Discrete Distribution Research Articles

A novel Gauss–Jacobi quadrature for multiscale Boltzmann solvers

In this paper, we introduce a novel Gauss–Jacobi quadrature rule designed for infinite intervals, which is specifically applied to the velocity discretization in multi-scale Boltzmann solvers. Our method utilizes a newly formulated bell-shaped weight function for numerical integration. We establish the relationship between this new quadrature and the classical Gauss–Jacobi, as well as the Gauss–Hermite quadrature rules, and we compare the resulting discrete velocity distributions with several commonly used methods. Additionally, we validate the performance of our method through numerical simulations of flows with various Knudsen numbers. The proposed quadrature provides fresh insights into velocity space discretization.

Axisymmetric phase-field-based lattice Boltzmann model for incompressible two-phase flow with phase change

In this work, a phase-field-based lattice Boltzmann equation (LBE) model for axisymmetric two-phase flow with phase change is proposed. Two sets of discrete particle distribution functions are employed to match the conserved Allen–Cahn equation and the hydrodynamic equations with phase change effect, respectively. Since phase change occurs at the interface, the divergence-free condition of the velocity field is no longer satisfied due to mass transfer, and the conserved Allen–Cahn equation needs to be equipped with a source term dependent on the phase change model. To deal with these, a novel source term in the hydrodynamic LBE is delicately designed to recover the correct target governing equations. Meanwhile, the LBE for the Allen–Cahn equation is modified with a discrete force term to model mass transfer. In particular, an additional correction term is added into the hydrodynamic LBE to reduce the spurious velocity and improve numerical stability. Several axisymmetric benchmark multiphase problems with phase change, including bubble growing in superheated liquid, D2 law, film boiling, bubble rising in superheated liquid under gravity, and droplet impact on a hot surface, have been conducted to test the performance of the proposed model. Numerical results agree well with analytical solutions and available published data in the literature.

Waiting time representation of discrete distributions

We discuss a representation of any probability distribution on the set of non-negative integers as a waiting time distribution in a sequence of independent Bernoulli trials. Several associated results are derived and illustrated by examples. Multivariate extensions are briefly treated as well.

Correlation between individual phase constitutive properties and plastic heterogeneities in advanced-high strength dual-phase steels

The present study comprehensively investigates the individual phase constitutive properties and plastic heterogeneities in advanced high-strength steels (AHSS), particularly DP590 and DP780 dual-phase (DP) steels. A machine learning-based model is implemented to identify the ferrite and martensite phases in the microstructures of DP590 and DP780. Then, the constitutive properties of ferrite and martensite phases are successfully obtained through a hybrid approach of in-situ neutron diffraction coupled with the crystal plasticity finite element method (CPFEM). The distinct microstructures between DP590 and DP780 result in different macroscopic and microscopic properties among the two materials. Owing to different martensite volume fractions (Vm) in DP590 (Vm = 8.3 %) and DP780 (Vm = 35.4 %), a noticeable dependency of plastic heterogeneities during deformation on martensite fraction and its spatial distribution is revealed. Compared to DP590, the deformed microstructure of DP780 exhibits a more heterogeneous distribution of stress and strain fields, along with significant formation of plastic strain localization leading to a remarkable increase in strain partitioning index. It shows that a lower fraction of martensite with its discrete distribution decreases martensite ability to hinder the ferrite deformation, thus strain localization is primarily concentrated within the ferrite phase as the predominant failure mode in DP590. In contrast, a higher martensite fraction in DP780 causes more pronounced strain localization which occurs in the ferrite and at the ferrite/martensite interface. In addition, interconnect distribution between martensite islands enhances the inhibition of martensite to ferrite deformation, thereby high strain gradient at their interface leads to prevalence of ferrite/martensite interface decohesion in DP780.

A note on statistical distances for discrete log-concave measures

In this note we explore how standard statistical distances are equivalent for discrete log-concave distributions. Distances include total variation distance, Wasserstein distance, and f-divergences.

