Probability Distribution Of Number Research Articles

A Stochastic Statistical Model for U.S. Outbreak-Level Tornado Occurrence Based on the Large-Scale Environment

Abstract Tornado outbreaks—when multiple tornadoes occur within a short period of time—are rare yet impactful events. Here we developed a two-part stochastic tornado outbreak index for the contiguous United States (CONUS). The first component produces a probability map for outbreak tornado occurrence based on spatially resolved values of convective precipitation, storm relative helicity (SRH), and convective available potential energy. The second part of the index provides a probability distribution for the total number of tornadoes given the outbreak tornado probability map. Together these two components allow stochastic simulation of location and number of tornadoes that is consistent with environmental conditions. Storm report data from the Storm Prediction Center for the 1979–2021 period are used to train the model and evaluate its performance. In the first component, the probability of an outbreak-level tornado is most sensitive to SRH changes. In the second component, the total number of CONUS tornadoes depends on the sum and gridpoint maximum of the probability map. Overall, the tornado outbreak index represents the climatology, seasonal cycle, and interannual variability of tornado outbreak activity well, particularly over regions and seasons when tornado outbreaks occur most often. We found that El Niño–Southern Oscillation (ENSO) modulates the tornado outbreak index such that La Niña is associated with enhanced U.S. tornado outbreak activity over the Ohio River Valley and Tennessee River Valley regions during January–March, similar to the behavior seen in storm report data. We also found an upward trend in U.S. tornado outbreak activity during winter and spring for the 1979–2021 period using both observations and the index. Significance Statement Tornado outbreaks are when multiple tornadoes happen in a short time span. Because of the rare, sporadic nature of tornadoes, it can be challenging to use observational tornado reports directly to assess how climate affects tornado and tornado outbreak activity. Here, we developed a statistical model that produces a U.S. map of the likelihood that an outbreak-level tornado would occur based on environmental conditions. In addition, using that likelihood map, the model predicts a range of how many tornadoes could occur in these events. We found that “storm relative helicity” (a proxy for potential rotation in a storm’s updraft) is especially important for predicting outbreak tornado likelihood, and the sum and maximum value of the likelihood map is important for predicting total numbers for an event. Overall, this model can represent the typical behavior and fluctuations in tornado outbreak activity well. Both the tornado outbreak model and observations agree that the state of sea surface temperature in the tropical Pacific (El Niño–Southern Oscillation) is linked to tornado outbreak activity over the Ohio River Valley and Tennessee River Valley in winter through early spring and that there are upward trends in tornado outbreak activity.

Entropy of the Canonical Occupancy (Macro) State in the Quantum Measurement Theory.

The paper analyzes the probability distribution of the occupancy numbers and the entropy of a system at the equilibrium composed by an arbitrary number of non-interacting bosons. The probability distribution is obtained through two approaches: one involves tracing out the environment from a bosonic eigenstate of the combined environment and system of interest (the empirical approach), while the other involves tracing out the environment from the mixed state of the combined environment and system of interest (the Bayesian approach). In the thermodynamic limit, the two coincide and are equal to the multinomial distribution. Furthermore, the paper proposes to identify the physical entropy of the bosonic system with the Shannon entropy of the occupancy numbers, fixing certain contradictions that arise in the classical analysis of thermodynamic entropy. Finally, by leveraging an information-theoretic inequality between the entropy of the multinomial distribution and the entropy of the multivariate hypergeometric distribution, Bayesianism of information theory and empiricism of statistical mechanics are integrated into a common "infomechanical" framework.

The reproduction number and its probability distribution for stochastic viral dynamics.

We consider stochastic models of individual infected cells. The reproduction number, R, is understood as a random variable representing the number of new cells infected by one initial infected cell in an otherwise susceptible (target cell) population. Variability in R results partly from heterogeneity in the viral burst size (the number of viral progeny generated from an infected cell during its lifetime), which depends on the distribution of cellular lifetimes and on the mechanism of virion release. We analyse viral dynamics models with an eclipse phase: the period of time after a cell is infected but before it is capable of releasing virions. The duration of the eclipse, or the subsequent infectious, phase is non-exponential, but composed of stages. We derive the probability distribution of the reproduction number for these viral dynamics models, and show it is a negative binomial distribution in the case of constant viral release from infectious cells, and under the assumption of an excess of target cells. In a deterministic model, the ultimate in-host establishment or extinction of the viral infection depends entirely on whether the mean reproduction number is greater than, or less than, one, respectively. Here, the probability of extinction is determined by the probability distribution of R, not simply its mean value. In particular, we show that in some cases the probability of infection is not an increasing function of the mean reproduction number.

