去趋势波动分析的一致性
Consistency of detrended fluctuation analysis
摘要：
去趋势波动分析（DFA）中的标度函数$F(s)$标度为$F(s)\sim s^{H}$，用于Hurst指数为$H$的随机过程。我们证明了这两个平稳随机过程与$0<H<1$，和非平稳随机过程与$1<H<2$的标度律。对于$H<0.5$，我们观察到使用渐近（幂律）自相关函数（ACF）产生$F(s)\sim s^{1/2}$。我们还证明了DFA中的涨落函数期望等于：i）ACF的加权和ii）二阶结构函数的加权和。这些结果使我们能够计算精确的有限大小的偏差的信号缩放，以及研究DFA的信号，没有幂律统计。我们用例子说明了这一点，我们发现，以前建议的修改DFA将增加赫斯特指数$H>1$的信号的偏差。作为新理论的最后一个应用，我们提出了一个估计器$\hat F(s)$，可以处理丢失的数据在定期采样的时间序列，而不需要插值方案。在弱正则性条件下，$\hat F(s)$在期望上等于无间隙情况下的涨落函数$F(s)$。
英文摘要：
The scaling function $F(s)$ in detrended fluctuation analysis (DFA) scales as $F(s)\sim s^{H}$ for stochastic processes with Hurst exponents $H$. We prove this scaling law for both stationary stochastic processes with $0<H<1$, and non-stationary stochastic processes with $1<H<2$. For $H<0.5$ we observe that using the asymptotic (power-law) auto-correlation function (ACF) yield $F(s)\sim s^{1/2}$. We also show that the fluctuation function in DFA is equal in expectation to: i) A weighted sum of the ACF ii) A weighted sum of the second order structure function. These results enable us to compute the exact finite-size bias for signals that are scaling, as well as studying DFA for signals that do not have power-law statistics. We illustrate this with examples, where we find that a previous suggested modified DFA will increase the bias for signals with Hurst exponents $H>1$. As a final application of the new theory, we present an estimator $\hat F(s)$ that can handle missing data in regularly sampled time series without the need for interpolation schemes. Under mild regularity conditions, $\hat F(s)$ is equal in expectation to the fluctuation function $F(s)$ in the gap-free case.
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