Fractional Brownian motion, fractional Gaussian noise, and Tsallis permutation entropy

2008 ◽  
Vol 387 (24) ◽  
pp. 6057-6068 ◽  
Author(s):  
L. Zunino ◽  
D.G. Pérez ◽  
A. Kowalski ◽  
M.T. Martín ◽  
M. Garavaglia ◽  
...  
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Gaussian Noise ◽  
Permutation Entropy ◽  
Fractional Gaussian Noise

Permutation entropy of fractional Brownian motion and fractional Gaussian noise

2008 ◽  
Vol 372 (27-28) ◽  
pp. 4768-4774 ◽  
Author(s):  
L. Zunino ◽  
D.G. Pérez ◽  
M.T. Martín ◽  
M. Garavaglia ◽  
A. Plastino ◽  
...  
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Gaussian Noise ◽  
Permutation Entropy ◽  
Fractional Gaussian Noise

scholarly journals Correlation Properties of (Discrete) Fractional Gaussian Noise and Fractional Brownian Motion

2015 ◽  
Vol 2015 ◽  
pp. 1-7 ◽  
Author(s):  
Didier Delignières
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Gaussian Noise ◽  
Smooth Transition ◽  
Theoretical Interpretation ◽  
Fractional Gaussian Noise ◽  
Straight Line ◽  
Central Interest ◽  
Correlation Properties ◽  
Lower Limits

The fractional Gaussian noise/fractional Brownian motion framework (fGn/fBm) has been widely used for modeling and interpreting physiological and behavioral data. The concept of 1/fnoise, reflecting a kind of optimal complexity in the underlying systems, is of central interest in this approach. It is generally considered that fGn and fBm represent a continuum, punctuated by the boundary of “ideal” 1/fnoise. In the present paper, we focus on the correlation properties of discrete-time versions of these processes (dfGn and dfBm). We especially derive a new analytical expression of the autocorrelation function (ACF) of dfBm. We analyze the limit behavior of dfGn and dfBm when they approach their upper and lower limits, respectively. We show that, asHapproaches 1, the ACF of dfGn tends towards 1 at all lags, suggesting that dfGn series tend towards straight line. Conversely, asHapproaches 0, the ACF of dfBm tends towards 0 at all lags, suggesting that dfBm series tend towards white noise. These results reveal a severe breakdown of correlation properties around the 1/fboundary and challenge the idea of a smooth transition between dfGn and dfBm processes. We discuss the implications of these findings for the application of the dfGn/dfBm model to experimental series, in terms of theoretical interpretation and modeling.

scholarly journals On the Behaviour of Weighted Permutation Entropy on Fractional Brownian Motion in the Univariate and Multivariate Setting

2021 ◽  
Vol 34 (1) ◽  
Author(s):  
Marisa Mohr ◽  
Florian Wilhelm ◽  
Ralf Möller
Keyword(s):  
Time Series ◽  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Real World ◽  
Multivariate Time Series ◽  
Permutation Entropy ◽  
Qualitative Behaviour ◽  
Univariate Time Series ◽  
Real World Applications ◽  
Real World Problems

The estimation of the qualitative behaviour of fractional Brownian motion is an important topic for modelling real-world applications. Permutation entropy is a well-known approach to quantify the complexity of univariate time series in a scalar-valued representation. As an extension often used for outlier detection, weighted permutation entropy takes amplitudes within time series into account. As many real-world problems deal with multivariate time series, these measures need to be extended though. First, we introduce multivariate weighted permutation entropy, which is consistent with standard multivariate extensions of permutation entropy. Second, we investigate the behaviour of weighted permutation entropy on both univariate and multivariate fractional Brownian motion and show revealing results.

scholarly journals On the law of the solution to a stochastic heat equation with fractional noise in time

