Analytical representation of Gaussian processes in the $\mathcal{A}-\mathcal{T}$ plane

Abstract

Closed-form expressions, parametrized by the Hurst exponent $H$ and the length $n$ of a time series, are derived for paths of fractional Brownian motion (fBm) and fractional Gaussian noise (fGn) in the $\mathcal{A}-\mathcal{T}$ plane, composed of the fraction of turning points $\mathcal{T}$ and the Abbe value $\mathcal{A}$. The exact formula for ${\mathcal{A}}_{\mathrm{fBm}}$ is expressed via Riemann $\zeta$ and Hurwitz $\zeta$ functions. A very accurate approximation, yielding a simple exponential form, is obtained. Finite-size effects, introduced by the deviation of fGn's variance from unity, and asymptotic cases are discussed. Expressions for $\mathcal{T}$ for fBm, fGn, and differentiated fGn are also presented. The same methodology, valid for any Gaussian process, is applied to autoregressive moving average processes, for which regions of availability of the $\mathcal{A}-\mathcal{T}$ plane are derived and given in analytic form. Locations in the $\mathcal{A}-\mathcal{T}$ plane of some real-world examples as well as generated data are discussed for illustration.

arXiv comment14 pages, 12 figures, 1 table

References in corpus (19)

• Unified Schemes for Radio-Loud Active Galactic Nuclei
• Practical implementation of nonlinear time series methods: The TISEAN package
• SparsePak: A Formatted Fiber Field Unit for The WIYN Telescope Bench Spectrograph. I. Design, Construction, and Calibration
• Multifractal Detrended Fluctuation Analysis of Sunspot Time Series
• Determination of the Hurst Exponent by Use of Wavelet Transforms
• Description of stochastic and chaotic series using visibility graphs
• Detrending Moving Average variance: a derivation of the scaling law
• Comparative performance of selected variability detection techniques in photometric time series
• Characterizing Time Series via Complexity-Entropy Curves
• Generalized arcsine laws for fractional Brownian motion
• Detecting Variability in Massive Astronomical Time-Series Data I: application of an infinite Gaussian mixture model
• On the relationship between the Hurst exponent, the ratio of the mean square successive difference to the variance, and the number of turning points
• Arbitrary-order Hilbert spectral analysis and intermittency in solar wind density fluctuations
• On continuous-time autoregressive fractionally integrated moving average processes
• Emergence of self-similarity in football dynamics
• Searching transients in large-scale surveys. A method based on the Abbe value
• Wavelet and R/S analysis of the X-ray flickering of cataclysmic variables
• Machine learning techniques to select Be star candidates. An application in the OGLE-IV Gaia south ecliptic pole field
• The Hurst exponents of Nitzschia sp. diatom trajectories observed by light microscopy

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