scholarly journals Investigating Information Geometry in Classical and Quantum Systems through Information Length

Entropy ◽  
2018 ◽  
Vol 20 (8) ◽  
pp. 574 ◽  
Author(s):  
Eun-jin Kim
Keyword(s):  
Stochastic Processes ◽  
Information Geometry ◽  
Quantum Systems ◽  
Probability Density Functions ◽  
Time Dependent ◽  
Density Functions ◽  
Mathematical Tool ◽  
Initial Condition ◽  
Disciplinary Boundaries ◽  
Information Change

Stochastic processes are ubiquitous in nature and laboratories, and play a major role across traditional disciplinary boundaries. These stochastic processes are described by different variables and are thus very system-specific. In order to elucidate underlying principles governing different phenomena, it is extremely valuable to utilise a mathematical tool that is not specific to a particular system. We provide such a tool based on information geometry by quantifying the similarity and disparity between Probability Density Functions (PDFs) by a metric such that the distance between two PDFs increases with the disparity between them. Specifically, we invoke the information length L(t) to quantify information change associated with a time-dependent PDF that depends on time. L(t) is uniquely defined as a function of time for a given initial condition. We demonstrate the utility of L(t) in understanding information change and attractor structure in classical and quantum systems.

scholarly journals Some Illustrations of Information Geometry in Biology and Physics

2012 ◽  
pp. 287-315 ◽  
Author(s):  
C. T. J. Dodson
Keyword(s):  
Amino Acid ◽  
State Space ◽  
Information Geometry ◽  
Probability Density Functions ◽  
Density Functions ◽  
Binary Variables ◽  
Information Theoretic ◽  
Precise Meaning ◽  
Rate Processes ◽  
Intuitive Sense

Many real processes have stochastic features which seem to be representable in some intuitive sense as `close to Poisson’, `nearly random’, `nearly uniform’ or with binary variables `nearly independent’. Each of those particular reference states, defined by an equation, is unstable in the formal sense, but it is passed through or hovered about by the observed process. Information geometry gives precise meaning for nearness and neighbourhood in a state space of processes, naturally quantifying proximity of a process to a particular state via an information theoretic metric structure on smoothly parametrized families of probability density functions. We illustrate some aspects of the methodology through case studies: inhomogeneous statistical evolutionary rate processes for epidemics, amino acid spacings along protein chains, constrained disordering of crystals, distinguishing nearby signal distributions and testing pseudorandom number generators.

DYNAMIC PROBABILISTIC FORECASTING WITH UNCERTAINTY

2021 ◽  
Author(s):  
FRED ESPEN BENTH ◽  
GLEDA KUTROLLI ◽  
SILVANA STEFANI
Keyword(s):  
Stochastic Processes ◽  
Model Uncertainty ◽  
Empirical Studies ◽  
Probability Density Functions ◽  
Density Functions ◽  
Option Prices ◽  
Probabilistic Forecasting ◽  
New Class ◽  
Probability Densities ◽  
Equity Indices

In this paper, we introduce a dynamical model for the time evolution of probability density functions incorporating uncertainty in the parameters. The uncertainty follows stochastic processes, thereby defining a new class of stochastic processes with values in the space of probability densities. The purpose is to quantify uncertainty that can be used for probabilistic forecasting. Starting from a set of traded prices of equity indices, we do some empirical studies. We apply our dynamic probabilistic forecasting to option pricing, where our proposed notion of model uncertainty reduces to uncertainty on future volatility. A distribution of option prices follows, reflecting the uncertainty on the distribution of the underlying prices. We associate measures of model uncertainty of prices in the sense of Cont.

Nonholonomic Motion Planning Using Diffusion of Workspace Density Functions

2003 ◽  
Author(s):  
Yu Zhou ◽  
Gregory S. Chirikjian
Keyword(s):  
Planck Equation ◽  
Probability Density Functions ◽  
Time Dependent ◽  
Density Functions ◽  
Value Of Time ◽  
Nonholonomic Mobile Robots ◽  
Planning Algorithm ◽  
The Fourier Transform ◽  
Nonholonomic Motion Planning ◽  
Closed Form Approximation

