Time-dependent probability density functions and information geometry in stochastic logistic and Gompertz models

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.

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Investigating Information Geometry in Classical and Quantum Systems through Information Length

Entropy ◽

10.3390/e20080574 ◽

2018 ◽

Vol 20 (8) ◽

pp. 574 ◽

Cited By ~ 17

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.

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Mie scattering measurements of scalar probability density functions in compressible mixing layers

10.2514/6.1991-1686 ◽

1991 ◽

Cited By ~ 3

Author(s):

N. MESSERSMITH ◽

J. DUTTON ◽

H. KRIER

Keyword(s):

Probability Density ◽

Mie Scattering ◽

Probability Density Functions ◽

Mixing Layers ◽

Density Functions ◽

Scattering Measurements

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Summary Statistics of Implied Probability Density Functions and their Properties

SSRN Electronic Journal ◽

10.2139/ssrn.314392 ◽

2002 ◽

Cited By ~ 1

Author(s):

Damien P.G. Lynch ◽

Nikolaos Panigirtzoglou

Keyword(s):

Probability Density ◽

Probability Density Functions ◽

Density Functions ◽

Summary Statistics

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Modeling Diameter Distributions with Six Probability Density Functions in Pinus halepensis Mill. Plantations Using Low-Density Airborne Laser Scanning Data in Aragón (Northeast Spain)

Remote Sensing ◽

10.3390/rs13122307 ◽

2021 ◽

Vol 13 (12) ◽

pp. 2307

Author(s):

J. Javier Gorgoso-Varela ◽

Rafael Alonso Ponce ◽

Francisco Rodríguez-Puerta

Keyword(s):

Probability Density ◽

Linear Models ◽

Pinus Halepensis ◽

Beta Function ◽

Probability Density Functions ◽

Lidar Data ◽

Density Functions ◽

Minimum Diameter ◽

Location Parameters ◽

Diameter Distributions

The diameter distributions of trees in 50 temporary sample plots (TSPs) established in Pinus halepensis Mill. stands were recovered from LiDAR metrics by using six probability density functions (PDFs): the Weibull (2P and 3P), Johnson’s SB, beta, generalized beta and gamma-2P functions. The parameters were recovered from the first and the second moments of the distributions (mean and variance, respectively) by using parameter recovery models (PRM). Linear models were used to predict both moments from LiDAR data. In recovering the functions, the location parameters of the distributions were predetermined as the minimum diameter inventoried, and scale parameters were established as the maximum diameters predicted from LiDAR metrics. The Kolmogorov–Smirnov (KS) statistic (Dn), number of acceptances by the KS test, the Cramér von Misses (W2) statistic, bias and mean square error (MSE) were used to evaluate the goodness of fits. The fits for the six recovered functions were compared with the fits to all measured data from 58 TSPs (LiDAR metrics could only be extracted from 50 of the plots). In the fitting phase, the location parameters were fixed at a suitable value determined according to the forestry literature (0.75·dmin). The linear models used to recover the two moments of the distributions and the maximum diameters determined from LiDAR data were accurate, with R2 values of 0.750, 0.724 and 0.873 for dg, dmed and dmax. Reasonable results were obtained with all six recovered functions. The goodness-of-fit statistics indicated that the beta function was the most accurate, followed by the generalized beta function. The Weibull-3P function provided the poorest fits and the Weibull-2P and Johnson’s SB also yielded poor fits to the data.

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A synergy of the velocity gradients technique and the probability density functions for identifying gravitational collapse in self-absorbing media

Monthly Notices of the Royal Astronomical Society ◽

10.1093/mnras/stab087 ◽

2021 ◽

Vol 502 (2) ◽

pp. 1768-1784

Author(s):

Yue Hu ◽

A Lazarian

Keyword(s):

Magnetic Fields ◽

Probability Density ◽

Column Density ◽

Mass Density ◽

Probability Density Functions ◽

Emission Lines ◽

Density Functions ◽

Star Forming ◽

Velocity Gradients ◽

Self Gravity

ABSTRACT The velocity gradients technique (VGT) and the probability density functions (PDFs) of mass density are tools to study turbulence, magnetic fields, and self-gravity in molecular clouds. However, self-absorption can significantly make the observed intensity different from the column density structures. In this work, we study the effects of self-absorption on the VGT and the intensity PDFs utilizing three synthetic emission lines of CO isotopologues 12CO (1–0), 13CO (1–0), and C18O (1–0). We confirm that the performance of VGT is insensitive to the radiative transfer effect. We numerically show the possibility of constructing 3D magnetic fields tomography through VGT. We find that the intensity PDFs change their shape from the pure lognormal to a distribution that exhibits a power-law tail depending on the optical depth for supersonic turbulence. We conclude the change of CO isotopologues’ intensity PDFs can be independent of self-gravity, which makes the intensity PDFs less reliable in identifying gravitational collapsing regions. We compute the intensity PDFs for a star-forming region NGC 1333 and find the change of intensity PDFs in observation agrees with our numerical results. The synergy of VGT and the column density PDFs confirms that the self-gravitating gas occupies a large volume in NGC 1333.

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