Information Geometry of Nonlinear Stochastic Systems

Hollerbach, Rainer; Dimanche, Donovan; Kim, Eun Jin

doi:10.3390/e20080550

Cited by 12 publications

(28 citation statements)

References 40 publications

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“…For a nonlinear process, the scaling analysis is much harder due to the unavailability of an exact analytical solution. In [22,28], we utilized both semi-analytical and numerical approaches to study an nth order nonlinear process governed by ∂ t x = −γx n + ξ (n = 3, 5, 7) and showed that L(t → ∞) ≡ L ∞ ∝ D − n−1 3n−1 in the limit of D 0 ≫ D. If we use p(x, t → ∞) ∝ exp [− γ (n+1)D x n+1 ] and let ǫ be the width of the stationary PDF, we obtain ǫ ∝ D 1 n+1 and thus L ∞ ∝ ǫ − n 2 −1 3n−1 . This implies that D F ∼ 1 + n 2 −1 3n−1 , which increases with n for n ≫ 1.…”

Section: Discussionmentioning

confidence: 99%

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Scalings and fractals in information geometry: Ornstein–Uhlenbeck processes

Oxley¹,

Kim²

2018

J. Stat. Mech.

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We propose a new methodology to understand a stochastic process from the perspective of information geometry by investigating power-law scaling and fractals in the evolution of information. Specifically, we employ the Ornstein-Uhlenbeck process where an initial Probability Density Function (PDF) with a given width ǫ 0 and mean value y 0 relaxes into a stationary PDF with a width ǫ, set by the strength of a stochastic noise. By utilizing the information length L which quantifies the accumulative information change, we investigate the scaling of L with ǫ. When ǫ = ǫ 0 , the movement of a PDF leads to a robust power-law scaling with the fractal dimension D F = 2. In general when ǫ = ǫ 0 , D F = 2 is possible in the limit of a large time when the movement of a PDF is a main process for information change (e.g. y 0 ≫ ǫ ≫ ǫ 0). We discuss the physical meaning of different scalings due to PDF movement, diffusion and entropy change as well as implications of our finding for understanding a main process responsible for the evolution of information.

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Section: Discussionmentioning

confidence: 99%

“…Far from equilibrium where a PDF continuously changes, we can use time t as a parameter and generalize dl in Eq. (1) above to dL [11,[19][20][21][22][23][24][25][26][27][28] by considering an infinitesimal distance between the two PDFs at time t and t + dt as dL = dt/τ (t) where τ (t) is (time-dependent) correlation time of p(x, t)…”

Section: Introductionmentioning

confidence: 99%

Scalings and fractals in information geometry: Ornstein–Uhlenbeck processes

Oxley¹,

Kim²

2018

J. Stat. Mech.

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“…Previous numerical solutions of various FP equations in both one-dimensional and two-dimensional have been implemented based on the CN approach [21][22][23][24]. Hollerbach, Dimanche, and Kim [21] investigated the effect of different orders of nonlinear interaction on the geometric structure by considering the case when the equilibrium is a stable point. Jacquet, Kim, and Hollerbach [22] presented the time-evolution of probability density functions (PDFs) in a toy model of self-organized shear flows.…”

Section: Introductionmentioning

confidence: 99%

Investigation of Phase-Locked Loop Statistics via Numerical Implementation of the Fokker–Planck Equation

Jwo

2020

Applied Sciences

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The goal of this paper is to explore the effect of various parameters on the information geometric structure of the phase-locked loop (PLL) statistics, both transient and stationary. Comprehensive treatment on the behavior of PLL statistics will be given. The behavior of the phase-error statistics of the first-order PLL, in the presence of additive white Gaussian noise (WGN) is investigated through solving the differential equations known as the Fokker–Planck (FP) equation using the implicit Crank–Nicolson finite-difference method. The PLL is one of the most commonly used circuits in electrical engineering. A full knowledge of probability density functions (PDFs) of the phase-error statistics becomes essential in understanding the PLLs. Several illustrative examples are presented to provide profound insights on understanding the PLL statistics both qualitatively and quantitatively. Results covered include the transient and stationary statistics for the nonmodulo-2π probability density function, modulo-2π probability density function, and cycle slipping density function, of the phase error. Various numerical settings of PLL parameters are involved, including the detuning factor and signal-to-noise ratio (SNR). The results presented in this paper elucidate the link between various parameters and the information geometry of the phase-error statistics and form a basis for future investigation on PLL designs.

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“…In this paper, we extend [9,10] to investigate the time-evolution of PDFs to elucidate the effects of different initial conditions and correlation times. A particular interest will be to understand the information change in the relaxation of an initial PDF to a stationary PDF by using the information length L [11][12][13][14][15][16][17][18][19][20][21]. In the case of a stochastic variable x and time-dependent PDF p(x, t), L is defined by…”

Section: Introductionmentioning

confidence: 99%

Information geometry in a reduced model of self-organised shear flows without the uniform coloured noise approximation

Kim¹,

Jacquet²,

Hollerbach³

2019

J. Stat. Mech.

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We investigate information geometry in a toy model of self-organised shear flows, where a bimodal PDF of x with two peaks signifying the formation of mean shear gradients is induced by a finite memory time γ −1 of a stochastic forcing f . We calculate time-dependent Probability Density Functions (PDFs) for different values of the correlation time γ −1 and amplitude D of the stochastic forcing, and identify the parameter space for unimodal and bimodal stationary PDFs. By comparing results with those obtained under the Uniform Coloured Noise Approximation (UCNA) in Jacquet, Kim & Hollerbach (Entropy 20, 613, 2018), we find that UCNA tends to favor the formation of a bimodal PDF of x for given parameter values γ −1 and D. We map out attractor structure associated with unimodal and bimodal PDFs of x by measuring the total information length L ∞ = L(t → ∞) against the location x 0 of a narrow initial PDF of x. Here L(t) represents the total number of statistically different states that a system passes through in time. We examine the validity of the UCNA from the perspective of information change and show how to fine-tune an initial joint PDF of x and f to achieve a better agreement with UCNA results.

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Information Geometry of Nonlinear Stochastic Systems

Cited by 12 publications

References 40 publications

Scalings and fractals in information geometry: Ornstein–Uhlenbeck processes

Scalings and fractals in information geometry: Ornstein–Uhlenbeck processes

Investigation of Phase-Locked Loop Statistics via Numerical Implementation of the Fokker–Planck Equation

Information geometry in a reduced model of self-organised shear flows without the uniform coloured noise approximation

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