Discrete Distribution Research Articles

On the Discretization of the Weibull-G Family of Distributions: Properties, Parameter Estimates, and Applications of a New Discrete Distribution

In this article, we introduce a new three-parameter distribution called the discrete Weibull exponential (DWE) distribution, based on the use of a discretization technique for the Weibull-G family of distributions. This distribution is noteworthy, as its probability mass function presents both symmetric and asymmetric shapes. In addition, its related hazard function is tractable, exhibiting a wide range of shapes, including increasing, increasing–constant, uniform, monotonically increasing, and reversed J-shaped. We also discuss some of the properties of the proposed distribution, such as the moments, moment-generating function, dispersion index, Rényi entropy, and order statistics. The maximum likelihood method is employed to estimate the model’s unknown parameters, and these estimates are evaluated through simulation studies. Additionally, the effectiveness of the model is examined by applying it to three real data sets. The results demonstrate that, in comparison to the other considered distributions, the proposed distribution provides a better fit to the data.

The Navier-Stokes equation and a fully developed turbulence

In fairly general conditions we give explicit (smooth) solutions for the potential flow. We show that, rigorously speaking, the equations of the fluid mechanics have not rotational solutions. However, within the usual approximations of an incompressible fluid and an isentropic flow, the Navier-Stokes equation has approximate vorticial (rotational) solutions, generated by viscosity. However, in general, these vortices are unstable, and a discrete distribution of vorticial solutions is not in me chanical equilibrium; it forms an unstable vorticial liquid. On the other hand, these solutions may exhibit turbulent, fluctuating instabilities for large variations of the velocity over short distances. We represent a fully developed turbulence as a homogeneous, isotropic and highly-fluctuating distribution of singular centres of turbulence. A regular mean flow can be included. In these circumstances the Navier-Stokes equation exhibits three time scales. The equations of the mean flow can be disentangled from the equations of the fluctuating part, which is reduced to a vanishing inertial term. This latter equation is not satisfied after averaging out the temporal fluctuations. For a homogeneous and isotropic distribution of non-singular turbulence centres the equation for the inertial term is satisfied trivially, i.e. both the average fluctuating velocity and the average fluctuating inertial term are zero. If the velocity is singular at the turbulence centres, we are left with a quasi-ideal classical gas of singularities, or a solution of singularities (solute) in quasi thermal equilibrium in the background fluid (solvent). This is an example of an emergent dynamics. We give three examples of vorticial liquids.

Probabilistic aspects of Jacobi theta functions

In this note we deduce well-known modular identities for Jacobi theta functions using the spectral representations associated with the real valued Brownian motion taking values on $[-1,+1]$. We consider two cases: (i) reflection at $-1$ and $+1$, (ii) killing at $-1$ and $+1$. It is seen that these two representations give, in a sense, most compact forms of the modular theta-function identities. We study also discrete Gaussian distributions generated by theta functions, and derive, in particular, addition formulas for discrete Gaussian variables.

Bounding the $$L^1$$-Distance Between One-Dimensional Continuous and Discrete Distributions via Stein’s Method

Bounding the $$L^1$$-Distance Between One-Dimensional Continuous and Discrete Distributions via Stein’s Method

Gas-Kinetic Unified Algorithm for Aerodynamics Covering Various Flow Regimes by Computable Modeling of Boltzmann Equation

The gas-kinetic unified algorithm (GKUA), to solve the modeling of the Boltzmann equation, has been developed to study the aerothermodynamics problems with the effects of wall activation energy covering various flow regimes. The unified velocity distribution function equation could be accordingly presented on the basis of the Boltzmann-Shakhov model. To remove the dependence of the distribution function on velocity space, the conservational discrete velocity ordinate method has been developed for hypervelocity flows. The gas-kinetic finite difference scheme is constructed to directly solve the discrete velocity distribution functions by the operator splitting technique. The discrete velocity numerical integration method with the Gauss-type weight function has been developed to evaluate the macroscopic flow variables. Specially, to model the real physical process between gas molecules and the surface, the Maxwell-type gas-surface interaction model has been presented by the MD (molecular dynamic) simulation to obtain the energy adaptability coefficient. The multi-processing domain decomposition strategy and parallel implementation of high parallel efficiency and expansibility designed for the gas-kinetic numerical method is presented with good load balance and data communication efficiency. To validate the accuracy and feasibility of the present algorithm, the supersonic flows past two-dimensional circular cylinder are simulated covering various flow regimes. The results are in good agreement with the related theoretical, DSMC (Direct Simulation Monte Carlo), N-S (Navier-Stokes), and experimental data. The hypersonic reentry flows with the effects of wall activation energy around the Tianzhou-5 cargo spacecraft are simulated by the present GKUA and the massive parallel strategy. It has been confirmed that the present algorithm from the gas-kinetic point of view probably provides a promising approach to resolve the hypersonic aerothermodynamic problems with the complete spectrum of flow regimes during the re-entry and disintegration of the large-scale spacecraft.

