Shape restriction of the multi-dimensional Bernstein prior for density functions

Kurtosis is a commonly used descriptive statistics. Kurtosis “Coefficient of excess” is critically reviewed in different aspects and is called as, measuring the fatness of the tails of the density functions, concentration towards the central value, scattering away from the target point or degree of peakedness of probability distribution. Kurtosis is referred to the shape of the distribution but many distributions having same kurtosis value may have different shapes while Kurtosis may exist when peak of a distribution is not in existence. Through extensive study of kurtosis on several distributions, Wu (2002) introduced a new measure called “W-Peakedness” that offers a fine capture of distribution shape to provide an intuitive measure of peakedness of the distribution which is inversely proportional to the standard deviation of the distribution. In this paper the work is extended for different others continuous probability distributions. Empirical results through simulation illustrate the proposed method to evaluate kurtosis by W-peakedness

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

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

Interactive design of probability density functions for shape grammars

ACM Transactions on Graphics ◽

10.1145/2816795.2818069 ◽

2015 ◽

Vol 34 (6) ◽

pp. 1-13 ◽

Cited By ~ 9

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

Download Full-text

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

Geosciences ◽

10.3390/geosciences11080322 ◽

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.

Download Full-text