**1. Introduction**

A ternary Cantor set is a set built by removing the middle part of a series when divided into three parts and repeating this process with the remaining shorter segments. It is the prototype of a fractal [1]. A fractal is a geometric object that has similar statistical properties to itself on all scales. If a fractal object is successively magnified, it looks similar or exactly like the original shape of the fractal. A similar pattern exhibited at increasingly smaller scales is often known in fractal mathematics as self-similarity [2,3]. In time series, self-similar phenomena describe the event in which the dependence in the time series decays more slowly than an exponential decay. Typically, it follows a power-like decay [4]. Scaling methods exist for quantifying the power-law exponent of the decay function such as Rescaled Range Analysis (R/S), Detrended Fluctuation Analysis (DFA) and the Truncated Lévy Flight (TLF).

The Rescaled Range Analysis (R/S) method by Hurst subdivides integrated time series into adjacent segmen<sup>t</sup> sizes and examines the range (R) of the integrated fluctuations. Then, a measure of dispersion, usually standard deviation (S), is determined as a function of segmen<sup>t</sup> size. A power law governs the approximate relationship between the Rescaled Range Analysis' statistic (R/S) and the segmen<sup>t</sup> size [5].

The Detrended Fluctuation Analysis (DFA) by Peng et al. (1994) is a technique that quantifies the same power-law exponent of the R/S method. Addressing difficulties in determining correct power-law exponents of the R/S method in non-stationary time series resulted in the introduction of the DFA. Unlike the R/S method, the DFA uses a local detrending approach (usually linear regression) in the segments of the integrated series. For time series with higher-order trends, polynomial fit replaces the linear regression approach of the DFA [6]. This provides its power-law exponents' protection against effects of nonstationarity and pollution of time series by external signals while eliminating spurious detection of long memory [7]. Empirical evidence has shown that the DFA performs well compared to other variance scaling methods including the R/S methods when estimating power-law exponents.

**Citation:** Mariani, C.M.; Kubin, W.; Asante, P.K.; Guthrie, J.A.; Tweneboah, O.K. Relationship between Continuum of Hurst Exponents of Noise-like Time Series and the Cantor Set. *Entropy* **2021**, *23*, 1505. https://doi.org/10.3390/ e23111505

Academic Editors: Ryszard Kutner, Christophe Schinckus and H. Eugene Stanley

Received: 20 October 2021 Accepted: 10 November 2021 Published: 13 November 2021

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Usually, characterizing stochastic processes empirically requires the study of determining asymptotic probability density distributions (pdf) and temporal correlations. Brownian motion models the evolution of a particle's position over time with the assumption that the movement of the particle follows a diffusive process with Gaussian distribution. This model did not describe accurately real-world time series because kurtosis of associated pdf is greater than that of the Gaussian distribution [8,9]. The Truncated Lévy Flight (TLF) model originated as a means to address the difficulties of the Brownian motion for working in long-range correlation scales. The scaling exponent (0 < *α* ≤ 2) of the TLF measures the memory behavior in time series that follows a diffusion process with Gaussian and non-Gaussian distributions [10].

In [11], a clear comparison was made between DNA and economics by the authors, showing the underlining similarities that allow researchers to model seemingly different phenomena using the same or slightly modified models. In the same manner, these variance scaling models have the added advantage of being used to model long memory effects in different fields where stochastic processes occur [2,7]. Thus, be it DNA sequencing, financial markets, geophysical time series etc., scaling methods have been used to detect long/short memory behaviors.

Scaling approaches serve as means of characterizing the dependence of observations separated in time series dominated by stochastic properties. Applications with DFA have been done in DNA sequences [6,12,13], neural oscillations [14], detection of speech pathology [15], heartbeat fluctuation in different sleep stages [16], describing cloud breaking [17], gearbox fault diagnosis [18], analysis of fetal cardiac data [19], streamflow in the Yellow River Basin in China [20], evaluation near infrared spectra of green and roasted coffee samples [21], just to mention a few.

Empirical evidence has shown that the DFA has the tendency of overestimating the scaling exponent [2,22]. We have not come accross any literature at the moment that describes a definite approach in the segmen<sup>t</sup> division step of the DFA algorithm. However, we observe that estimates of power-law exponents are influenced by the scale of choice [23,24]. Our goal in this work is to propose a definite non-overlapping segmen<sup>t</sup> division approach in the DFA algorithm (CDFA) that utilizes the theory of the ternary Cantor set. We show that using this approach we are able to rightly determine the correct scaling exponent to detect the memory behavior of the time series as well as reducing the over-fitting nature of the DFA. This approach has the advantage of generalizing the segmen<sup>t</sup> division step in the DFA algorithm. The Hurst exponents obtained from the CDFA method are then compared with the exponents of the DFA and the TLF on real time series.

In Section 2, we present proof of the relationship between the continuum of Hurst exponents of the DFA and Cantor set. We also present the scaling methods TLF, DFA and CDFA in this section. In Section 3, we present results and discussions from our investigation noting that for noise-like time series, anti-persistence, white noise and persistence behavior in time series imply 0 ≤ *H* < 0.5, *H* = 0.5 and 0.5 < *H* ≤ 1 respectively. The overestimation of DFA's Hurst exponent decreasing with the Cantor scales is also discussed in this section. Section 4 concludes the paper.