On the development of new cosine-based probabilistic methods with applications to univariate and bivariate analyses of the wind speed energy

So far in the literature, a number of probability distributions have been successfully implemented for analyzing the wind speed and energy data sets. However, there is no published work on modeling and analyzing the wind speed and energy data sets with probability distributions that are introduced using trigonometric functions. In the existing literature, there is also a lack of studies on implementing the bivariate trigonometric-based probability distributions for modeling the wind speed and energy data sets. In this paper, we take up a meaningful effort to cover these interesting research gaps. Thus, we first incorporate a cosine function and introduce a new univariate probability distributional method, namely, a univariate modified cosine-G (UMC-G) family. Using the UMC-G method, a new probability distribution called a univariate modified cosine-Weibull (UMC-Weibull) distribution is studied. We apply the UMC-Weibull distribution for analyzing the wind energy data set taken from the weather station at Sotavento Galicia, Spain. Furthermore, we also introduce a bivariate version of the UMC-G method using the Farlie–Gumble–Morgenstern copula approach. The proposed bivariate distributional method is called a bivariate modified cosine-G (BMC-G) family. A special member of the BMC-G distributions called a bivariate modified cosine-Weibull (BMC-Weibull) distribution is introduced. We apply the BMC-Weibull distribution for analyzing the bivariate data set representing the wind speed and energy taken from the weather station at Sotavento Galicia. Using different statistical tools, we observe that the UMC-Weibull and BMC-Weibull are the best-suited models for analyzing the wind speed and energy data sets.

Net-Baryon Probability Distributions from Lattice Simulations

We use the results of lattice simulations of the net-baryon number density at imaginary baryon chemical potential in Nf=2 QCD to construct the equation of state of dense and hot strong-interacting matter both above the Roberge–Weiss temperature T>TRW and below the critical temperature T<Tc. For these cases, we also evaluate probability distributions of the net-baryon number, as well as the respective cumulants and moments. The consequences of the asymptotic behavior of these probability distributions for the problem of reconstruction of the net-baryon probability distributions from cumulants are discussed.

Experimental study on impact toughness of multi-combination hybrid fiber-reinforced magnesium phosphate cement-based materials

In order to improve the brittleness of magnesium phosphate cement-based materials (MPC) and overcome the limitations of single fiber-reinforced cement-based materials, this study investigated the mechanical properties of hybrid fiber-reinforced magnesium phosphate cement-based composite materials (HFRMPC) by incorporating four types of fibers: hooked-end steel fiber (HSF), micro steel fiber (MSF), polyvinyl alcohol fiber (PVAF), and basalt fiber (BF). Cylindrical specimens were prepared by single and double hybridization of the fibers and subjected to drop hammer impact tests to analyze the effects of different fiber hybridization methods on the impact resistance of these materials. The two-parameter Weibull distribution was used to analyze the test results to predict failure probabilities and to investigate the fiber reinforcement and toughening mechanisms in MPC through microscopic structural analysis. The results showed that the impact resistance of the fiber hybridization group was much better than that of the single fiber group, and the impact resistance increased continuously with increasing fiber content. The fibers' impact resistance enhancement effects were in the following order: MSF > PVAF > BF. The fiber combination methods were ranked as follows from best to worst: HSF + PVAF > HSF + BF > HSF + MSF > PVAF + BF. The 1.5% HSF+0.6% PVAF hybridization had the best impact resistance performance, with an increase in initial cracking and impact energy dissipation at the failure of 7748.6% and 8921.2%, respectively, compared to the M0. The probability distribution of impact number of initial cracking (N1) and failure (N2) of the specimens followed the double-parameter Weibull distribution well. The multiscale strengthening and toughening zone composed of fibers with different geometric dimensions and elastic modulus was crucial to the material's performance, as the fibers acted in progressive briding under external impact loads, dissipating impact energy through the failure morphology of pull-out and breaking, significantly improving the HFRMPC's impact toughness. A comprehensive evaluation of the four indicators of impact resistance and manufacturing costs for various hybrid fiber compositions found that the combination of 1.5% HSF and 0.6% PVAF exhibits the most outstanding overall performance.

The neutron number probability distribution in coupled lumped assemblies

The validity of the gamma distribution in describing the neutron number probability distribution function for both isolated and coupled multiplying assemblies when constrained to reproduce the true mean and variance is investigated in lumped geometry by numerical comparison with kinetic Monte Carlo simulations. The mean and variance are obtained from numerical solution of moment equations constructed from the relevant forward Master equation with assembly coupling coefficients obtained from a view factor method. Numerical results for a two-group, two coupled assemblies model, with static and dynamic reactivity insertion, show that except for subcritical assemblies, the gamma distribution well-approximates the number distribution. Differences in the fast and thermal neutron population shapes are explained in terms of effective source strengths due to downscatter and coupling.

