Wavelet versus detrended fluctuation analysis of multifractal structures
Wavelet versus detrended fluctuation analysis of multifractal structures
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DOI：
10.1103/physreve.74.016103
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发表时间：
2006-07-01
影响因子：
2.4
通讯作者：
Drozdz, Stanislaw
中科院分区：
文献类型：
--
作者：
Oswiecimka, Pawel;Kwapien, Jaroslaw;Drozdz, Stanislaw
我们比较研究了多重分形去趋势波动分析(MFDFA)和小波变换模极大值(WTMM)方法在正确检测数据的单分维和多重分维特征方面的适用性。我们通过使用根据几个著名的精确可解的数学模型产生的不同类型的人工信号来量化这两种方法的性能：单分支分数布朗运动、两分支Levy飞行和不同类型的多重分形二项式级联。我们的结果表明，在大多数情况下，在不知道过程的先验性质的情况下，应该推荐选择MFDFA。特别是，WTMM给出了具有不同赫斯特指数值的分数布朗运动的有偏结果，表明了虚假的多重分形性。在某些情况下，如果应用不同的小波，WTMM也可以给出不同的结果。我们不排除在实际数据分析中使用WTMM，但是，虽然可以以更自动的方式应用MFDFA，但WTMM的应用必须谨慎。在我们工作的第二部分，我们对来自美国和德国股市的经验数据进行了类似的分析。对于这一数据，两种方法都检测到了关于广义f(α)的丰富的多重分形性，但MFDFA表明，这种多重分形性比WTMM的情况更差。
We perform a comparative study of applicability of the multifractal detrended fluctuation analysis (MFDFA) and the wavelet transform modulus maxima (WTMM) method in proper detecting of monofractal and multifractal character of data. We quantify the performance of both methods by using different sorts of artificial signals generated according to a few well-known exactly soluble mathematical models: monofractal fractional Brownian motion, bifractal Levy flights, and different sorts of multifractal binomial cascades. Our results show that in the majority of situations in which one does not know a priori the fractal properties of a process, choosing MFDFA should be recommended. In particular, WTMM gives biased outcomes for the fractional Brownian motion with different values of Hurst exponent, indicating spurious multifractality. In some cases WTMM can also give different results if one applies different wavelets. We do not exclude using WTMM in real data analysis, but it occurs that while one may apply MFDFA in a more automatic fashion, WTMM must be applied with care. In the second part of our work, we perform an analogous analysis on empirical data coming from the American and from the German stock market. For this data both methods detect rich multifractality in terms of broad f(alpha), but MFDFA suggests that this multifractality is poorer than in the case of WTMM.