Multiscale multifractal analysis of traffic signals to uncover richer structures

Multifractal detrended fluctuation analysis (MF-DFA) is the most popular method to detect multifractal characteristics of considerable signals such as traffic signals. When fractal properties vary from point to point along the series, it leads to multifractality. In this study, we concentrate not only on the fact that traffic signals have multifractal properties, but also that such properties depend on the time scale in which the multifractality is computed. Via the multiscale multifractal analysis (MMA), traffic signals appear to be far more complex and contain more information which MF-DFA cannot explore by using a fixed time scale. More importantly, we do not have to avoid data sets with crossovers or narrow the investigated time scales, which may lead to biased results. Instead, the Hurst surface provides a spectrum of local scaling exponents at different scale ranges, which helps us to easily position these crossovers. Through comparing Hurst surfaces for signals before and after removing periodical trends, we find periodicities of traffic signals are the main source of the crossovers. Besides, the Hurst surface of the weekday series behaves differently from that of the weekend series. Results also show that multifractality of traffic signals is mainly due to both broad probability density function and correlations. The effects of data loss are also discussed, which suggests that we should carefully handle MMA results when the percentage of data loss is larger than 40%.