2.1.1. Asymmetric Multifractal Spectrum Parameters

The multifractal characteristics of time series can be described not only by the generalized Hurst exponent *H*(*q*) but also by the multifractal scaling exponent *τ*(*q*), and their relationship can be expressed as *τ*(*q*) = *qH*(*q*) − 1. In the case that *τ*(*q*) and *q* are linearly related, the analyzed time series is monofractal. In the case that *τ*(*q*) and *q* have a nonlinear relationship, the analyzed time series is multifractal. Additionally, it is significant to note that the stronger their nonlinear relationship is, the stronger the multifractal characteristics are [46].

Moreover, using the multifractal (singularity) spectrum *f*(*α*) can also describe multifractional characteristics of time series. The multifractal spectrum is obtained by applying the first-order Legendre transform [39,46]:

$$
\alpha = d\tau(q) / dq,\tag{5}
$$

$$f(\mathfrak{a}) = q\mathfrak{a} - \mathfrak{r}(q),\tag{6}$$

where *α* is the singularity strength (also known as the Hölder exponent) that characterizes singularities in the time series. The interpretation of *α* is as follows: If *α* = 1, then the distribution of the time series data is uniform. If *α* < 1, then the singularity degree is larger. On the other hand, if *α* > 1, then the singularity degree is smaller. The multifractal spectrum *f*(*α*) denotes the singularity content [46,47].

To analyze and make a solid understanding of the multifractal characteristics of a time series, a set of the asymmetric multifractal spectrum parameters (*α*0, Δ*α*, *A*) has been suggested. More specifically, the maximum of the multifractal spectrum *f*(*α*) is used to detect the correlation behavior in terms of persistence and anti-persistence. The spectrum *α*<sup>0</sup> gives the maximum *f*(*α*), i.e., *f*(*α*0) = 1. At this spectrum, the measure provides information about the central tendency of the multifractal spectrum. If *α*<sup>0</sup> < 0.5, then the correlations in the time series exhibit anti-persistent behavior (i.e., an increase is very likely to be followed by a decrease), if *α*<sup>0</sup> > 0.5, then the correlations in the time series exhibit persistent behavior (i.e., an increase is very likely to be followed by an increase, and a decrease is very likely to be followed by a decrease), whereas if *α*<sup>0</sup> = 0.5, then the time series displays characteristics of a standard non-correlated sequence [39,47,48]. By looking into the spectrum width, one can quantitatively detect the time series multifractality. Specifically, the width of the spectrum is estimated by the equation Δ*α* = *αmax* − *αmin*, and it reflects the degree of multifractality of the time series. The larger values of Δ*α* are, the stronger the degree is and the more severe the fluctuations in the time series are. On the contrary, the smaller the values of Δ*α*, the more the time series is close to a monofractal behavior, indicating less significant fluctuations in the time series. The spectrum width should be equal to zero for a completely monofractal time series [39,49,50]. The dominance of small or large fluctuations is also an interesting characteristic of time series. This information can be extracted from the skew asymmetry of the multifractal spectrum, which is defined by the equation [51] *<sup>A</sup>* <sup>=</sup> *<sup>L</sup>*−*<sup>R</sup> <sup>R</sup>*+*<sup>L</sup>* <sup>=</sup> <sup>−</sup>Δ*<sup>S</sup> <sup>W</sup>* , where *<sup>R</sup>* <sup>=</sup> *<sup>α</sup>max* <sup>−</sup> *<sup>α</sup>*0, *<sup>L</sup>* <sup>=</sup> *<sup>α</sup>*<sup>0</sup> <sup>−</sup> *<sup>α</sup>min*, <sup>Δ</sup>*<sup>S</sup>* <sup>=</sup> *<sup>R</sup>* <sup>−</sup> *<sup>L</sup>*, and *W* = *R* + *L* = Δ*α* = *αmax* − *αmin*. If *A* > 0 (*L* > *R*), the spectrum is left-skewed, which means that the scaling behavior of large fluctuations dominates the multifractal behavior. On the contrary, if *A* < 0 (*L* < *R*), then the spectrum is right-skewed, where the scaling behavior of small fluctuations dominates. The case of *A* = 0 indicates that the shape of multifractal spectra is symmetric [46,51].

Another multifractal spectrum asymmetry metric is the so-called truncation, defined as Δ*f*(*a*) = *f*(*αmin*) − *f*(*αmax*) [49,52]. If Δ*f*(*a*) < 0, the multifractal spectrum is righttruncated, i.e., it has a long left tail, indicating that the multifractal structure in the time series is insensitive to the local fluctuations with small magnitudes. In other words, the time series is less multifractal, closer to monofractal, for the small fluctuations than for the large fluctuations. If Δ*f*(*a*) > 0, the multifractal spectrum is left-truncated, i.e., it has a long right tail, indicating that the multifractal structure is then insensitive to the local fluctuations with large magnitudes. It has to be noted that, very often, truncation and skew

asymmetries are directly related so that a left-skewed spectrum is also right-truncated, and a right-skewed is left-truncated. The absolute value of truncation, also known as "C-value", C − value = |Δ*f*(*a*)| = | *f*(*αmin*) − *f*(*αmax*)| [49,52], indicates the degree of the truncation asymmetry, which also provides interesting information as C-values are known to illustrate the systems' underlying undulation or instability. The degree of undulation or instability becomes minimum when the C-value presents the smallest value (≈ 0) [49,52].