2015 ◽  
Vol 23 (3) ◽  
Author(s):  
Solesne Bourguin ◽  
Ciprian A. Tudor
Keyword(s):  
Brownian Motion ◽  
Heat Equation ◽  
Fractional Brownian Motion ◽  
Gaussian Noise ◽  
Stochastic Heat Equation ◽  
Bifractional Brownian Motion ◽  
Fractional Noise ◽  
The Stochastic Heat Equation ◽  
The Law

AbstractWe study the law of the solution to the stochastic heat equation with additive Gaussian noise which behaves as the fractional Brownian motion in time and is white in space. We prove a decomposition of the solution in terms of the bifractional Brownian motion. Our result is an extension of a result by Swanson.

scholarly journals EMPIRICAL MODE DECOMPOSITION: APPLICATIONS ON SIGNAL AND IMAGE PROCESSING

2009 ◽  
Vol 01 (01) ◽  
pp. 125-175 ◽  
Author(s):  
JEAN-CLAUDE NUNES ◽  
ÉRIC DELÉCHELLE
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Texture Analysis ◽  
Empirical Mode Decomposition ◽  
Gaussian Noise ◽  
Hurst Exponent ◽  
Analytical Approach ◽  
Monogenic Signal ◽  
Mode Decomposition ◽  
Local Properties

In this paper, we propose some recent works on data analysis and synthesis based on Empirical Mode Decomposition (EMD). Firstly, a direct 2D extension of original Huang EMD algorithm with application to texture analysis, and fractional Brownian motion synthesis. Secondly, an analytical version of EMD based on PDE in 1D-space is presented. We proposed an extension in 2D-case of the so-called "sifting process" used in the original Huang's EMD. The 2D-sifting process is performed in two steps: extrema detection (by neighboring window or morphological operators) and surface interpolation by splines (thin plate splines or multigrid B-splines). We propose a multiscale segmentation approach by using the zero-crossings from each 2D-intrinsic mode function (IMF) obtained by 2D-EMD. We apply the Hilbert–Huang transform (which consists of two parts: (a) Empirical mode decomposition, and (b) the Hilbert spectral analysis) to texture analysis. We analyze each 2D-IMF obtained by 2D-EMD by studying local properties (amplitude, phase, isotropy, and orientation) extracted from the monogenic signal of each one of them. The monogenic signal proposed by Felsberg et al. is a 2D-generalization of the analytic signal, where the Riesz transform replaces the Hilbert transform. These local properties are obtained by the structure multivector such as proposed by Felsberg and Sommer. We present numerical simulations of fractional Brownian textures. Recent works published by Flandrin et al. relate that, in the case of fractional Gaussian noise (fGn), EMD acts essentially as a dyadic filter bank that can be compared to wavelet decompositions. Moreover, in the context of fGn identification, Flandrin et al. show that variance progression across IMFs is related to Hurst exponent H through a scaling law. Starting with these results, we proposed an algorithm to generate fGn, and fractional Brownian motion (fBm) of Hurst exponent H from IMFs obtained from EMD of a White noise, i.e., ordinary Gaussian noise (fGn with H = 1/2). Deléchelle et al. proposed an analytical approach (formulated as a partial differential equation (PDE)) for sifting process. This PDE-based approach is applied on signals. The analytical approach has a behavior similar to that of the EMD proposed by Huang.

High order Anderson parabolic model driven by rough noise in space

2021 ◽  
pp. 2150052
Author(s):  
Qiyong Cao ◽  
Hongjun Gao
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Gaussian Noise ◽  
Existence And Uniqueness ◽  
Space Variable ◽  
Index Formula ◽  
Hurst Index ◽  
Parabolic Model ◽  
Time And Space ◽  
Model Driven

In this paper, we concern the fourth parabolic model on [Formula: see text] driven by a multiplicative Gaussian noise which behaves like fractional Brownian motion in time and space with Hurst index [Formula: see text] and [Formula: see text], respectively. The existence and uniqueness of mild solution in Skorohod sense are proved, and the weak intermittency is obtained by estimating [Formula: see text]th ([Formula: see text]) moment of the solution. Moreover, the Hölder continuity can be obtained for the time and space variable.

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