基于经验模态分解的修正去趋势波动分析
Modified detrended fluctuation analysis based on empirical mode decomposition
摘要：
去趋势波动分析（DFA）是研究非平稳时间序列的幂律长期相关性的一种简单而有效的方法。我们提出了通过经验模式分解（EMD）来确定局部趋势，并通过去除基于EMD的局部趋势来执行去趋势操作，从而给出了基于EMD的DFA方法。同样，我们也提出了一种改进的多重分形DFA算法，称为基于EMD的MFDFA。基于分数阶布朗运动和乘性级联过程的大量数值实验评估了基于EMD的DFA和MFDFA方法的性能。我们发现，当时间序列具有强相关性时，基于EMD的DFA方法比经典DFA方法更适合确定Hurst指数;当去趋势波动的矩阶数$q$为正时，基于EMD的MFDFA方法比传统MFDFA方法更适合确定Hurst指数。我们将基于经验模态分解的MFDFA应用于上证综合指数的一分钟数据，证实了多重分形的存在。
英文摘要：
Detrended fluctuation analysis (DFA) is a simple but very efficient method for investigating the power-law long-term correlations of non-stationary time series, in which a detrending step is necessary to obtain the local fluctuations at different timescales. We propose to determine the local trends through empirical mode decomposition (EMD) and perform the detrending operation by removing the EMD-based local trends, which gives an EMD-based DFA method. Similarly, we also propose a modified multifractal DFA algorithm, called an EMD-based MFDFA. The performance of the EMD-based DFA and MFDFA methods is assessed with extensive numerical experiments based on fractional Brownian motion and multiplicative cascading process. We find that the EMD-based DFA method performs better than the classic DFA method in the determination of the Hurst index when the time series is strongly anticorrelated and the EMD-based MFDFA method outperforms the traditional MFDFA method when the moment order $q$ of the detrended fluctuations is positive. We apply the EMD-based MFDFA to the one-minute data of Shanghai Stock Exchange Composite index, and the presence of multifractality is confirmed.
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