A Novel Discrete Linear-Exponential Distribution for Modeling Physical and Medical Data

In real-life data, count data are considered more significant in different fields. In this article, a new form of the one-parameter discrete linear-exponential distribution is derived based on the survival function as a discretization technique. An extensive study of this distribution is conducted under its new form, including characteristic functions and statistical properties. It is shown that this distribution is appropriate for modeling over-dispersed count data. Moreover, its probability mass function is right-skewed with different shapes. The unknown model parameter is estimated using the maximum likelihood method, with more attention given to Bayesian estimation methods. The Bayesian estimator is computed based on three different loss functions: a square error loss function, a linear exponential loss function, and a generalized entropy loss function. The simulation study is implemented to examine the distribution’s behavior and compare the classical and Bayesian estimation methods, which indicated that the Bayesian method under the generalized entropy loss function with positive weight is the best for all sample sizes with the minimum mean squared errors. Finally, the discrete linear-exponential distribution proves its efficiency in fitting discrete physical and medical lifetime count data in real-life against other related distributions.

Minimizing distance between distribution functions: discrete counterparts to continuous random variables with applications in non-life insurance and stochastic reliability

In this work, we propose a novel family of procedures for deriving a discrete counterpart to a continuous probability distribution. They are based on a class of distances between cumulative distribution functions, including the Cramér, the Cramér-von Mises, and the Anderson-Darling distances as particular cases. The discrete counterpart is defined and derived as the random variable which minimizes its distance to the assigned continuous probability distribution among all the discrete random variables supported on the set of integers (or positive integers). Applications are provided with reference to the exponential and the normal distributions, among others; the discrete counterparts are derived, and their main properties are discussed, also in comparison with the one obtained through an existing discretization technique based on the preservation of the cumulative distribution function at integer values. Parameter estimation for these discrete analogs is discussed, along with an analysis of two real datasets, where they are compared in terms of goodness-of-fit with some popular discrete distributions. Furthermore, in order to highlight the effectiveness and the benefits derived from the proposed discretization procedures, we illustrate two practical applications in actuarial science and in reliability engineering. In the former case, the problem of determining the distribution of the total claims amount for a non-life insurance portfolio is considered, where the claim sizes can be modelled as iid random variables, and the number of claims is random as well. Actuaries use a recursive calculation method based on Panjer's formula, which requires an appropriate discretization of the individual claim distribution, and therefore the proposed procedures can be used. Since we consider two simple cases where the distribution of the total claims amount is analytically acquirable, the efficacy of the discretization procedures in the final approximation can be easily assessed and turns out to be satisfactory, especially when compared to the existing discretization. The latter case considers the determination of the reliability parameter for a complex stress-strength model. Here, the approximation by discretization is compared to Monte Carlo simulation and shown to be relevant: with a comparable if not smaller computational effort, discretization leads to similar results as simulation. Such discretizations can also naturally be applied to more complex problems such as scenario generation in stochastic programming. R code for this article is provided as supplementary material.

A novel 4-DOF marine stern bearing support model considering discrete distribution effects

To address the issue of the traditional support model's inability to accurately describe the non-uniform dynamics behavior inside the bearing, this study proposes a novel 4-degree-of-freedom (DOF) bearing support model that incorporates the discrete distribution effects induced by the bi-directional misalignment and bending deformation of the journal. Additionally, a coupling algorithm is designed to study the interaction behaviors between the marine stern bearing lubrication performance and the shaft dynamics. Moreover, a full-size marine shaft alignment test rig for measuring dynamic bearing loads is designed to verify the validity of the present model. On this basis, the effects of different bearing models, rotational speeds, external loads, and bush elastic modulus on the shaft dynamic alignment and the static and dynamic characteristics of the stern bearing are investigated. The experimental results indicate that the 4-DOF model, which takes into account discrete distribution, exhibits a more precise prediction of the alignment characteristics of the shaft system, with a forecast error for the stern bearing load of 1.32%. Heavy loads or low rotational speeds significantly deteriorate the edge-loading effect and increase the stiffness and damping coefficient of the bearings. The bearing lubrication performance directly affects the alignment performance of the shaft system, particularly in low-speed conditions. A softer bush material can increase the minimum oil film thickness, carrying region, and damping coefficient, mitigate the phenomenon of edge-loading, and facilitate the vibration control of the shaft. Research results have made a great contribution to marine shaft alignment optimization and structure design.

