Contact and chord length distributions of the Poisson Voronoi tessellation

1992 ◽  
Vol 29 (2) ◽  
pp. 467-471 ◽  
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.

Contact and chord length distributions of the Poisson Voronoi tessellation

1992 ◽  
Vol 29 (02) ◽  
pp. 467-471 ◽  
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.

scholarly journals Contact and Chord Length Distribution Functions of the Poisson-Voronoi Tessellation in High Dimensions

2010 ◽  
Vol 42 (1) ◽  
pp. 48-68 ◽  
Author(s):  
L. Muche
Keyword(s):  
Voronoi Tessellation ◽  
Length Distribution ◽  
Distribution Functions ◽  
Chord Length ◽  
Dimensional Euclidean Space ◽  
Spherical Contact ◽  
Chord Length Distribution ◽  
Special Cases ◽  
Contact Distribution ◽  
Contact Distribution Function

In this paper we present formulae for contact distributions of a Voronoi tessellation generated by a homogeneous Poisson point process in the d-dimensional Euclidean space. Expressions are given for the probability density functions and moments of the linear and spherical contact distributions. They are double and simple integral formulae, which are tractable for numerical evaluation and for large d. The special cases d = 2 and d = 3 are investigated in detail, while, for d = 3, the moments of the spherical contact distribution function are expressed by standard functions. Also, the closely related chord length distribution functions are considered.

Contact and chord length distribution of a stationary Voronoi tessellation

1998 ◽  
Vol 30 (03) ◽  
pp. 603-618 ◽  
Author(s):  
Lothar Heinrich
Keyword(s):  
Point Process ◽  
Voronoi Tessellation ◽  
Pair Correlation ◽  
Length Distribution ◽  
Distribution Functions ◽  
Chord Length ◽  
Closed Form Expression ◽  
Voronoi Tessellations ◽  
Chord Length Distribution ◽  
Contact Distribution

We give formulae for different types of contact distribution functions for stationary (not necessarily Poisson) Voronoi tessellations in ℝ d in terms of the Palm void probabilities of the generating point process. Moreover, using the well-known relationship between the linear contact distribution and the chord length distribution we derive a closed form expression for the mean chord length in terms of the two-point Palm distribution and the pair correlation function of the generating point process. The results obtained are specified for Voronoi tessellations generated by Poisson cluster and Gibbsian processes, respectively.

Contact and chord length distribution of a stationary Voronoi tessellation

1998 ◽  
Vol 30 (3) ◽  
pp. 603-618 ◽  
Author(s):  
Lothar Heinrich
Keyword(s):  
Point Process ◽  
Voronoi Tessellation ◽  
Pair Correlation ◽  
Length Distribution ◽  
Distribution Functions ◽  
Chord Length ◽  
Closed Form Expression ◽  
Voronoi Tessellations ◽  
Chord Length Distribution ◽  
Contact Distribution

We give formulae for different types of contact distribution functions for stationary (not necessarily Poisson) Voronoi tessellations in ℝd in terms of the Palm void probabilities of the generating point process. Moreover, using the well-known relationship between the linear contact distribution and the chord length distribution we derive a closed form expression for the mean chord length in terms of the two-point Palm distribution and the pair correlation function of the generating point process. The results obtained are specified for Voronoi tessellations generated by Poisson cluster and Gibbsian processes, respectively.

scholarly journals Comments on Copula Functions and Their Relationship to Probability Density Functions

2020 ◽  
pp. 1115-1122
Author(s):  
Ahmed AL-Adilee ◽  
Ola Hassan
Keyword(s):  
Probability Density ◽  
Statistical Inference ◽  
Joint Distribution ◽  
Probability Density Functions ◽  
Distribution Functions ◽  
Density Functions ◽  
Copula Functions ◽  
Statistical Inferences ◽  
Dependence Structures

Copulas are very efficient functions in the field of statistics and specially in statistical inference. They are fundamental tools in the study of dependence structures and deriving their properties. These reasons motivated us to examine and show  various types of copula functions and their families. Also, we separately explain each method that is used to construct each copula in detail with different examples. There are various outcomes that show the copulas and their densities with respect to the joint distribution functions. The aim is to make copulas available to new researchers and readers who are interested in the modern phenomenon of statistical inferences.

scholarly journals Contact and Chord Length Distribution Functions of the Poisson-Voronoi Tessellation in High Dimensions

