Statistical Distribution Function Research Articles

Beautiful Gini

AbstractYou may very well be familiar with the Gini Coefficient, also known as the Gini index: a quantitative gauge with which socioeconomic inequality is measured, e.g. income disparity and wealth disparity. However, you may not know that the Gini Coefficient is an exquisite mathematical object. Enter this review paper—whose aim is to showcase (some of) the mathematical beauty and riches of the Gini Coefficient. The paper does so, in a completely self-contained manner, by illuminating the Gini Coefficient from various perspectives: Euclidean geometry vs. grid geometry; maxima and minima of random variables; statistical distribution functions; the residual lifetime and the total lifetime of renewal processes; increasing and decreasing failure rates; socioeconomic divergence from perfect equality; and weighted differences of statistical distribution functions. Together, these different perspectives offer a deep and comprehensive understanding of the Gini Coefficient. In turn, a profound understanding of the Gini Coefficient may lead to novel ‘Gini applications’ in science and engineering—such as recently established in the multidisciplinary field of restart research.

Моделирование оценки риска пожара на резервуаре хранения нефтепродуктов

The purpose of the study is to assess the consequences of emergency situations in real danger, it is necessary to carry out studies to determine the degree of risk of accidents in the form of fires and explosions on technological equipment for processing and storage of petroleum products. When operating oil tanks, it is recommended to use the «fault tree» method to predict failures or failures’. This method is a logical-probabilistic model of the causal relationship of a tank fire with defects in its elements and other events. This will allow you to use the modeling method to determine the probability of an event using an analytical or statistical distribution function of a random variable representing a real event. Using the «fault tree», you can calculate the probability of a «failure» event and uptime. In the article, a reliability model is constructed, which is a logical-probabilistic model of cause-and-effect relationships of a tank fire with malfunctions of its elements and other events.

Tuning up the Kolmogorov–Smirnov test for testing Benford’s law

. For moderately large and large numbers of data points (n≥100), the Kolmogorov-Smirnov test is too conservative for testing Benford’s law. Moreover, the asymptotic cumulative distribution function of the Kolmogorov statistic shows unacceptable large deviations, up to about 35%, from the ones obtained in Monte Carlo simulations. Such deviations can be reduced to a level below 0.5% if an appropriate linear transformation of the argument of the Kolmogorov cumulative function is performed.

Assessment method for determining rock brittleness based on statistical damage constitutive relations

The brittleness index of rocks is able to provide crucial guidance to the operation of the drilling and hydraulic fracturing. The definition and evaluation method of the rock brittleness has not yet been unified and standardized due to their diversity. We therefore develop an evaluation method of the rock brittleness based on statistical damage relation. A set of uniaxial compression tests on different rock samples are conducted, and corresponding experimental data is collected and analyzed. Then we establish piecewise damage constitutive models based on different combinations of statistical distribution functions, including power function distribution, Weibull distribution, lognormal distribution and logistic distribution. A new evaluation method of rock brittleness based on energy method and the piecewise statistical damage constitutive model is proposed, and the evaluation results show that the increase of damage variable of peak strain will undermine rock brittleness, and the mineral composition contents have influence on the brittleness of rocks. Comparison work between this proposed method and previous brittleness criteria shows that the brittleness index B43 exhibits enhanced stability and consistency in rock brittleness. This study presents a novel method to studying rock brittleness, enhancing current evaluation methods and deepening our understanding of the rock index. Regarding practical application, some field operations involved with the process of drilling or fracturing lead the continuous damage evolution of rocks under loading conditions. The brittleness index from the proposed evaluation method is comparatively reliable and practical for on-site drilling and fracturing.

The dissipative Generalized Hydrodynamic equations and their numerical solution

“Generalized Hydrodynamics” (GHD) stands for a model that describes one-dimensional integrable systems in quantum physics, such as ultra-cold atoms or spin chains. Mathematically, GHD corresponds to nonlinear equations of kinetic type, where the main unknown, a statistical distribution function f(t,z,θ), lives in a phase space which is constituted by a one-dimensional position variable z, and a one-dimensional “kinetic” variable θ, actually a wave-vector, called “rapidity”. Two key features of GHD equations are first a non-local and nonlinear coupling in the advection term, and second an infinite set of conserved quantities, which prevent the system from thermalizing. To go beyond this, we consider the dissipative GHD equations, which are obtained by supplementing the right-hand side of the GHD equations with a non-local and nonlinear diffusion operator or a Boltzmann-type collision integral. In this paper, we deal with new high-order numerical methods to efficiently solve these kinetic equations. In particular, we devise novel backward semi-Lagrangian methods for solving the advective part (the so-called Vlasov equation) by using a high-order time-Taylor series expansion for the advection fields, whose successive time derivatives are obtained by a recursive procedure. This high-order temporal approximation of the advection fields is used to design new implicit/explicit Runge–Kutta semi-Lagrangian methods, which are compared to Adams–Moulton semi-Lagrangian schemes. For solving the source terms, constituted by the diffusion and collision operators, we use and compare different numerical methods of the literature.

