ESTIMATION AND SIMULATION OF FRACTAL STOCHASTIC POINT PROCESSES

Abstract

We investigate the properties of fractal stochastic point processes (FSPPs). First, we define FSPPs and develop several mathematical formulations for these processes, showing that over a broad range of conditions they converge to a particular form of FSPP. We then provide examples of a wide variety of phenomena for which they serve as suitable models. We proceed to examine the analytical properties of two useful fractal dimension estimators for FSPPs, based on the second-order properties of the points. Finally, we simulate several FSPPs, each with three specified values of the fractal dimension. Analysis and simulation reveal that a variety of factors confound the estimate of the fractal dimension, including the finite length of the simulation, structure or type of FSPP employed, and fluctuations inherent in any FSPP. We conclude that for segments of FSPPs with as many as 106 points, the fractal dimension can be estimated only to within ±0.1.