The concept of nociplastic pain—where to from here?

Abstract Nociplastic pain, a third mechanistic pain descriptor in addition to nociceptive and neuropathic pain, was adopted in 2017 by the International Association for the Study of Pain (IASP). It is defined as “pain that arises from altered nociception” not fully explained by nociceptive or neuropathic pain mechanisms. Peripheral and/or central sensitization, manifesting as allodynia and hyperalgesia, is typically present, although not specific for nociplastic pain. Criteria for possible nociplastic pain manifesting in the musculoskeletal system define a minimum of 4 conditions: (1) pain duration of more than 3 months; (2) regional, multifocal or widespread rather than discrete distribution of pain; (3) pain cannot entirely be explained by nociceptive or neuropathic mechanisms; and (4) clinical signs of pain hypersensitivity present in the region of pain. Educational endeavors and field testing of criteria are needed. Pharmacological treatment guidelines, based on the three pain types, need to be developed. Currently pharmacological treatments of nociplastic pain resemble those of neuropathic; however, opioids should be avoided. A major challenge is to unravel pathophysiological mechanisms driving altered nociception in patients suffering from nociplastic pain. Examples from fibromyalgia would include pathophysiology of the peripheral as well as central nervous system, such as autoreactive antibodies acting at the level of the dorsal root ganglia and aberrant cerebral pain processing, including altered brain network architecture. Understanding pathophysiological mechanisms and their interactions is a prerequisite for the development of diagnostic tests allowing for individualized treatments and development of new strategies for prevention and treatment.

Uncertain multi‐objective programming approach for planning supplementary irrigation areas in rainfed agricultural regions

AbstractRainfed agriculture is crucial for ensuring global food and water security, and supplementary irrigation is an effective means of improving the economic benefits in rainfed agricultural regions. This study proposes a novel uncertain optimization approach to optimize supplementary irrigation areas in rainfed agricultural regions. The approach incorporates multi‐objective linear programming, interval linear programming, fuzzy goal programming and stochastic expected value programming. In the proposed interval multi‐fuzzy goal stochastic expected‐value programming, uncertainties are expressed in the form of discrete intervals, probability distributions and fuzzy goals. This approach, which considers the randomness of precipitation during the optimization process and allocates limited irrigation water resources to different subareas and crops, was applied to a case study of crop irrigation area planning in Guyuan City, Ningxia Hui Autonomous Region, northwestern China. The maximum economic benefits and the minimum sum of the Gini coefficients among the different subareas and crops were regarded as the planning objectives, and a series of optimal irrigation areas with different crops and subareas under different water levels were obtained. The optimization results revealed that vegetables and fruits consumed large amounts of irrigation water to increase their economic benefits. In addition, allocating a large amount of irrigation water to wheat is essential to decrease the Gini coefficient and meet food security constraints, particularly at extremely low water levels. Compared with current management, the optimized irrigation area decreased by 30%, the crop water deficit index increased by 9%, the economic benefits increased by 3% and the total Gini coefficient of crops decreased by 17%, indicating that the optimization approach could fairly and reasonably allocate irrigation water resources. Our research provides a mathematical approach for decision‐makers to plan supplementary irrigation areas in rainfed agricultural regions.