2010 ◽  
Vol 42 (01) ◽  
pp. 48-68 ◽  
Author(s):  
L. Muche
Keyword(s):  
Voronoi Tessellation ◽  
Length Distribution ◽  
Distribution Functions ◽  
Chord Length ◽  
Dimensional Euclidean Space ◽  
Spherical Contact ◽  
Chord Length Distribution ◽  
Special Cases ◽  
Contact Distribution ◽  
Contact Distribution Function

In this paper we present formulae for contact distributions of a Voronoi tessellation generated by a homogeneous Poisson point process in the d-dimensional Euclidean space. Expressions are given for the probability density functions and moments of the linear and spherical contact distributions. They are double and simple integral formulae, which are tractable for numerical evaluation and for large d. The special cases d = 2 and d = 3 are investigated in detail, while, for d = 3, the moments of the spherical contact distribution function are expressed by standard functions. Also, the closely related chord length distribution functions are considered.

scholarly journals An application of theoretical probability distributions, to the study of PM10 and PM2.5 time series in Athens, Greece

2013 ◽  
Vol 8 (3) ◽  
pp. 241-251
Keyword(s):  
Probability Density ◽  
Concentration Distribution ◽  
Goodness Of Fit ◽  
Probability Density Functions ◽  
Distribution Functions ◽  
Density Functions ◽  
Inverse Gaussian ◽  
High Concentration ◽  
Pm10 And Pm2.5 ◽  
Pearson Type

Probability density functions (pdf) have been used in the analysis of the distribution of pollutant data, for examining the frequency of high concentration events. There have been very few studies on the concentration distribution of PM in urban areas. The distribution of PM concentrations has an impact on human health effects and the setting of PM regulations. Eight probability distribution functions were fitted to measured concentrations of PM10 and PM2.5 in order to determine the shape of the concentration distribution. The “goodness-of-fit” of the probability density functions, to the data, was evaluated, using various statistical indices (including Chi-square and Kolmogorov-Smirnov tests). The evaluation was conducted for two separate years and the results indicated that the Pearson type VI pdf provided a better fit to the measured data. Other functions exhibiting high accuracy of fit were the inverse Gaussian, the lognormal and Pearson type V. The possibility to use probability density functions for predicting the daily high concentration percentiles to less than everyday sampling scenarios is also shown. The differences in the distribution of concentrations under these scenarios are important for regulatory compliance. When trying to detect the high concentrations there is significant possibility of missing the events and thus, underestimating the number of exceedances occurred. Significant deviations from actual daily measurements of PM10 and PM2.5 concentration percentiles were observed, when infrequent sampling scenarios were examined. The differences were higher for the 1-in- 6 sampling schedules and reached 2.8% for mean PM10 and 8% for PM2.5 while for the maximum concentrations the respective differences were 21.3% and 31.9%. Differences between the frequency distributions of everyday and non-everyday sampled concentrations were observed, while lognormal and inverse Gaussian functions provided a better approximation of the upper percentiles. Fitting infrequent data on continuous probability functions for the improvement of the approximation to the real statistical values provided good results regarding the 90th percentile, which corresponds to the E.U. provision of 35 annual exceedances of 24-h limit PM10 values. In the case of the extreme 98th and 99th percentiles, the method provided satisfactory results for both the PM10 infrequent sampling scenarios.

scholarly journals Investigation of Fuel Atomization with Density Functions

2017 ◽  
Vol 62 (1) ◽  
pp. 33 ◽  
Author(s):  
András Urbán ◽  
Viktor Józsa
Keyword(s):  
Probability Density ◽  
Size Distribution ◽  
Probability Density Functions ◽  
Distribution Functions ◽  
Density Functions ◽  
Fuel Atomization ◽  
Measurement Results ◽  
Combustion Technology ◽  
Phase Doppler Anemometer ◽  
Semi Empirical

Atomization involves mass, energy, and impulse transfer, in such a complex way that the overall process can only be described by empirical and semi-empirical correlations to date. The phenomenon of atomization is used in numerous applications, e.g., in combustion technology and metallurgy. However, many formulae are available in the literature to derive mean diameters of the spray, size distribution functions are barely discussed. Based on the measurement results performed earlier by a Phase Doppler Anemometer, twenty probability density functions were evaluated and seven are discussed in detail over the course of the present paper. The atomization pressure was varied, and characteristic regimes of the spray were measured. Interestingly, the analysis showed that not only the three most commonly used probability density functions (Nukiyama-Tanasawa, Rosin-Rammler, and Gamma) are eligible for describing the size distribution of the spray.

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