Wang Jing, Shang Pengjian, Cui Xingran
Department of Mathematics, School of Science, Beijing Jiaotong University, Beijing 100044, People's Republic of China and Division of Interdisciplinary Medicine and Biotechnology, Department of Medicine, Beth Israel Deaconess Medical Center/Harvard Medical School, Boston, Massachusetts 02215, USA.
Department of Mathematics, School of Science, Beijing Jiaotong University, Beijing 100044, People's Republic of China.
Phys Rev E Stat Nonlin Soft Matter Phys. 2014 Mar;89(3):032916. doi: 10.1103/PhysRevE.89.032916. Epub 2014 Mar 24.
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%.
多重分形去趋势波动分析（MF-DFA）是检测交通信号等大量信号多重分形特征最常用的方法。当分形特性沿序列逐点变化时，就会导致多重分形性。在本研究中，我们不仅关注交通信号具有多重分形特性这一事实，还关注这些特性取决于计算多重分形性所使用的时间尺度。通过多尺度多重分形分析（MMA），交通信号似乎要复杂得多，并且包含更多MF-DFA使用固定时间尺度无法探索的信息。更重要的是，我们不必避开具有交叉点的数据集或缩小所研究的时间尺度，因为这可能会导致有偏差的结果。相反，赫斯特曲面提供了不同尺度范围内的局部标度指数谱，这有助于我们轻松定位这些交叉点。通过比较去除周期性趋势前后信号的赫斯特曲面，我们发现交通信号的周期性是交叉点的主要来源。此外，工作日序列的赫斯特曲面与周末序列的表现不同。结果还表明，交通信号的多重分形性主要是由于宽概率密度函数和相关性。我们还讨论了数据丢失的影响，这表明当数据丢失百分比大于40%时，我们应该谨慎处理MMA结果。