Probabilistic model for Padel games dynamics

This study applies complexity sciences to analyze the game of Padel. Data from 18 professional matches were collected, and the probability distributions of the total number of shots and the probability distribution of rallies’ duration were analyzed. Based on these empirical observations and previous reports, a probabilistic model with two parameters was proposed to describe the game dynamics. One of them controls the probability of making a shot and the other probability of doing it offensively. The model also considers the offensive advantage of the team serving the ball. Using this model, an analytical expression for the probability distribution of the total number of shots was obtained and fit to the data. The results reveal that the complex dynamics of Padel can be effectively approximated as a stochastic process governed by simple probabilistic rules.

Phase diagram detection via Gaussian fitting of number probability distribution

We investigate the number probability density function that characterizes sub-portions of a quantum many-body system with globally conserved number of particles. We put forward a linear fitting protocol capable of mapping out the ground-state phase diagram of the rich one-dimensional extended Bose-Hubbard model: The results are quantitatively comparable with more sophisticated traditional and machine learning techniques. We argue that the studied quantity should be considered among the most informative bipartite properties, being moreover readily accessible in atomic gases experiments.

Reliability and lifetime estimation of bioregenerative life support system based on 370-day closed human experiment of lunar palace 1 and Monte Carlo simulation

It is extremely important to accurately estimate reliability and lifetime of Bioregenerative Life Support System (BLSS) for its proper design, building and operation. However, this problem remains unsolved due to data deficiency of long-period prototype experiments. In this paper, the first ground-based experimental prototype of BLSS in China, Lunar Palace 1 (LP1), was selected as representative to estimate the reliability and lifetime of BLSS, because the number and time of each unit failure of LP1 was accurately recorded during the 370-day closed human experiment of LP1. Based on these time-series data, failure number probability distribution function of each unit and overall LP1 were formulated through parameter estimation and composite probability distribution function derived from units’ series and parallel connection. Then, the influence of each unit failure on overall BLSS reliability and lifetime was determined correspondingly through sensitivity analysis and Monte Carlo simulations. Finally, the numerous pseudo-random numbers obeying the overall failure probability distribution function of LP1 were generated to estimate the reliability and lifetime of BLSS through maximum likelihood estimation and Monte Carlo simulations. The results demonstrate that the failure possibility of five units, water treatment unit, mineral element supply unit, LED light source unit, atmosphere management unit and temperature and humidity control unit have greater impact on reliability and lifetime of BLSS, and the mean lifetime of BLSS was 19112.37 days (about 52.4 years) with 95% confidence interval of [17367.11, 20672.68] days (about [47.58, 56.64] years) under normal operation and maintenance of BLSS. This research can lay a theoretical foundation for design, building and operation of BLSS with high reliability and long lifetime in the future.

Assessment of the residential Finnish wolf population combines DNA captures, citizen observations and mortality data using a Bayesian state-space model

Assessment of the Finnish wolf population relies on multiple sources of information. This paper describes how Bayesian inference is used to pool the information contained in different data sets (point observations, non-invasive genetics, known mortalities) for the estimation of the number of territories occupied by family packs and pairs. The output of the assessment model is a joint probability distribution, which describes current knowledge about the number of wolves within each territory. The joint distribution can be used to derive probability distributions for the total number of wolves in all territories and for the pack status within each territory. Most of the data set comprises of both voluntary-provided point observations and DNA samples provided by volunteers and research personnel. The new method reduces the role of expert judgement in the assessment process, providing increased transparency and repeatability.

Winding number statistics of a parametric chiral unitary random matrix ensemble* *Dedicated to the Memory of Fritz Haake.

The winding number is a concept in complex analysis which has, in the presence of chiral symmetry, a physics interpretation as the topological index belonging to gapped phases of fermions. We study statistical properties of this topological quantity. To this end, we set up a random matrix model for a chiral unitary system with a parametric dependence. We analytically calculate the discrete probability distribution of the winding numbers, as well as the parametric correlations functions of the winding number density. Moreover, we address aspects of universality for the two-point function of the winding number density by identifying a proper unfolding procedure. We conjecture the unfolded two-point function to be universal.

