Stochastic Methods and Non-Fractal Applications

Abstract

Our everyday environment abounds with examples of stochastic phenomena, objects or activities that are governed by probabilistic laws [COX65]. These range from observations evident to any observer, such as the number of people in a supermarket checkout queue or the time taken to drive to work on a particular day, to more latent situations, such as the energy levels of subatomic particles. Practitioners of statistics and operational research have, for decades, used computers to create models of such activity in order to gain useful insights into the behaviour of systems that depend on statistical or probabilistic laws. More recently, stochastic methods have been used to model certain natural phenomena in a visually convincing way. Images depicting simulations of the structures of, for example, plants [PRUS90] and other life forms [KAAN91], marble [PERL85], clouds [VOSS85], mountainous tenain [SAUP88] and the boundaries of cities [BATT91] have become familiar. Many researchers use standard “random number generators” to reproduce such effects; not all of them appreciate the implications of what they are doing. The purpose of this tutorial is to describe some of the underlying statistical theory and to show its application to a selection of techniques in computer graphics. The development will be explanatory rather than theoretically rigorous. The intention is to give end users an understanding of the methods they are using without converting them into statistical experts. Statistics is similar to many other technical subjects in that much of its mystique is concerned with its terminology. Many terms will be highlighted when introduced.