Study on the shear strength and damage constitutive model of the contact surface between PVA fiber-enhanced cement mortar and concrete

In this study, a polyvinyl alcohol (PVA) fiber-enhanced cement mortar (PFCM) with high workability, early strength, and high strength was developed. The influences of water content (w), strength, and surface roughness (Rd) of concrete on the shear strength of its contact surface with PFCM were experimentally investigated, followed by the establishment of damage constitutive model. The PFCM was found to be suitable for spraying and achieved a compressive strength of 102.8 MPa and a flexural strength of 19.6 MPa at 28 days. At 3–7 days, its compressive and flexural strengths reached about 79–80% and 83–90% of the 28-day strength, respectively. The shear load and fracture energy release rate of the PFCM and Concrete contact surfaces is between 6.13 kN and 12.09 kN and 1.13 J/m2 to 2.20 J/m2, respectively, both increasing with the increasing w, strength and Rd of concrete. The increased w of concrete from 0.0% to 5.5% could reduce the porosity and capillary pore content of PFCM by about 43% and 50.4%, respectively, and the increased Rd could enhance the interlocking effect between PFCM and concrete. The damage function proposed based on the Najar energy theory could reflect the evolution characteristics of shearing damage of PFCM and Concrete contact surface, and the damage constitutive model proposed by combining the Weibull statistical distribution function and Guo Zhenhai model could reflect the shear mechanical behavior of the PFCM and Concrete contact surface.

Eulerian–Eulerian RSTM-PDF Modeling of Turbulent Particulate Flow

A novel 3D computational model was developed for the turbulent particulate two-phase flow simulation in the rectangular channel. The model is based on the Eulerian approach applied to 3D Reynolds-averaged Navier–Stokes modeling and statistical Probability Distribution Function method. The uniqueness of the method lies in the direct calculation of normal and transverse components of the Reynolds stresses for both gas and particles. Two cases were examined: a conventional channel flow and grid-generated turbulence flow. The obtained numerical results have been verified and validated by the experimental data, received from the turbulent particle dispersion test. The computed values of the particles’ turbulent dispersion and the maximum value of the particulate concentration distribution show good agreement with the experimental results. The examples are ranged from coal and other bulk material pneumatic transport, vertical fluidized beds, coal gasifiers, and chemical reactors.

Approximation for the distribution function of non-central complex Wilks statistic

In this article, we use the saddlepoint approximation method to compute the cumulative distribution function of non-central complex Wilks statistic. This approximation does not require conditions on the dimension, degrees of freedom, and non-centrality matrix. We also compare our results with known approximations to show the accuracy of our approximation. For applications, we compute the power functions of the Wilks statistic in the complex multivariate analysis of variance and the probability of detection of the Wilks detector in signal detection in noise model.

Fitting Statistical Distribution Functions of Air Pollutant Concentration in Different Urban Locations in Malaysia

Ground level ozone is an unstable substance in the atmosphere which reacts with nitrogen dioxide (NOx) in the presence of sunlight normally found near ground level. Ground level ozone is known as one of the major air quality issues worldwide. The presence of the NOx volatile organic compounds (VOCs) can influence the concentration levels of ground level ozone. This pollutant can affect crops and human health. The main aim of this research is to find the best fit distribution for urban monitoring stations in Kuala Terengganu, Kota Bharu and Alor Setar. Secondary data from 2011 to 2015 used in this study was obtained from the Department of Environment (DOE). This research used the central fitting probability distribution which is between gamma, Nakagami, lognormal and log logistic distributions. Meanwhile, the method of moments was used to estimate the parameters. The best distribution represented the monitoring station and predicted the return period of the concentration. The results show that the Nakagami distribution represented the Kuala Terengganu station from 2011 to 2015. In Kota Bharu however, the gamma distribution fit better compared to other distributions in 2013. The gamma distribution seems to fit the data from 2011 to 2014 whereas in 2015, the Nakagami distribution fit better than other distributions for the Alor Setar station. In addition, no return period was predicted for concentrations above 0.10 ppm found at the Kuala Terengganu monitoring station.

