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

Abstract

<jats:p>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.</jats:p>

Relationship between Continuum of Hurst Exponents of Noise-like Time Series and the Cantor Set

“…For the monofractal and multifractal noise-like time series, we observe that H c α is approximately equal to 1 while Hα exceeds 1. This deviates from the inverse relationship between the Hurst exponents and the scaling exponents of the TLF for Gaussian noise as discussed in the paper [4]. This highlights the overestimation of the Hurst exponent of the DFA approach that happens in practice.…”

“…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.…”

“…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).…”

Determining the background driving process of the Ornstein-Uhlenbeck model

“…Detrended fluctuation analysis. Peng et al [38] proposed the Detrended Fluctuation Analysis (DFA) while examining a sequence of DNA nucleotides to study the self-similarity [35] and long-range dependence of time series. From the moment it was submitted to date, DFA has become a widely used method for determining fractal scaling properties and detecting long-range correlations in non-stationary time series.…”

Methods for Mathematical Analysis of Simulated and Real Fractal Processes with Application in Cardiology

“…A number of studies conducted in recent years have shown that many systems in nature generate time series with fractal behaviour [1][2][3][4][5][6]. Examples of such time series are: series of intervals between consecutive heartbeats [2,3,7], economic data on exchange rates [4], electricity prices [5], geophysical data on temperature, precipitation, tides [6,[8][9][10], etc. For the analysis of this type of processes, methods are applied that allow the determination of the global characteristics of the processes as well as the peculiarities of their local structures.…”