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

Maria C Mariani,Osei K Tweneboah,Peter K Asante,William Kubin,Maria P Beccar-Varela,Sebastian Jaroszewicz,Hector Gonzalez-Huizar

doi:10.3390/math8071046

Abstract

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.

Highlights

In recent years, researchers have developed several techniques to predict and forecast real-world events or processes that are seemingly difficult to predict due to their complex nature
From our numerical results we propose a mathematical relationship between the self-similar models and the Lévy model
This suggests that there exist an inverse relationship between the Diffusion Entropy Analysis (DEA) parameter δ and the Lévy parameter α, i.e.,: δ≈

Summary

Introduction

Researchers have developed several techniques to predict and forecast real-world events or processes that are seemingly difficult to predict due to their complex nature. Such studies have produced major improvements in predictions related to financial market crashes, volcanic activities, climates, heartbeats, etc. Mandelbrot (1982), in the article “The Fractal Geometry of Nature”, describes the surprisingly excellent resemblance between real landscapes and geometrically generated landscapes using simple fractal models [1] He pointed out that fractals can exhibit self-similarity in enormous ways. October 2001 and USA (S&P500) from 2 January 1991 to 25 October 2001 This data was first used for big data analysis in [16] and recently used in [7].

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