Marginal and joint failure importance for K-terminal network edges under counting process

Importance measures have been applied extensively in various fields such as reliability optimization, failure diagnosis, and risk analysis. Traditional importance measures are insufficient for networks because they only quantify the contribution of each edge's reliability to network reliability. The counting process is a kind of stochastic process that can count the number of edge failures appeared in a time of period, which requires less information than knowing all edge reliabilities. In the context of edge failures subject to a counting process, this paper investigates importance measures for a given binary K-terminal network including n binary edges. This paper develops some formulas to compute the joint failure importance (JFI) and the marginal failure importance (MFI). The MFI quantifies the changes of network failure probability caused by the change of edge state, while the JFI evaluates the interaction effect between two edges regarding network failure probability. Their values, as functions of time t, depend on the probability distribution of the total number of edge failures at time t and the network structure. Additionally, we present a Monte-Carlo algorithm to approximate the values of the MFI and JFI. Finally, a numerical example concerning the road network is provided to demonstrate the computation methods of MFI and JFI, whose edge failures are subject to a saturated nonhomogeneous Poisson process. The numerical results provide further insights for the road network regarding the importance of edges.

Secondary droplet size distribution upon breakup of a sub-milimeter droplet in high speed cross flow

The droplet breakup behaviors under the action of shock wave are frequently observed in liquid-fueled detonation-based engines. In this work, the secondary droplet size distribution after the breakup of a sub-milimeter droplet under high velocity cross flow conditions (the gas flow velocity is 105.64 m/s) was experimentally determined on a horizontal shock tube with the developed digital in-line holography technique. During the droplet breakup process, the spatial (two-dimensional, three-dimensional) and droplet size distributions of two characteristic stages (early and late stages) were majorly concerned based on the reconstructed droplet cloud field. At the early stage, the liquid-mist field appeared as the barrel shape, and the probability distribution of the droplet number in the cross-stream direction presented as “M shape”. While the secondary droplet cloud field appeared as the conical shape, and the probability distribution presented as the parabolic shape at the late stage. Both the probability distribution of the droplet number along the flow direction increased firstly then decreased at the early and late stages with the increasing of distance. In addition, the peak position of the droplet number probability distribution also changed with time. The secondary droplets mainly distributed in the range of 0∼20 μm for the early and late stages, and its number accounted for more than 93%. Meanwhile, the corresponding volume probabilities changed from 12% to 72%, due to the consistent stripping process. In addition, the droplet size distribution could be well described by the log-normal and root-normal distribution curves at both stages. The study of the spatial and droplet size distributions is believed to be of importance for corresponding droplet breakup modeling.

Vortex structure of excitation fields in a supercooled glass-forming liquid and its relationship with relaxations

In the framework of the dynamic facilitation model, a glass former is structured with local cooperative excitations in space and time when approaching the glass transition, which is closely related to dynamic relaxations. We investigate the structure of excitation fields in supercooled ${\mathrm{Cu}}_{50}{\mathrm{Zr}}_{50}$ glass former, and reveal that the excitation fields feature the vortexlike structure beyond the well-known stringlike excitations. The vortex is developed from the stringlike excitations and frequently annihilates via the interactions between vortex and antivortex. The region favored by vortices exhibits percolationlike morphology. We find that the probability distributions of vortex core number are characterized by the discrete multiple mode, which is very analogous to the quantized mode of the slow-\ensuremath{\beta} type relaxation. Furthermore, the relaxation time derived from the formation rate of vortices as well as the corresponding activation energy coincide with the characteristic values of the slow-\ensuremath{\beta} relaxation in metallic glasses. Therefore, it is concluded that the vortex excitation is the origin of the slow-\ensuremath{\beta} relaxation in dynamic space. We further find that the activation energy and relaxation time associated with individual excitations agree well with the Johari-Goldstein \ensuremath{\beta} relaxation. It manifests a more intrinsic structural sign of the Johari-Goldstein \ensuremath{\beta} relaxation than the interpretation based on stringlike excitations.

Early field planning using optimisation and considering uncertainties

During early phases of oil field development, field planners must decide upon the optimal number of wells and optimal field plateau rate, usually by performing sensitivity studies. These design choices are then “frozen” in subsequent development stages. However, they often end up being suboptimal when the field is built and produced and the uncertainty is reduced. In this work, we employ non-linear numerical optimisation, latin hypercube sampling and the Schwartz & Smith oil price model to compute probability distributions of the optimal number of wells, plateau rate and project value. We also employ an analytical model to compute production profiles and project value and consider uncertainties in in-place oil volume, well productivity and oil price. Then, we study how do these distributions change from early field planning until when the field is abandoned, when uncertainties are reduced to a minimum. The variation in time of the in-place oil volume uncertainty is modelled with a random walk. The well productivity is a step function altered randomly after production startup. The actual oil price trajectory is picked randomly from possible trajectories computed with the Schwartz and Smith model. The results show that the distributions of the optimal number of wells, plateau rate and project value depend greatly on the uncertainties in the input data. Field designs based on the average of the distributions during the early phase are profitable, but suboptimal. A potential upside of such designs is that they entail less capital investment and therefore less financial risk when compared against the optimal field design. • Robust approach to consider uncertainties when computing optimal field design. • The optimal design strongly depends on the uncertainties in input parameters. • Uncertainties in input parameters are considered to vary during the field’s life. • The value of perfect information is modest for the cases studied in this paper.

