoretical results for stationary random functions will emerge. Visualization tools and Freshman Seminars at the University of Washington will introduce undergraduate students to mathematical and statistical modeling, to fractals, long-memory dependence, and self-similarity, and to the natural beauty and diversity of the Pacific Northwest. Many natural phenomena such as mountains, clouds, ferns and rocks have features that look alike across scales. A little piece might indeed resemble the whole if the scale is not disclosed. Over the past decades, this notion of self-similarity has emerged as a popular and fruitful theme in a vast number of scientific studies. This research develops statistical tests, which allow scientists to assess whether their observations and data are compatible with classical assumptions of self-similarity. Alternative models allow for distinct behaviors at different scales and might inspire and encourage new theories across disciplines. The research is paired with the development of visualization and simulation tools as well as Freshman Seminars, through which students are introduced to mathematical and statistical modeling, to fractals, long-memory dependence, and self-similarity, and to the natural beauty and diversity of the Pacific Northwest.