Statistical Modeling and Design of Discrete-Time Chaotic Processes: Advanced Finite-Dimensional Tools and Applications

摘要

With the aim of explaining the formal development behind the chaos-based modeling of network traffic and other similar phenomena, here we generalize the tools presented in the companion paper (Setti et al., 2002) to the case of piecewise-affine Markov maps with a possibly infinite, but countable number of Markov intervals. Since, in doing so, we keep the dimensionality of the space of the observables finite, we still obtain a finite tensor-based framework. Nevertheless, the increased complexity of the model forces the use of tensors of functions whose handling is greatly simplified by extensive z transformation. With this, a systematic procedure is devised to write analytical expressions for the tensors that take into account the joint probability assignments needed to compute any-order expectations. As an example of use, this machinery is finally applied to the study of self-similarity of quantized processes both in the analysis of higher order phenomena as well as in the analysis and design of second-order self-similar sources suitable for artificial network traffic generation.