Estimation Surface Runoff Characteristics of Hishkaro Basin, Duhok Governorate Using Some Hydrological Approaches

An analysis of some hydrological characteristics specifically surface runoff of Hishkaro stream basin located in Duhok Governorate-Kurdistan Iraq was conducted. The available historical recorded rainfall data in Duhok and Zawita meteorological stations were used to calculate maximum daily rainfall using probability distribution for number of return periods. Natural resources conservation service method (NRCS) is used to estimate daily surface runoff at the outlet of the basin. Then runoff coefficient, monthly and annual surface runoff volume were predicted.  A maximum daily surface runoff for different probability distribution and number of return period was calculated too which is an essential parameter in the design process of any future required hydraulic structures on Hishkaro stream

Differential evolution for early-phase offshore oilfield design considering uncertainties in initial oil-in-place and well productivity

• Differential evolution optimization of number of wells, plateau rate and NPV. • Multi-objective optimization considering ultimate recovery factor and NPV. • The NPV distribution average can be estimated with the average of the variables. • Method provides decision support to field planners in early field development. During the early phases of offshore oil field development, field planners must decide upon general design features such as the required number of wells and maximum oil processing capacity (field plateau rate), usually by performing sensitivity studies. These design choices are then locked in subsequent development stages and often prevent from achieving optimal field designs in later planning stages when more information is available and uncertainties are reduced. In the present study, we propose using numerical optimization of net present value (NPV) to advice field planners when deciding on the appropriate number of wells, maximum oil processing capacity (plateau rate) in a Brazilian offshore oil field. Differential Evolution (DE) is employed for solving the optimization models. The uncertainties considered are well productivity and initial oil-in-place, handled by (1) using the mean of the distributions and (2) Monte Carlo simulation. A multi-objective optimization was also formulated and solved including ultimate recovery factor in addition to net present value. The proposed method successfully computes probability distributions of optimal number of wells, plateau rate and NPV. If one wishes to compute the mean of such distributions only, for most cases it is adequate to run an optimization using the mean of the input values instead of performing Monte Carlo sampling. The multi-objective optimization allows to find field designs with high ultimate recovery factor and high NPV. In this case, the value of NPV is similar to the optimum NPV value when optimizing NPV only. The methods described could provide decision support to field planners in early stages of field development.

Time-dependent probability distribution for number of infection in a stochastic SIS model: case study COVID-19.

We derive the time-dependent probability distribution for the number of infected individuals at a given time in a stochastic Susceptible-Infected-Susceptible (SIS) epidemic model. The mean, variance, skewness, and kurtosis of the distribution are obtained as a function of time. We study the effect of noise intensity on the distribution and later derive and analyze the effect of changes in the transmission and recovery rates of the disease. Our analysis reveals that the time-dependent probability density function exists if the basic reproduction number is greater than one. It converges to the Dirac delta function on the long run (entirely concentrated on zero) as the basic reproduction number tends to one from above. The result is applied using published COVID-19 parameters and also applied to analyze the probability distribution of the aggregate number of COVID-19 cases in the United States for the period: January 22, 2020-March 23, 2021. Findings show that the distribution shifts concentration to the right until it concentrates entirely on the carrying infection capacity as the infection growth rate increases or the recovery rate reduces. The disease eradication and disease persistence thresholds are calculated.

On estimation of the PDF and the CDF of the one-parameter polynomial exponential family of distributions

In this article, we have considered the estimation of the probability density function and cumulative distribution function of the one-parameter polynomial exponential family of distributions. A number of probability distributions like the exponential, Lindley, length-biased Lindley and Sujatha are particular cases. Two estimators—maximum likelihood and uniformly minimum variance unbiased estimators of the probability density function and cumulative distribution function of the family have been discussed. The estimation issues of the length-biased Lindley and Sujatha distribution have been considered in detail. The estimators have been compared in mean squared error sense. Monte Carlo simulations and real data analysis are performed to compare the performances of the proposed estimators.