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.

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 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

scholarly journals The Impact of Probability Density Functions Assessment on Model Performance for Slope Stability Analysis

Geosciences ◽  
2021 ◽  
Vol 11 (8) ◽  
pp. 322
Author(s):  
Evelina Volpe ◽  
Luca Ciabatta ◽  
Diana Salciarini ◽  
Stefania Camici ◽  
Elisabetta Cattoni ◽  
...  
Keyword(s):  
Probability Density ◽  
Central Italy ◽  
Model Performance ◽  
Probability Density Functions ◽  
Slope Instability ◽  
Complex Nature ◽  
Density Functions ◽  
Umbria Region ◽  
Input Variables ◽  
The Impact

The development of forecasting models for the evaluation of potential slope instability after rainfall events represents an important issue for the scientific community. This topic has received considerable impetus due to the climate change effect on territories, as several studies demonstrate that an increase in global warming can significantly influence the landslide activity and stability conditions of natural and artificial slopes. A consolidated approach in evaluating rainfall-induced landslide hazard is based on the integration of rainfall forecasts and physically based (PB) predictive models through deterministic laws. However, considering the complex nature of the processes and the high variability of the random quantities involved, probabilistic approaches are recommended in order to obtain reliable predictions. A crucial aspect of the stochastic approach is represented by the definition of appropriate probability density functions (pdfs) to model the uncertainty of the input variables as this may have an important effect on the evaluation of the probability of failure (PoF). The role of the pdf definition on reliability analysis is discussed through a comparison of PoF maps generated using Monte Carlo (MC) simulations performed over a study area located in the Umbria region of central Italy. The study revealed that the use of uniform pdfs for the random input variables, often considered when a detailed geotechnical characterization for the soil is not available, could be inappropriate.

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