Abstract
In this paper, we have modified the Detrended Fluctuation Analysis (DFA) using the ternary Cantor set. We propose a modification of the DFA algorithm, Cantor DFA (CDFA), which uses the Cantor set theory of base 3 as a scale for segment sizes in the DFA algorithm. An investigation of the phenomena generated from the proof using real-world time series based on the theory of the Cantor set is also conducted. This new approach helps reduce the overestimation problem of the Hurst exponent of DFA by comparing it with its inverse relationship with of the Truncated Lévy Flight (TLF). CDFA is also able to correctly predict the memory behavior of time series.
Highlights
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
Cantor Detrended Fluctuation Analysis (CDFA) In this subsection, we prove that the subspace [Hmin, Hmax] of Hurst exponents is homeomorphic to [0, 1] of the Cantor set
We have proposed a modification to the DFA algorithm by utilizing the theory of the tenary Cantor set in the segment division step
Summary
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]. Self-similar phenomena describe the event in which the dependence in the time series decays more slowly than an exponential decay. 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)
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