This paper introduces a trajectory planning algorithm for nonholonomic mobile robots which operate in an environment with obstacles. An important feature of our approach is that the planning domain is the workspace of the mobile robot rather than its configuration space. The basic idea is to imagine the robot being subjected to Brownian motion forcing, and to generate evolving probability density functions (PDF) that describe all attainable positions and orientations of the robot at a given value of time. By planning a path that optimizes the value of this PDF at each instant in time, we generate a feasible trajectory. The PDF of robot pose can be constructed by solving the corresponding Fokker-Planck equation using the Fourier transform for SE(N). A closed-form approximation of the resulting time-dependent PDF is then used to plan a trajectory based on the observation that the evolution of this “workspace density” is a diffusion process. Examples are provided to illustrate the algorithm.

scholarly journals OPTIMAL CONTROL OF PROBABILITY DENSITY FUNCTIONS OF STOCHASTIC PROCESSES

2010 ◽  
Vol 15 (4) ◽  
pp. 393-407 ◽  
Author(s):  
Mario Annunziato ◽  
Alfio Borzì
Keyword(s):  
Optimal Control ◽  
Stochastic Processes ◽  
Probability Density ◽  
Control Strategy ◽  
Time Windows ◽  
Probability Density Functions ◽  
Density Functions ◽  
Optimal Control Strategy ◽  
Control Laws ◽  
Fokker Planck

A Fokker‐Planck framework for the formulation of an optimal control strategy of stochastic processes is presented. Within this strategy, the control objectives are defined based on the probability density functions of the stochastic processes. The optimal control is obtained as the minimizer of the objective under the constraint given by the Fokker‐Planck model. Representative stochastic processes are considered with different control laws and with the purpose of attaining a final target configuration or tracking a desired trajectory. In this latter case, a receding‐horizon algorithm over a sequence of time windows is implemented.

scholarly journals Time-Dependent Probability Density Functions and Attractor Structure in Self-Organised Shear Flows

Entropy ◽  
2018 ◽  
Vol 20 (8) ◽  
pp. 613 ◽  
Author(s):  
Quentin Jacquet ◽  
Eun-jin Kim ◽  
Rainer Hollerbach
Keyword(s):  
Probability Density ◽  
Shear Flows ◽  
Numerical Solutions ◽  
Planck Equation ◽  
Probability Density Functions ◽  
Time Dependent ◽  
Parameter Study ◽  
Density Functions ◽  
Stochastic Forcing ◽  
Mean Values

We report the time-evolution of Probability Density Functions (PDFs) in a toy model of self-organised shear flows, where the formation of shear flows is induced by a finite memory time of a stochastic forcing, manifested by the emergence of a bimodal PDF with the two peaks representing non-zero mean values of a shear flow. Using theoretical analyses of limiting cases, as well as numerical solutions of the full Fokker–Planck equation, we present a thorough parameter study of PDFs for different values of the correlation time and amplitude of stochastic forcing. From time-dependent PDFs, we calculate the information length ( L ), which is the total number of statistically different states that a system passes through in time and utilise it to understand the information geometry associated with the formation of bimodal or unimodal PDFs. We identify the difference between the relaxation and build-up of the shear gradient in view of information change and discuss the total information length ( L ∞ = L ( t → ∞ ) ) which maps out the underlying attractor structures, highlighting a unique property of L ∞ which depends on the trajectory/history of a PDF’s evolution.

scholarly journals A time-dependent probabilistic fatigue analysis method considering stochastic loadings and strength degradation

2018 ◽  
Vol 10 (7) ◽  
pp. 168781401878556 ◽  
Author(s):  
Chunbo Su ◽  
Shui Yu ◽  
Zhonglai Wang ◽  
Zafar Tayyab
Keyword(s):  
Density Estimation ◽  
Failure Probability ◽  
Safety Margin ◽  
Estimation Method ◽  
Kernel Density ◽  
Strength Degradation ◽  
Probability Density Functions ◽  
Time Dependent ◽  
Density Functions ◽  
Dependent Failure

This article proposes two strategies for time-dependent probabilistic fatigue analysis considering stochastic loadings and strength degradation based on the failure transformation and multi-dimensional kernel density estimation method. The time-dependent safety margin function is first established to describe the limit state of the time-dependent failure probability for mechatronics equipment with stochastic loadings and strength degradation. Considering the effective safety margin points and the corresponding number of the load cycles, two strategies for transforming the time-dependent failure probability calculation to the static reliability calculation are then proposed. Multi-dimensional kernel density estimation method is finally employed to build the probability density functions and the reliability is estimated based on the probability density functions. An engineering case of a filtering gear reducer is presented to validate the effectiveness of the proposed methods both in computational efficiency and accuracy.

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