Generalization of Higuchi’s fractal dimension for multifractal analysis of time series with limited length

Abstract

There exist several methodologies for the multifractal characterization of nonstationary time series. However, when applied to sequences of limited length, these methods often tend to overestimate the actual multifractal properties. To address this aspect, we introduce here a generalization of Higuchi’s estimator of the fractal dimension as a new way to characterize the multifractal spectrum of univariate time series or sequences of relatively short length. This multifractal Higuchi dimension analysis (MF-HDA) method considers the order-q moments of the partition function provided by the length of the time series graph at different levels of subsampling. The results obtained for different types of stochastic processes, a classical multifractal model, and various real-world examples of word length series from fictional texts demonstrate that MF-HDA provides a reliable estimate of the multifractal spectrum already for moderate time series lengths. Practical advantages as well as disadvantages of the new approach as compared to other state-of-the-art methods of multifractal analysis are discussed, highlighting the particular potentials of MF-HDA to distinguish mono- from multifractal dynamics based on relatively short sequences.

Appendices

Appendix A: The case $q = 0$

In multifractal analysis, it is common that the scaling exponent $d_q$ is not well defined when $q \rightarrow {} 0$. In our case, this value cannot be directly determined by means of the generalized curve lengths (Eq. (7)) due to the presence of a divergence in the exponent. More formally, we have

$$\begin{aligned}&\lim _{q\rightarrow 0} {\mathcal {L}}(q, k) \nonumber \\&\quad = \lim _{q\rightarrow 0} \frac{N-1}{k^2} \Bigg \{ \sum _{n=1}^{N_b} \langle (\Delta X_n(k))^q \rangle P_n(\Delta X(k)) \Bigg \}^{1/q} \nonumber \\&\qquad \sim \lim _{q\rightarrow 0} k ^ {- d_q}. \end{aligned}$$

(19)

Using some algebraic operations applied to the latter equation, which are omitted here for brevity, and applying L’H${\hat{o}}$spital’s rule, we find that a logarithmic transformation is required in order to determine the scaling exponent $d_0$ as

$$\begin{aligned} {\mathcal {L}}(0, k) \equiv \frac{N-1}{k^2} \exp { \Bigg ( E [ \langle \ln \{ \Delta X(k) \} \rangle ] \Bigg ) } \sim k^{-d_0}, \end{aligned}$$

(20)

where $E [ \langle \ln \{ \Delta X(k) \} \rangle ]= \sum _{n=1}^{N_b} \langle \ln \{ \Delta X(k)\} \rangle P_n(\Delta X(k))$.

Appendix B: Regularization effect of removing the lower percentile of absolute increments

As discussed in Sect. 3, numerical instabilities can appear in the evaluation of the moments of the generalized curve length ${\mathcal {L}}(k,q)$ (Eq. 7), especially for $q<-1$. In this case, we have suggested that the local mean could be replaced by a one-sided trimmed version $\langle \Delta X_{n'}(k)\rangle _{>p_r}$ in Eq. (7), where $n'$ represents the $n'$–th interval of a new equiprobable partition in which we have removed the r–th percentile $p_r$ of the empirical distribution of the absolute increments. We note that numerical experiments with both, one-sided (asymmetrically) and two-sided (symmetrically) trimmed means revealed no qualitative differences in the resulting estimates (not shown), while removing the uppermost percentiles (i.e., very large increments) appears unnecessary since those values have no negative effects on the stability of the numerical estimates of the generalized curve lengths.

To address the problem of selecting a specific percentile to be removed, we focus here on just one statistical property, the confidence interval (CI) of the estimated slope (i.e., the scaling exponent $d_q$ in Eq. (8)) in the linear regression of the double-logarithmic generalized curve length versus scale relationship, at a certain confidence level (here, ${\gamma }=0.05$), and for the most negative value of q. For two-sided confidence intervals, the CI width (CIW) (measured in units of the associated standard error) is given by ${\hat{I}}_{q,p_{r}} \equiv I_{q,p_{r}}/S_{d_q}$, with $S_{d_{q}}$ being the standard error of the estimated slope $d_q$ [56].

Figure 11 shows the behavior of the rescaled CIW (in units of ${\hat{I}}_{p_{r}=0}$) as a function of the removed percentile $p_r$, for some of the simulated stochastic processes and real-world data sets discussed in Sections 4 and 6, respectively, for $q=-5$. The results show that, as the removed percentile is increased, the CIW decreases in such a way that, for fractional Gaussian noises with $H=0.3$, $H=0.5$ and $H=0.75$, the rescaled CIW has decayed by more than one half of its initial value when $p_r=1$, while for the world length data (exemplified here by the ULY book) the observed decay is slower. For practical purposes, we suggest that a criterion for selecting the value of the percentile to be removed should consider empirically a value of the percentile for which the rescaled CIW has stabilized, that is, even if higher percentiles are removed, there are no substantial further changes. We observe that $p_r\approx 5$ and $p_r \approx 12$ would be desirable in the cases of the simulated fractional Gaussian noises and word length data, respectively.

Fig. 11

a Behavior of the rescaled confidence interval in dependence on the removed percentile $p_r$ of the distribution of absolute increments for $q=-5$ for fractional Gaussian noises and word length data (ULY book). b Dependence of the rescaled confidence interval on the sequence length N for $q=-5$ in the case of fractional Gaussian noises. As N increases, ${\hat{I}}/{\hat{I}}_{p_r=0}$ decreases and becomes independent of the system size. Error bars indicate the standard deviation estimated from 10 independent realizations

While the suggested strategy presents just a first attempt to improving the practical estimation of the generalized Higuchi’s fractal dimensions, we emphasize that there may be cases in which the rescaled CIW may behave in a more unstable way with increasing percentile $p_r$. In such cases, additional numerical tests with larger $p_r$ values may become necessary to determine a reliable value leading to sufficiently stable estimates.