A new two-parameter over-dispersed discrete distribution with mathematical properties, estimation, regression model and applications

This paper focuses on the derivation of a new two-parameter discrete probability distribution. The new model is derived by mixing Poisson and Loai distributions and is named “Poisson Loai Distribution”. The paper explores various mathematical properties of the new model, introducing a count-regression model based on this distribution. The parameters of the model are estimated using the maximum likelihood estimation method. A comprehensive simulation study is utilized to assess the behavior of derived estimators. The importance of the proposed distribution is confirmed through the analysis of three real datasets. It is found that the suggested distribution has the greatest match when compared to all rival distributions, and it may be a viable alternative for assessing dispersed count data.

Feasibility analysis of the Sanchez-Pomraning method to treat the particle size distribution in particle-dispersed fuel

The non-uniform size of particles appears in the particle-dispersed fuel generally, but this phenomenon is usually neglected. This study aims to evaluate the feasibility of the Sanchez-Pomraning method to solve the particle size distribution problem in particle-dispersed fuel. The Pu-rich agglomerate phenomenon in the MOX fuel and manufacturing tolerance in the Fully Ceramic Microencapsulated (FCM) fuel are calculated by ALPHA-DH code, which implements the Sanchez-Pomraning method based on the MOC transport calculation and improved subgroup resonance calculation. Numerical results show that the Sanchez-Pomraning method has the capability to treat the particle size distribution effect accurately, where the differences between the Sanchez-Pomraning method and the Monte Carlo method are generally lower than 160 pcm. For the Pu-rich agglomerate phenomenon, the results show that ignoring the Pu-rich agglomerate effect introduces the reactivity difference of around 600 pcm, and ignoring the size distribution of the Pu agglomerate will cause the reactivity of up to 200 pcm in the extra-high or extra-low Pu enrichment. The more discrete size distribution and the smaller Pu-spots, the more significant influence will occur in the MOX fuel. For the FCM fuel, the influence brought by the TRISO particle size distribution can be ignored when only TRISO particles are filled. However, ignoring the size distribution when TRISO particles and poison particles are both added into the FCM fuel introduces the differences reaching 600–1000 pcm. These results demonstrate the importance of considering the particle size distribution for analyzing the MOX fuel and FCM fuel in practice. And the Sanchez-Pomraning method is reliable to analyze the effect of particle size distribution.

Analysis of leukemia and forest fires data using new Poisson Quasi-Shanker distribution

In this study, a new two-parameter discrete distribution is introduced by mixing Poisson and Quasi-Shanker distributions. The new model is named Poisson Quasi-Shanker-(PQS) distribution. Some important mathematical properties of the proposed distribution are derived including factorial moments, moment-generating function, variance, and dispersion index. The parameters of PQS distribution are estimated using the maximum likelihood estimation approach. A Monte Carlo simulation is carried out to illustrate the behavior of derived estimators. The applicability of the new model is assessed using two datasets from different areas. It is concluded the proposed distribution is more flexible as compared to the considered competitive distributions.

Efficient First-Order Algorithms for Large-Scale, Non-Smooth Maximum Entropy Models with Application to Wildfire Science.

Maximum entropy (MaxEnt) models are a class of statistical models that use the maximum entropy principle to estimate probability distributions from data. Due to the size of modern data sets, MaxEnt models need efficient optimization algorithms to scale well for big data applications. State-of-the-art algorithms for MaxEnt models, however, were not originally designed to handle big data sets; these algorithms either rely on technical devices that may yield unreliable numerical results, scale poorly, or require smoothness assumptions that many practical MaxEnt models lack. In this paper, we present novel optimization algorithms that overcome the shortcomings of state-of-the-art algorithms for training large-scale, non-smooth MaxEnt models. Our proposed first-order algorithms leverage the Kullback-Leibler divergence to train large-scale and non-smooth MaxEnt models efficiently. For MaxEnt models with discrete probability distribution of n elements built from samples, each containing m features, the stepsize parameter estimation and iterations in our algorithms scale on the order of O(mn) operations and can be trivially parallelized. Moreover, the strong ℓ1 convexity of the Kullback-Leibler divergence allows for larger stepsize parameters, thereby speeding up the convergence rate of our algorithms. To illustrate the efficiency of our novel algorithms, we consider the problem of estimating probabilities of fire occurrences as a function of ecological features in the Western US MTBS-Interagency wildfire data set. Our numerical results show that our algorithms outperform the state of the art by one order of magnitude and yield results that agree with physical models of wildfire occurrence and previous statistical analyses of wildfire drivers.