Improving Word Embedding on Malayalam Corpus

Abstract: NLP is natural language processing or neuro linguis-tic programming.Natural languages like malayalam are highly inflectional and agglutinative in nature.This is problematic whendealing with nlp based malayalam applications.So that inorder toimprove performance of malayalam nlp based applications, wordembedding improvement on malayalam corpus is needed.The improvement is based on converting the words contained inthe malayalam corpus into a standardised means removingall inflectional parts in the words in the existing malayalamcorpus ie taking root words only.All that needed is a stemmer.Inthis project i have used a malayalam morphological analyserfor taking root words of all words in the existing malayalam corpus.The advanatge of removing inflectional parts from allwords is that we can reduce the sparsity in the existing malayalam corpus.Also there will be a high hike in frequency of wordsin the resulting corpus,then the space and time complexity of wordembedding representation of the existing corpus willdecreases.According to zipfs law by increasing frequency ofwords performance of neural word embedding will increases. Zipfs Law is a discrete probability distribution that tells you the probability of encountering a word in a given corpus.By applying zipfs law am proposing there will be improvement on malyalam wordembedding.Here using fasttext, word embeddingsare performed and capture dense word vector representation ofthe malayalam corpus with dimensionality reduction from thesparse word co-occurence matrix.The improvement is mainly used for wordnet,analogy,ontology based malayalam applications.Index Terms—Morphological Analyzer,Zipfs law, Preprocessing, Testing, Training

Discrete Joint Random Variables in Fréchet-Weibull Distribution: A Comprehensive Mathematical Framework with Simulations, Goodness-of-Fit Analysis, and Informed Decision-Making

This paper introduces a novel four-parameter discrete bivariate distribution, termed the bivariate discretized Fréchet–Weibull distribution (BDFWD), with marginals derived from the discretized Fréchet–Weibull distribution. Several statistical and reliability properties are thoroughly examined, including the joint cumulative distribution function, joint probability mass function, joint survival function, bivariate hazard rate function, and bivariate reversed hazard rate function, all presented in straightforward forms. Additionally, properties such as moments and their related concepts, the stress–strength model, total positivity of order 2, positive quadrant dependence, and the median are examined. The BDFWD is capable of modeling asymmetric dispersion data across various forms of hazard rate shapes and kurtosis. Following the introduction of the mathematical and statistical frameworks of the BDFWD, the maximum likelihood estimation approach is employed to estimate the model parameters. A simulation study is also conducted to investigate the behavior of the generated estimators. To demonstrate the capability and flexibility of the BDFWD, three distinct datasets are analyzed from various fields, including football score records, recurrence times to infection for kidney dialysis patients, and student marks from two internal examination statistical papers. The study confirms that the BDFWD outperforms competitive distributions in terms of efficiency across various discrete data applications.

Asymptotic Properties of a Statistical Estimator of the Jeffreys Divergence: The Case of Discrete Distributions

We investigate the asymptotic properties of the plug-in estimator for the Jeffreys divergence, the symmetric variant of the Kullback–Leibler (KL) divergence. This study focuses specifically on the divergence between discrete distributions. Traditionally, estimators rely on two independent samples corresponding to two distinct conditions. However, we propose a one-sample estimator where the condition results from a random event. We establish the estimator’s asymptotic unbiasedness (law of large numbers) and asymptotic normality (central limit theorem). Although the results are expected, the proofs require additional technical work due to the randomness of the conditions.

Discrete modulation continuous variable quantum key distribution under fast fading channel

After decades of development, continuous variable quantum key distribution technology still cannot meet people’s needs for secure communication over long distances. Free-space quantum communication provides a new way for secure communication over long distances. However, because the communication connected in free space is inevitably affected by atmospheric turbulence, the transmission rate of the channel changes rapidly according to the probability distribution, making it difficult for the communication personnel to determine the transmission rate of the channel. At the same time, discrete modulated continuously variable quantum key Distribution protocol (DM-CVQKD) has certain advantages over Gauss modulated continuously variable quantum key Distribution protocol (GM-CVQKD) in the communication distance. Therefore, this paper analyzes the key rate of DM-CVQKD in this case (that is, in the fast attenuation channel), proves that it can resist the collective attack, and observes the relationship between the parameters and the key rate in the fast attenuation channel by numerical simulation and control variables. Finally, through the analysis of numerical simulation results, it is found that DM-CVQKD can still maintain a high key rate under certain conditions, which proves that long-distance free space quantum communication is feasible under certain conditions. At the same time, it was found that the performance of DM-CVQKD was affected by the extreme case where Eve completely controlled the instantaneous transmittance.

