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.

Distances in Gaussian point sets

1985 ◽  
Vol 97 (3) ◽  
pp. 515-524 ◽  
Author(s):  
Peter Clifford ◽  
N. J. B. Green
Keyword(s):  
Probability Density ◽  
Generating Functions ◽  
Joint Distribution ◽  
Probability Density Functions ◽  
Density Functions ◽  
Point Sets ◽  
Point Configuration ◽  
Moment Generating Functions ◽  
Gaussian Point ◽  
Normally Distributed

AbstractThe joint distribution of the n(n− l)/2 distances between n normally distributed points in d dimensions is studied. Moment generating functions and probability density functions are obtained. It is shown that when n = d the squared distances are jointly exponentially distributed subject only to the constraint that a valid n point configuration is prescribed. In the case n = d = 3 the distributions of the ordered distance are obtained explicitly.

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 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.

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

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.

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

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.

scholarly journals Interactive design of probability density functions for shape grammars

2015 ◽  
Vol 34 (6) ◽  
pp. 1-13 ◽  
Author(s):  
Minh Dang ◽  
Stefan Lienhard ◽  
Duygu Ceylan ◽  
Boris Neubert ◽  
Peter Wonka ◽  
...  
Keyword(s):  
Probability Density ◽  
Probability Density Functions ◽  
Interactive Design ◽  
Density Functions ◽  
Shape Grammars
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