Research Center for Econophysics, East China University of Science and Technology, Shanghai 200237, China.
Department of Finance, East China University of Science and Technology, Shanghai 200237, China.
Phys Rev E. 2017 Nov;96(5-1):052201. doi: 10.1103/PhysRevE.96.052201. Epub 2017 Nov 3.
In the canonical framework, we propose an alternative approach for the multifractal analysis based on the detrending moving average method (MF-DMA). We define a canonical measure such that the multifractal mass exponent τ(q) is related to the partition function and the multifractal spectrum f(α) can be directly determined. The performances of the direct determination approach and the traditional approach of the MF-DMA are compared based on three synthetic multifractal and monofractal measures generated from the one-dimensional p-model, the two-dimensional p-model, and the fractional Brownian motions. We find that both approaches have comparable performances to unveil the fractal and multifractal nature. In other words, without loss of accuracy, the multifractal spectrum f(α) can be directly determined using the new approach with less computation cost. We also apply the new MF-DMA approach to the volatility time series of stock prices and confirm the presence of multifractality.
在规范框架中，我们提出了一种基于去趋势移动平均方法（MF-DMA）的多重分形分析的替代方法。我们定义了一个规范测度，使得多重分形质量指数 τ(q)与分区函数有关，并且可以直接确定多重分形谱 f(α)。我们基于一维 p-模型、二维 p-模型和分数布朗运动生成的三个合成多重分形和单分形测度，比较了直接确定方法和 MF-DMA 传统方法的性能。我们发现这两种方法都具有揭示分形和多重分形性质的可比性。换句话说，在不损失准确性的情况下，可以使用新方法以更少的计算成本直接确定多重分形谱 f(α)。我们还将新的 MF-DMA 方法应用于股票价格的波动率时间序列，并证实了多重分形的存在。
