Fractal Analysis of Time-Series Data Sets: Methods and Challenges

Many methods exist for quantifying the fractal characteristics of a structure via a fractal dimension. As a traditional example, a fractal dimension of a spatial fractal structure may be quantified via a box-counting fractal analysis that probes a manner in which the structure fills space. However, such spatial analyses generally are not well-suited for the analysis of so-called “time-series” fractals, which may exhibit exact or statistical self-affinity but which inherently lack well-defined spatial characteristics. In this chapter, we introduce and investigate a variety of fractal analysis techniques directed to time-series structures. We investigate the fidelity of such techniques by applying each technique to sets of computer-generated time-series data sets with well-defined fractal characteristics. Additionally, we investigate the inherent challenges in quantifying fractal characteristics (and indeed of verifying the presence of such fractal characteristics) in time-series traces modeled to resemble physical data sets.