A new family of copulas based on probability generating functions

Abstract We propose a method to obtain a new class of copulas using a probability generating function (PGF) of positive-integer valued random variable. Some existing copulas in the literature are sub-families of the proposed copulas. Various dependence measures and invariant property of the tail dependence coefficient under PGF transformation are also discussed. We propose an algorithm for generating random numbers from the PGF copula. The bivariate concavity properties, such as Schur concavity and quasi-concavity, associated with the PGF copula are studied. Two new generalized FGM copulas are introduced using PGFs of geometric and discrete Mittag-Leffler distributions. The proposed two copulas improved the Spearman’s rho of FGM copula by (−0.3333, 0.4751) and (−0.3333, 0.9573). Finally, we analyse a real dataset to illustrate the practical application of the proposed copulas.

Strategi Harga Penawaran Tender pada Proyek Konstruksi Dengan Memperhitungkan Faktor Resiko

To get a construction project through a process called a tender (auction). This process is very crucial in determining the party who will carry out the construction project. For this reason, a strategy is needed, one of which is determining the offering price. The bid price really determines the possibility of winning a construction project as well as the size of the profit or profit on the construction project. One of the problems that is often encountered is determining the offering price which should not be too high or too low. The aim of this research is to analyze the mark up value in construction project tenders at the Jambi City Electronic Procurement Institute (LPSE) using the Friedman Method, Gates Method, and Ackoff & Sasieni Method bidding strategy approaches and compare the strategic approaches that produce optimum mark up values so that can be used as an alternative in determining the offering price. The data used in this research is auction data for construction work that has been completed at Electronic Procurement Institute (LPSE) Jambi in 2022 using three statistical approaches, namely multi discrete distribution, multi normal distribution and single normal distribution. In this research it can be concluded that the Friedman Method and Ackoff & Sasieni Method produce the lowest optimum mark up value, while the Gates Method produces the highest optimum mark up value.

Taming numerical imprecision by adapting the KL divergence to negative probabilities

The Kullback–Leibler (KL) divergence is frequently used in data science. For discrete distributions on large state spaces, approximations of probability vectors may result in a few small negative entries, rendering the KL divergence undefined. We address this problem by introducing a parameterized family of substitute divergence measures, the shifted KL (sKL) divergence measures. Our approach is generic and does not increase the computational overhead. We show that the sKL divergence shares important theoretical properties with the KL divergence and discuss how its shift parameters should be chosen. If Gaussian noise is added to a probability vector, we prove that the average sKL divergence converges to the KL divergence for small enough noise. We also show that our method solves the problem of negative entries in an application from computational oncology, the optimization of Mutual Hazard Networks for cancer progression using tensor-train approximations.

When is the discrete Weibull distribution infinitely divisible?

For the discrete Weibull probability distribution we prove that it is only infinitely divisible if the shape parameter lies in the range 0<β≤1 . The proof is based on some results of Steutel and van Harn (2004). For this case we construct the corresponding compound Poisson distribution and thus the related Lévy process.

Mutations in Ras homolog family member A in patients with peripheral T-cell lymphoma and implications for personalized medicine.

Genome sequencing has revealed frequent mutations in Ras homolog family member A (RHOA) among various cancers with unique aberrant profiles and pathogenic effects, especially in peripheral T-cell lymphoma (PTCL). The discrete positional distribution and types of RHOA amino acid substitutions vary according to the tumor type, thereby leading to different functional and biological properties, which provide new insight into the molecular pathogenesis and potential targeted therapies for various tumors. However, the similarities and discrepancies in characteristics of RHOA mutations among various histologic subtypes of PTCL have not been fully elucidated. Herein we highlight the inconsistencies and complexities of the type and location of RHOA mutations and demonstrate the contribution of RHOA variants to the pathogenesis of PTCL by combining epigenetic abnormalities and activating multiple downstream pathways. The promising potential of targeting RHOA as a therapeutic modality is also outlined. This review provides new insight in the field of personalized medicine to improve the clinical outcomes for patients.

