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Open AccessArticle

Self-Similar Models: Relationship between the Diffusion Entropy Analysis, Detrended Fluctuation Analysis and Lévy Models

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Department of Mathematical Sciences, University of Texas at El Paso, El Paso, TX 79968, USA
2
Computational Science Program, University of Texas at El Paso, El Paso, TX 79968, USA
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Comisión Nacional de Energía Atómica, C1429BNP Buenos Aires, Argentina
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Department of Geological Sciences, University of Texas at El Paso, El Paso, TX 79968, USA
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Department of Seismology, Centro de Investigación Científica y de Educación Superior de Ensenada (CICESE), Ensenada 22860, Mexico
*
Author to whom correspondence should be addressed.
Mathematics 2020, 8(7), 1046; https://doi.org/10.3390/math8071046
Received: 7 May 2020 / Revised: 24 June 2020 / Accepted: 25 June 2020 / Published: 30 June 2020
Financial and geophysical data, like many other low and high frequency time series, are known to exhibit some memory effects. These memory effects may be long or short, permanent or temporal depending on the event that is being modeled. The purpose of this study is to investigate the memory effects characterized by the financial market closing values and volcanic eruption time series as well as to investigate the relation between the self-similar models used and the Lévy process. This paper uses highly effective scaling methods including Lévy processes, Detrended Fluctuation Analysis (DFA) and Diffusion Entropy Analysis (DEA) to examine long-range persistence behavior in time series by estimating their respective parameters. We use the parameter of the Lévy process ( α ) characterizing the data and the scaling parameters of DFA (H) and DEA ( δ ) characterizing the self-similar property to generate a relationship between the three (3) aforementioned scaling methods. Findings from the numerical simulations confirm the existence of long-range persistence (long-memory behavior) in both the financial and geophysical time series. Furthermore, the numerical results from this study indicates an approximate inverse relationship between the parameter of the Lévy process and the scaling parameters of the DFA and DEA (i.e., H , δ 1 α ), which we prove analytically.
Keywords: self-similar models; Lévy flight model; Diffusion Entropy Analysis; Detrended Fluctuation Analysis; long-range correlation self-similar models; Lévy flight model; Diffusion Entropy Analysis; Detrended Fluctuation Analysis; long-range correlation
MDPI and ACS Style

Mariani, M.C.; Kubin, W.; Asante, P.K.; Tweneboah, O.K.; Beccar-Varela, M.P.; Jaroszewicz, S.; Gonzalez-Huizar, H. Self-Similar Models: Relationship between the Diffusion Entropy Analysis, Detrended Fluctuation Analysis and Lévy Models. Mathematics 2020, 8, 1046.

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