**5. Conclusions**

In this work, we have proposed a modification to the DFA algorithm by utilizing the theory of the tenary Cantor set in the segmen<sup>t</sup> division step. The Cantor DFA (CDFA) has been compared to the *α* exponent of the truncated Lévy model and the Hurst exponent of the DFA. We have in addition proved that the interval of the Hurst exponent of the DFA is homeomorphic to the Cantor set. We confirm the results from the proof by illustrating the fractal phenomena exhibited by the Cantor set using real-world time series in Tables 1–3.

Our results from numerical simulations show that the CDFA generates better estimates of Hurst exponents. The CDFA proposed in this work automates the segmen<sup>t</sup> sizes in the DFA algorithm using the number base 3 theory of the Cantor set, where the time series is divided into multiples of 3 at each level. This modification helps to curb the overestimation problem of the Hurst exponent ( *H*) of the DFA by determining segmen<sup>t</sup> sizes based on the fractal phenomena depicted by the Cantor set while correctly predicting the memory behavior of the series in question.

The results are shown in Table 4 where the Hurst exponent of the CDFA is compared with that of the DFA and the scaling exponents (*α*) of the TLF. In [4], a relationship was established between the Hurst exponent of the DFA and the *α* exponent of the TLF. The CDFA is also shown to satisfy this relation, thus making it possible to extract the *α* exponent of the TLF from the Hurst exponent of the CDFA.

The CDFA approach can be applied to time series with odd lengths, time series whose lengths are not easily divisible by even numbers, time series whose lengths do not permit equal segmentation, etc. These kinds of series exist in several industries, including financial, geophysics, health and the like. Another application of the CDFA would be to act as a control model for the ordinary DFA to reduce the chances of overestimation of the Hurst exponent.

Since this is a modification of the DFA, there is the need to simulate CDFA with different data sets having varying characteristics for which the DFA has been shown to correctly detect their scaling behavior. An example will be DNA sequences, financial markets, etc., for further comparison of the model performance against the DFA.

For future work, we seek to investigate the robustness of the CDFA as stated earlier by simulating the model with data sets from different fields, including, but not limited to, DNA sequences, financial markets and geophysical data. In the case of "big data", we seek to extend the CDFA by "parallelizing" the sequential code of the CDFA (PCDFA) to improve its efficiency in simulation.

**Author Contributions:** Conceptualization, M.C.M., W.K., P.K.A. and O.K.T.; methodology, M.C.M., W.K., P.K.A., J.A.G. and O.K.T.; software, W.K.; validation, M.C.M. and O.K.T.; formal analysis, W.K.; data curation, W.K.; writing—original draft preparation, W.K.; writing—review and editing, W.K., P.K.A. and O.K.T.; visualization, W.K. and P.K.A.; supervision, M.C.M. and O.K.T. All authors have read and agreed to the published version of the manuscript.

**Funding:** The study was funded partially by the National Institute on Minority Health and Health Disparities (NIMHD) gran<sup>t</sup> (U54MD007592).

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** The data used in this work can be accessed at https://www.ntnu.edu/ inb/geri/software.

**Conflicts of Interest:** The authors declare no conflict of interest.
