*2.2. EDFA*

Signal properties can vary strongly between different parts of a dataset. This is observed, e.g., for transients from one state to another, when well-pronounced variations in the local mean value affect the scaling exponent. Considering datasets with and without such transients can lead to distinct results of DFA. In a recent study [27], we illustrated the effects of nonstationarity for several cases: low-frequency trend, intermittent dynamics, and nonstationarity in energy. Besides the case when time-varying dynamics occurs throughout the signal, the *α* exponent is also influenced by short-term failures of the recording equipment or artifacts. Such data segments provide distinct local standard deviations (2) compared to the averaged quantities. In order to characterize the differences in nonstationarity across the entire signal, we have proposed the following modification of the method, called EDFA [31,32]. Within this approach, a new measure

$$dF(n) = \max[F\_{\text{loc}}(n)] - \min[F\_{\text{loc}}(n)], \quad F\_{\text{loc}}(n) = \sqrt{\frac{1}{n} \sum\_{k=1}^{n} [y(k) - y\_n(k)]^2} \tag{4}$$

is introduced, where *Floc*(*n*) are the local standard deviations of the profile from the trend, which are estimated for each segment. The difference *dF*(*n*) contains information about the impact of signal inhomogeneity. If the properties of the signal vary insignificantly, *dF*(*n*) takes values close to zero. Otherwise, a wide distribution of *Floc*(*n*) appears, and *dF*(*n*) varies with *n*, exhibiting power-law behavior characterized by the scaling exponent *β*

$$dF(n) \sim n^{\beta}.\tag{5}$$

In this definition, *β* becomes highly sensitive to artifacts in experimental recordings. The existence of a single artifact can lead to a large *Floc*(*n*) associated with max[*Floc*(*n*)], and the latter reduces the stability of the EDFA method. In particular, the *dF*(*n*) dependence can show strong fluctuations with *n*. A more stable algorithm is based on the statistical analysis of *Floc*(*n*), and the use of the standard deviation *σ*(*Floc*(*n*)) as a measure of the signal inhomogeneity. Thus, here we propose to consider the dependence

$$
\sigma(F\_{loc}(n)) \sim n^{\beta}.\tag{6}
$$

Figure 1 shows both dependences (5) and (6) in a lg–lg plot for the case of 1/*f*-noise used as a simple example of a homogeneous process with power-law correlations. This figure confirms that the latter definition provides reduced variability in the estimated values. Thus, standard error of the *β* estimates decreases from 0.0038 for the definiton (5) to 0.0023 for the definiton (6).

For physiological datasets, differences are usually larger. Although the exponents *β* in Equations (5) and (6) may differ, we use the same designation (*β*) to quantify the impact of nonstationarity and a more stable algorithm based on *σ*(*Floc*(*n*)) for its evaluation. The *β* exponent can take as positive, as negative values [30]. Both *α* and *β* exponents describe different signal properties and are independent quantities.

**Figure 1.** Dependences described by Equations (5) and (6) in the lg–lg plot for 1/*f*-noise. The *β*exponent is estimated with the standard errors 0.0038 and 0.0023, respectively.
