The Poisson-Voronoi tessellation: relationships for edges

In a unified approach, this paper presents distributional properties of a Voronoi tessellation generated by a homogeneous Poisson point process in the Euclidean space of arbitrary dimension. Probability density functions and moments are given for characteristics of the ‘typical’ edge in lower-dimensional section hyperplanes (edge lengths, adjacent angles). We investigate relationships between edges and their neighbours, called Poisson points or centres; namely angular distributions, distances, and positions of neighbours relative to the edge. The approach is analytical, and the results are given partly explicitly and partly as integral expressions, which are suitable for the numerical calculations also presented.

Download Full-text

Modelling with star-shaped distributions

Dependence Modeling ◽

10.1515/demo-2020-0003 ◽

2020 ◽

Vol 8 (1) ◽

pp. 45-69

Author(s):

Eckhard Liebscher ◽

Wolf-Dieter Richter

Keyword(s):

Probabilistic Models ◽

Arbitrary Dimension ◽

Probability Density Functions ◽

Density Functions ◽

Wide Range ◽

Gaussian Density ◽

Spherical Distributions ◽

Data Points ◽

Non Gaussian ◽

General Method

AbstractWe prove and describe in great detail a general method for constructing a wide range of multivariate probability density functions. We introduce probabilistic models for a large variety of clouds of multivariate data points. In the present paper, the focus is on star-shaped distributions of an arbitrary dimension, where in case of spherical distributions dependence is modeled by a non-Gaussian density generating function.

Download Full-text

Contact and chord length distributions of the Poisson Voronoi tessellation

Journal of Applied Probability ◽

10.2307/3214584 ◽

1992 ◽

Vol 29 (2) ◽

pp. 467-471 ◽

Cited By ~ 22

Author(s):

Lutz Muche ◽

Dietrich Stoyan

Keyword(s):

Poisson Process ◽

Probability Density ◽

Voronoi Tessellation ◽

Probability Density Functions ◽

Distribution Functions ◽

Chord Length ◽

Density Functions ◽

Double Integral ◽

Contact Distribution ◽

Integral Formulae

This paper presents the form of some characteristics of the Voronoi tessellation which is generated by a stationary Poisson process in . Expressions are given for the spherical and linear contact distribution functions. These formulae lead to numerically tractable double-integral formulae for chord length probability density functions.

Download Full-text

On the Generalized Power Transformation of Left Truncated Normal Distribution

African Journal of Mathematics and Statistics Studies ◽

10.52589/ajmss-4cew6pbo ◽

2021 ◽

Vol 4 (2) ◽

pp. 101-116

Author(s):

Okoli C.O. ◽

Nwosu D.F. ◽

Osuji G.A. ◽

Nsiegbe N.A.

Keyword(s):

Normal Distribution ◽

Random Variable ◽

Simulation Method ◽

Probability Density Functions ◽

Density Functions ◽

Unified Approach ◽

Truncated Normal Distribution ◽

Optimal Value ◽

Truncated Normal ◽

Transformation Problems

In this study, we considered various transformation problems for a left-truncated normal distribution recently announced by several researchers and then possibly seek to establish a unified approach to such transformation problems for certain type of random variable and their associated probability density functions in the generalized setting. The results presented in this research, actually unify, improve and as well trivialized the results recently announced by these researchers in the literature, particularly for a random variable that follows a left-truncated normal distribution. Furthermore, we employed the concept of approximation theory to establish the existence of the optimal value y_max in the interval denoted by (σ_a,σ_b) ((σ_p,σ_q)) corresponding to the so-called interval of normality estimated by these authors in the literature using the Monte carol simulation method.

Download Full-text

Contact and chord length distributions of the Poisson Voronoi tessellation

Journal of Applied Probability ◽

10.1017/s0021900200043229 ◽

1992 ◽

Vol 29 (02) ◽

pp. 467-471 ◽

Cited By ~ 8

Author(s):

Lutz Muche ◽

Dietrich Stoyan

Keyword(s):

Poisson Process ◽

Probability Density ◽

Voronoi Tessellation ◽

Probability Density Functions ◽

Distribution Functions ◽

Chord Length ◽

Density Functions ◽

Double Integral ◽

Contact Distribution ◽

Integral Formulae

This paper presents the form of some characteristics of the Voronoi tessellation which is generated by a stationary Poisson process in . Expressions are given for the spherical and linear contact distribution functions. These formulae lead to numerically tractable double-integral formulae for chord length probability density functions.

Download Full-text

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

Download Full-text

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

Download Full-text

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.

Download Full-text

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.

Download Full-text