2018/09/30 by Jarosław Klamut, Ryszard Kutner, Tomasz Gubiec +1
Economics, Econometrics and Finance · Mathematics · Physics and Astronomy · #Complex Systems and Time Series Analysis #Detrended fluctuation analysis #Exponent #Fractal #Geology #Geometry #Hurst exponent #Legendre polynomials #Mandelbrot set #Mathematical analysis #Mathematics #Metastability #Multifractal system #Physics #Quantum mechanics #Series (stratigraphy) #Statistical Mechanics and Entropy #Statistical physics #Statistics #Theoretical and Computational Physics #q-fin.ST
paper · pdf · doi:10.1103/physreve.101.063303
published as Phys. Rev. E 101, 063303 (2020) · 24 pages, 13 figures
arxiv created 2020/04/25 · openalex publication_date 2020/06/08 · arxiv updated 2020/07/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Empirical time series of interevent or waiting times are investigated using a modified Multifractal Detrended Fluctuation Analysis operating on fluctuations of mean detrended dynamics. The core of the extended multifractal analysis is the nonmonotonic behavior of the generalized Hurst exponent h(q)-the fundamental exponent in the study of multifractals. The consequence of this behavior is the nonmonotonic behavior of the coarse Hölder exponent α(q) leading to multibranchedness of the spectrum of dimensions. The Legendre-Fenchel transform is used instead of the routinely used canonical Legendre (single-branched) contact transform. Thermodynamic consequences of the multibranched multifractality are revealed. These are directly expressed in the language of phase transitions between thermally stable, metastable, and unstable phases. These phase transitions are of the first and second orders according to Mandelbrot's modified Ehrenfest classification. The discovery of multibranchedness is tantamount in significance to extending multifractal analysis.