Cryogenic liquid nitrogen stimulation-induced cracking in gas-enriched shale reservoirs: A numerical investigation

Cryogenic shock-induced brittle cracking has promising application prospects in reservoir stimulation. In this paper, a fully coupled thermo-hydro-mechanical (T-H-M) model combined with the statistical meso-damage theory is proposed to investigate the trans-scale progressive cracking process of shale gas reservoirs stimulated by liquid nitrogen (LN2). In this theoretical framework, shale meso-heterogeneity in both thermo-mechanical and hydraulic properties is characterized by two independent statistical distribution functions; temperature-dependent rock properties and temperature- and pressure-dependent fluid thermodynamic properties are described using some experimental and theoretical formulas. The numerical model is implemented in a finite element context and subsequently validated against existing classical coupling results. Numerical results demonstrate that LN2 fracturing accompanied by the unique cryogenic shock effect leads to a complex cracking process involving multiple-point fracture initiation, asymmetric macrofracture propagation, and primary fracture bifurcation. Sensitivity analysis indicates that approximately equidistant or equilength radial fractures may be generated at a lower stress difference, lower permeability, or higher elastic modulus; both fracture initiation pressure and induced fracture morphology are more sensitive to injected fluid temperature relative to initial reservoir temperature; a larger thermal expansion coefficient is beneficial for inducing more fractures, but not applicable to the thermal conductivity. Preferable candidates for LN2 stimulation to generate multiple fractures in shale reservoirs are recommended. It is also found that compared with H2O and supercritical CO2, fracturing using LN2 can generate fractures with lower initiation pressure and greater complexity. Preliminary results confirm the potential of using LN2 fracturing to improve the stimulation effect of shale reservoirs.

Adequate Mathematical Models of the Cumulative Distribution Function of Order Statistics to Construct Accurate Tolerance Limits and Confidence Intervals of the Shortest Length or Equal Tails

The technique used here emphasizes pivotal quantities and ancillary statistics relevant for obtaining tolerance limits (or confidence intervals) for anticipated outcomes of applied stochastic models under parametric uncertainty and is applicable whenever the statistical problem is invariant under a group of transformations that acts transitively on the parameter space. It does not require the construction of any tables and is applicable whether the experimental data are complete or Type II censored. The exact tolerance limits on order statistics associated with sampling from underlying distributions can be found easily and quickly making tables, simulation, Monte-Carlo estimated percentiles, special computer programs, and approximation unnecessary. The proposed technique is based on a probability transformation and pivotal quantity averaging. It is conceptually simple and easy to use. The discussion is restricted to one-sided tolerance limits. Finally, we give practical numerical examples, where the proposed analytical methodology is illustrated in terms of the exponential distribution. Applications to other log-location-scale distributions could follow directly.

The reverse flow and amplification of heat in quantum-dot systems

For classical systems, heat is impossible to be transferred from a cool bath to a hot bath without the existence of any external agency. Here, we show that such a phenomenon can occur in quantum systems through a simple device that a quantum dot is embedded between the two reservoirs described by different statistical distribution functions. The reverse flow and amplification of heat in the device can be realized by regulating the energy levels of the quantum dot and the chemical potentials of reservoirs. It is of particular interest to discover that because of the existence of the quantum tunneling effect in the device, this new quantum phenomenon can be observed even when the chemical potential gradient and the temperature gradient are in the same direction. Moreover, the reverse heat flow and amplification coefficient of the quantum device are calculated. The novelty of this device is that the reverse flow of heat does not need externally driving force and this seemingly paradoxical phenomenon does not violate the laws of thermodynamics. It is further expounded that the quantum device has some practical applications. For example, the device can work as a micro/nano cooler. The performance characteristics of the cooler are revealed for different distribution functions. The coefficients of performance of the cooler operated at different conditions are calculated and the optimum selection criteria of key parameters are supplied.

3D Characterization of the Molecular Neighborhood of •OH Radical in High Temperature Water by MD Simulation and Voronoi Polyhedra.

