Scalings and fractals in information geometry: Ornstein-Uhlenbeck processes

William Oxley, Eun Jin Kim

Research output: Contribution to journalArticle

1 Downloads (Pure)

Abstract

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 ϵ0 and mean value y0 relaxes into a stationary PDF with a width ϵ, 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 ϵ. When ϵ = ϵ0, the movement of a PDF leads to a robust power-law scaling with the fractal dimension DF = 2. In general when ϵ ≠ ϵ0, DF = 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. y0 ≫ ϵ ≫ ϵ0). 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.

Original languageEnglish
Article number113401
JournalJournal of Statistical Mechanics: Theory and Experiment
Volume2018
Issue number11
DOIs
Publication statusPublished - 15 Nov 2018
Externally publishedYes

Fingerprint

Ornstein-Uhlenbeck process
Information Geometry
Ornstein-Uhlenbeck Process
Probability density function
Fractal
fractals
probability density functions
Scaling
scaling
geometry
deuterium fluorides
Power Law
scaling laws
Fractal Dimension
Mean Value
stochastic processes
Stochastic Processes
Quantify
Entropy
Geometry

Bibliographical note

This is an author-created, un-copyedited version of an article accepted for publication/published in Journal of Statistical Mechanics: Theory and Experiment. IOP Publishing Ltd is not responsible for any errors or omissions in this version of the manuscript or any version derived from it. The Version of Record is available online at 10.1088/1742-5468/aae851

Copyright © and Moral Rights are retained by the author(s) and/ or other copyright owners. A copy can be downloaded for personal non-commercial research or study, without prior permission or charge. This item cannot be reproduced or quoted extensively from without first obtaining permission in writing from the copyright holder(s). The content must not be changed in any way or sold commercially in any format or medium without the formal permission of the copyright holders.

Keywords

  • stochastic processes

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Statistics and Probability
  • Statistics, Probability and Uncertainty

Cite this

Scalings and fractals in information geometry : Ornstein-Uhlenbeck processes. / Oxley, William; Kim, Eun Jin.

In: Journal of Statistical Mechanics: Theory and Experiment, Vol. 2018, No. 11, 113401, 15.11.2018.

Research output: Contribution to journalArticle

@article{7937822520f5404fa933f3f2fd0b3041,
title = "Scalings and fractals in information geometry: Ornstein-Uhlenbeck processes",
abstract = "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 ϵ0 and mean value y0 relaxes into a stationary PDF with a width ϵ, 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 ϵ. When ϵ = ϵ0, the movement of a PDF leads to a robust power-law scaling with the fractal dimension DF = 2. In general when ϵ ≠ ϵ0, DF = 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. y0 ≫ ϵ ≫ ϵ0). 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.",
keywords = "stochastic processes",
author = "William Oxley and Kim, {Eun Jin}",
note = "This is an author-created, un-copyedited version of an article accepted for publication/published in Journal of Statistical Mechanics: Theory and Experiment. IOP Publishing Ltd is not responsible for any errors or omissions in this version of the manuscript or any version derived from it. The Version of Record is available online at 10.1088/1742-5468/aae851 Copyright {\circledC} and Moral Rights are retained by the author(s) and/ or other copyright owners. A copy can be downloaded for personal non-commercial research or study, without prior permission or charge. This item cannot be reproduced or quoted extensively from without first obtaining permission in writing from the copyright holder(s). The content must not be changed in any way or sold commercially in any format or medium without the formal permission of the copyright holders.",
year = "2018",
month = "11",
day = "15",
doi = "10.1088/1742-5468/aae851",
language = "English",
volume = "2018",
journal = "Journal of Statistical Mechanics: Theory and Experiment",
issn = "1742-5468",
publisher = "IOP Publishing",
number = "11",

}

TY - JOUR

T1 - Scalings and fractals in information geometry

T2 - Ornstein-Uhlenbeck processes

AU - Oxley, William

AU - Kim, Eun Jin

N1 - This is an author-created, un-copyedited version of an article accepted for publication/published in Journal of Statistical Mechanics: Theory and Experiment. IOP Publishing Ltd is not responsible for any errors or omissions in this version of the manuscript or any version derived from it. The Version of Record is available online at 10.1088/1742-5468/aae851 Copyright © and Moral Rights are retained by the author(s) and/ or other copyright owners. A copy can be downloaded for personal non-commercial research or study, without prior permission or charge. This item cannot be reproduced or quoted extensively from without first obtaining permission in writing from the copyright holder(s). The content must not be changed in any way or sold commercially in any format or medium without the formal permission of the copyright holders.

PY - 2018/11/15

Y1 - 2018/11/15

N2 - 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 ϵ0 and mean value y0 relaxes into a stationary PDF with a width ϵ, 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 ϵ. When ϵ = ϵ0, the movement of a PDF leads to a robust power-law scaling with the fractal dimension DF = 2. In general when ϵ ≠ ϵ0, DF = 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. y0 ≫ ϵ ≫ ϵ0). 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.

AB - 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 ϵ0 and mean value y0 relaxes into a stationary PDF with a width ϵ, 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 ϵ. When ϵ = ϵ0, the movement of a PDF leads to a robust power-law scaling with the fractal dimension DF = 2. In general when ϵ ≠ ϵ0, DF = 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. y0 ≫ ϵ ≫ ϵ0). 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.

KW - stochastic processes

UR - http://www.scopus.com/inward/record.url?scp=85057605389&partnerID=8YFLogxK

U2 - 10.1088/1742-5468/aae851

DO - 10.1088/1742-5468/aae851

M3 - Article

VL - 2018

JO - Journal of Statistical Mechanics: Theory and Experiment

JF - Journal of Statistical Mechanics: Theory and Experiment

SN - 1742-5468

IS - 11

M1 - 113401

ER -