Visualizing multifractal scaling behavior: A simple scaling heuristic

The self-similar or monofractal scaling behavior of a signal can be readily visualized in the time domain by plotting and comparing the signal at many different levels of resolution (i.e., time scales). No comparable visualization techniques appear to be available for qualitatively assessing the presence of multifractal scaling in a given signal. To help in developing an intuitive understanding of multifractals, we propose in this paper a simple coloring heuristic that gives rise to an effective, time domain-based visualization technique for multifractals. Our approach combines wavelet-based statistical denoising techniques with wavelet characterizations of local regularity for functions. We illustrate our heuristic with a number of examples, including the binomial measure, a simple and well-studied multifractal, and measured traces of actual data network traffic.