Understanding the properties of the •OH radical in aqueous environments is essential for biochemistry, atmospheric chemistry, and the development of green chemistry technologies. In particular, the technological applications involve knowledge of microsolvation of the •OH radical in high temperature water. In this study, the classical molecular dynamics (MD) simulation and the technique based on the construction of Voronoi polyhedra were used to provide 3D characteristics of the molecular vicinity of the aqueous hydroxyl radical (•OHaq). The statistical distribution functions of metric and topological features of solvation shells represented by the constructed Voronoi polyhedra are reported for several thermodynamic states of water, including the pressurized high-temperature liquid and supercritical fluid. Calculations showed a decisive influence of the water density on the geometrical properties of the •OH solvation shell in the sub- and supercritical region: with the decreasing density, the span and asymmetry of the solvation shell increase. We also showed that the 1D analysis based on the oxygen-oxygen radial distribution functions (RDFs) overestimates the solvation number of •OH and insufficiently reflects the influence of transformations in the hydrogen-bonded network of water on the structure of the solvation shell.

ON THE PROPERTIES OF TOPP-LEONE KUMARASWAMYWEIBUL DISTRIBUTION WITH APPLICATIONS TO BIOMEDICAL DATA

In this study, a new four-parameter lifetime distribution called the Topp Leone KumaraswamyWeibull distribution was derived using the Topp-Leone Kumaraswamy-G family of distributions.  The model includes several important sub-models as special cases such as Topp-Leone exponentiatedWeibull,Topp Leone Weibull, exponentiatedWeibull and Weibull distributions. An expansion for the probability distribution function was carried out which was used to derive some of the mathematical properties. Some mathematical properties of the distribution were presented such as moments, moment generating function, quantile function, survival function, hazard function as well as mean, 1st quartile, median and 3rd quartile. The probability distribution function of order statistics of the Topp-Leone KumaraswamyWeibull distribution was obtained. Estimation of the parameters by maximum likelihood estimation method was discussed.  Two real-life application of the distribution was presented and the analysis showed the fit and flexibility of the new distribution over some lifetime models considered. The analysis showed that the model is effective in fitting biomedical data.

Spatial monitoring of meteorological drought characteristics based on the NASA POWER precipitation product over various regions of Iran.

Drought directly impacts the human economy and society, so a proper understanding of its spatiotemporal characteristics in different time scales and return periods can be effective in its evaluation and risk warning. In this research, the spatiotemporal variation of drought characteristics in 70 investigated stations in Iran during 1981-2020 was examined, evaluated, and compared. The Standardized Precipitation Index (SPI) and Standardized Precipitation Evapotranspiration Index (SPEI) have been used on time scales of 1, 3, 6, 9, 12, and 24months to calculate the meteorological drought. Drought characteristics have been calculated through the run theory method, and the correlation between these characteristics has been checked. Statistical distribution functions have been used to calculate drought characteristics for the 10-, 20-, 50-, and 100-year return periods. Results show that the duration, severity, and peak of the drought in rainy areas increase as the return period increases. The drought features obtained from the SPI and SPEI show that the average value of severity obtained based on the SPI (43.5) is higher than that of the SPEI (40.9) while the average values of the peak are 3.9 and 2.6 for SPI and SPEI, respectively. Extreme drought was identified in 1990 in all regions of Iran. The highest severity in the current study is from 1999 to 2003. At the end of this period, Iran faced wet years. These results are evident on all time scales. The results obtained in this study can identify drought-prone regions and the beneficial use of water resources in the region.

Effect of uncertainties on block volume estimation

The combination of the aleatory nature of the rock mass structure and the epistemic errors related to the survey methods make rock mass characterization a challenge despite the remarkable evolution of the survey tools and the research on the subject. In particular, significant uncertainties affect block volume estimation: the need for simplification connected to the engineering approach to rockfall problems, for instance, risks to mask the ripple effect of uncertainties on the reliability of the results. Even considering a simplified shape of the block created by three sets of discontinuities (i.e., a prism), the uncertainties on the geometrical characteristics of the discontinuities (orientation, spacing, and persistence) greatly influence the resulting volume distribution. It is a fact that a single value of the volume cannot be representative of the rock mass: the In Situ Block Size Distribution (IBSD) should be built to describe the variability of block volumes. Many statistical distribution functions can be used for fitting spacing data (i.e., gamma, negative exponential, log-normal, Weibull). The choice of the function must follow a rigorous evaluation of the goodness of fit. This research aims to assess the influence of the uncertainties related to the discontinuities sets, with particular reference to spacing samples, on block volume estimation. Through numerical examples and a case study, this research shows that a reduction of uncertainty can be reached by rigorous statistical processing of the data.