Statistics of punctuation in experimental literature-The remarkable case of Finnegans Wake by James Joyce.

As the recent studies indicate, the structure imposed onto written texts by the presence of punctuation develops patterns which reveal certain characteristics of universality. In particular, based on a large collection of classic literary works, it has been evidenced that the distances between consecutive punctuation marks, measured in terms of the number of words, obey the discrete Weibull distribution-a discrete variant of a distribution often used in survival analysis. The present work extends the analysis of punctuation usage patterns to more experimental pieces of world literature. It turns out that the compliance of the the distances between punctuation marks with the discrete Weibull distribution typically applies here as well. However, some of the works by James Joyce are distinct in this regard-in the sense that the tails of the relevant distributions are significantly thicker and, consequently, the corresponding hazard functions are decreasing functions not observed in typical literary texts in prose. Finnegans Wake-the same one to which science owes the word quarks for the most fundamental constituents of matter-is particularly striking in this context. At the same time, in all the studied texts, the sentence lengths-representing the distances between sentence-ending punctuation marks-reveal more freedom and are not constrained by the discrete Weibull distribution. This freedom in some cases translates into long-range nonlinear correlations, which manifest themselves in multifractality. Again, a text particularly spectacular in terms of multifractality is Finnegans Wake.

On the mathematical quantification of inequality in probability distributions

A fundamental challenge in the study of probability distributions is the quantification of inequality that is inherently present in them. Some parts of the distribution are more probable and some others are not, and we are interested in the quantification of this inequality through the lens of mathematical diversity, which is a new approach to studying inequality. We offer a theoretical advance, based on case-based entropy and slope of diversity, which addresses inequality for arbitrary probability distributions through the concept of mathematical diversity. Our approach is useful in three important ways: (1) it offers a universal way to measure inequality in arbitrary probability distributions based purely on the entropic uncertainty that is inherent in them and nothing else; (2) it allows us to compare the degree of inequality of arbitrary parts of any distribution (not just tails) and entire distributions alike; and (3) it can glean out empirical rules similar to the 80/20 rule, not just for the power law but for any given distribution or its parts thereof. The techniques shown in this paper demonstrate a more general machinery to quantify inequality, compare the degree of inequality of parts or whole of general distributions, and prove or glean out empirical rules for general distributions based on mathematical diversity. We demonstrate the utility of this new machinery by applying it to the power law, the exponential and the geometric distributions. The 60 − 40 rule of restricted diversity states that 60 percent or more of cases following a power law (or more generally a right skewed distribution) reside within 40 percent or less of the lower bound of Shannon equivalent equi-probable (SEE) types as measured by case-based entropy. In this paper, we prove the 60 − 40 rule for power law distributions analytically. We also show that in all power law distributions, the second half of the distribution is at least 4 times more uniformly distributed as the first. Lastly, we also show a scale-free way of comparing probability distributions based on the idea of mathematical diversity of parts of a distribution. We use this comparison technique to compare the exponential and power law distribution, and obtain the exponential distribution as an entropic limit of the power law distribution. We also demonstrate that the machinery is applicable to discrete distributions by proving a general result regarding the comparison of parts of the geometric distribution.

Understanding Higher-Order Interactions in Information Space.

Methods used in topological data analysis naturally capture higher-order interactions in point cloud data embedded in a metric space. This methodology was recently extended to data living in an information space, by which we mean a space measured with an information theoretical distance. One such setting is a finite collection of discrete probability distributions embedded in the probability simplex measured with the relative entropy (Kullback-Leibler divergence). More generally, one can work with a Bregman divergence parameterized by a different notion of entropy. While theoretical algorithms exist for this setup, there is a paucity of implementations for exploring and comparing geometric-topological properties of various information spaces. The interest of this work is therefore twofold. First, we propose the first robust algorithms and software for geometric and topological data analysis in information space. Perhaps surprisingly, despite working with Bregman divergences, our design reuses robust libraries for the Euclidean case. Second, using the new software, we take the first steps towards understanding the geometric-topological structure of these spaces. In particular, we compare them with the more familiar spaces equipped with the Euclidean and Fisher metrics.