Моделирование гидрографа максимального стока как часть задачи трансформации паводка через гидроузел

Modeling ofthe maximum flow hydrographs during spring flood is interpretedas part of a more general task as extreme floods transit throw spillway structures accounting the cutting of maxima in flood control reservoirs. A general statement of the problem related to solving the problem of floods and related economic and environmental damages requiring carrying out complex compensatory and protective water management and hydraulic engineering measures is formulated. The solution of the general problem is not considered, but the authors’ link to the relevant publications is given. The presence of important infrastructure facilities, industrial enterprises and agricultural land can significantly reduce the class of structures. The method of modeling a maximum hydrograph using the Pearson type I differential distribution function for the cases of single-peak and two-peak hydrographs is proposed. Taking into account previous studies of research, the coefficients of completeness of hydrograph form and asymmetry are accepted as criteria for the correspondence of full-scale and simulated hydrographs. A theoretical justification is given and formulas are derived for the functions of maximum flow and flood volumes depending on time. The modeling algorithm is implemented in an Excel environment using built-in statistical beta distribution functions and an optimization procedure “solver”. Excel program is presented for a set of average parameter values, outlines of hydrographs are constructed depending on the shape coefficient, as well as nomograms of the relationship of the asymmetry coefficient with the shape coefficient depending on the ratio of the maximum flow rate and the volume of the flood. Methodology for a two-peak hydrograph is also considered, which is illustrated by the example of the Sursky hydroelectric complex on the Sura River.

Design Under Uncertainties of the Thermal Ablation Treatment of Skin Cancer

Abstract This computational work deals with the optimal design of the thermal ablation treatment of skin cancer, by considering uncertainties in the model parameters. The tumor and other tissues were heated by a laser. Nanoparticles were used to improve the effects of the heating procedure and to promote thermal damage localized in the region containing the tumor. Treatment protocols examined in this work involved one single heating session with different prespecified durations, where the design variables were considered as the volume fraction of nanoparticles in the epidermis and tumor, as well as the time variation of the incident laser fluence rate. The optimal design problems were solved with the Markov Chain Monte Carlo method, by applying a modified version of the Metropolis-Hastings algorithm with sampling by blocks of parameters. The two parameter blocks were given by the properties of the tissues and by the design variables. The prior for the volume fraction of nanoparticles was given by a truncated Gaussian distribution, while a noninformative Gaussian Markov random field prior was used for the time variation of the laser fluence rate. The posterior distributions of the design variables were estimated by taking into account uncertainties in the model parameters and the desired statistical distribution of the thermal damage in the region of interest. The stochastic simulations resulted in optimal thermal damages with small uncertainties, which closely followed their desired statistical distribution functions.

Kinetic consistent MHD algorithm for incompressible conductive fluids

The first ideas concerning the principle of operation of non-mechanical magneto-hydrodynamic pumping systems were originally created at the end of the 1960s during studies of efficient heat exchange systems in nuclear energy sources using liquid metals as coolant. Currently, these ideas are of increasing interest in the creation of many critical technologies that require precise flow control in distributed energy transfer systems, in particular for a new generation of environmentally friendly efficient nuclear power sources.In this regard, computational fluid dynamic methods are an effective instrument for studying complex energy transfer processes, the possibilities of which have been significantly expanded with the creation of modern high-performance computing systems and modern computing technologies.The purpose of this article is to describe an advanced method of the mathematical modeling of magneto-hydrodynamic processes based on kinetic and kinetically consistent models. The novelty of this approach lies in more advanced physical models derived from the Boltzmann kinetic equation with a complex-valued statistical distribution function that includes electromagnetic interactions. The possibility of applying this description to incompressible viscous media (liquid metals) in complex magneto-hydrodynamic systems is shown.The system of equations of the proposed method, a numerical algorithm and the results of the mathematical modeling of a test system for energy transfer using a magneto-hydrodynamic pump and an exchange cavity are presented.

Multiplicity distribution of backward shower particle in total disintegrated events revisited with Weibull distribution

The well-known statistical distribution function, Weibull distribution function, has been employed to study the multiplicity distribution and multiplicity moments for the backward shower particles produced in case of total disintegrated events of [Formula: see text]O–AgBr, [Formula: see text]Ne–AgBr and [Formula: see text]Si–AgBr interactions at (4.1–4.5) AGeV/c. Multiplicity fluctuation of backward shower particles in case of total disintegrated events by the method of scaled variance and the forward–backward asymmetry parameter had also been investigated.