An Evidential Reasoning Method of Comprehensive Evaluation of Water Quality Based on Gaussian Distribution

This study provides an evidential reasoning method for water quality evaluation based on Gaussian distribution to handle the problem of comprehensive water quality evaluation for a region across a period (multiple sections and multiple time points). The method turns the collection of observed water quality indicator values into a probability distribution of water quality grades by using the Gaussian distribution to compute the confidence assessment of water quality grades over one period. It eliminates the subjectivity involved in determining confidence levels and the problem of information loss during data fusion that arises with conventional approaches. The probability distribution of each assessment grade is then determined by repeatedly synthesizing evidence of the same water quality grade using the improved evidential reasoning synthesis rule. To avoid the subjectivity included in experience-based weight settings, principal component analysis (PCA) is utilized to calculate the weights of water quality indicators based on contribution rates and load coefficients. In the end, utility theory is presented to modify the discrete probability distribution of precise numerical expressions, offering thorough results for the evaluation of water quality and facilitating the comparison of various water quality grades. Using the Xiangjiang River Basin as a case study, the proposed evaluation method is contrasted with popular techniques for assessing water quality, including the Single-Factor Evaluation Method, the Fuzzy Comprehensive Evaluation Method, and the Evidential Reasoning Comprehensive Evaluation Method. The findings suggest that the evidence reasoning approach for evaluating water quality that is based on Gaussian distribution is more rational, accurate, and scientific. Additionally, empirical studies on the annual water quality trends in various regions, the upstream, midstream, and downstream trends, and the water quality trends during wet and dry periods are conducted using this method to assess and analyze changes in water quality in the Xiangjiang River Basin during the “11th Five-Year Plan” and “12th Five-Year Plan” periods. The analysis findings demonstrate that, even if the rate of progress has slowed, the Xiangjiang River Basin’s overall water quality has been steadily improving since management and protection measures were put in place. This shows that the preventive and control efforts implemented in the “11th Five-Year Plan” and “12th Five-Year Plan” periods were successful; nevertheless, carrying out the current tactics might only have a limited impact. As a result, more advanced and creative approaches are required to encourage the ongoing enhancement of the water quality in the Xiangjiang River Basin.

The limit law of maximum of discrete partial-sums distribution

The limit law of maximum of discrete partial-sums distribution

Pricing of a Binary Option Under a Mixed Exponential Jump Diffusion Model

This paper focuses on the pricing problem of binary options under stochastic interest rates, stochastic volatility, and a mixed exponential jump diffusion model. Considering the negative interest rates in the market in recent years, this paper assumes that the stochastic interest rate follows the Hull–White (HW) model. In addition, we assume that the stochastic volatility follows the Heston volatility model, and the price of the underlying asset follows the jump diffusion model in which the jumps follow the mixed exponential jump model. Considering these factors comprehensively, the mixed exponential jump diffusion of the Heston–HW (abbreviated as MEJ-Heston–HW) model is established. Using the idea of measure transformation, the pricing formula of binary call options is derived by the martingale method, eigenfunction, and Fourier transform. Finally, the effects of the volatility term and the parameters of the mixed-exponential jump diffusion model on the option price in the O-U process are analyzed. In the numerical simulation, compared with the double exponential jump Heston–HW (abbreviated as DEJ-Heston–HW) model and the Heston–HW model, the mixed exponential jump model is an extension of the double exponential jump model, which can approximate any distribution in the sense of weak convergence, including arbitrary discrete distributions, normal distributions, and various thick-tailed distributions. Therefore, the MEJ-Heston–HW model adopted in this paper can better describe the price of the underlying asset.

On the Conflation of Negative Binomial and Logarithmic Distributions

In recent decades, the study of discrete distributions has received increasing attention in the field of statistics, mainly because discrete distributions can model a wide range of count data. One common distribution used for modeling count data, for instance, is the negative binomial distribution (NBD), which performs well with over-dispersed data. In this paper, a new count distribution is introduced, called the conflation of negative binomial and logarithmic distributions, which is formed by conflating the negative binomial and logarithmic distributions, resulting in a distribution that possesses some of the properties of negative binomial and logarithmic distributions. The distribution has two parameters and is verified by a positive integer. Two modifications are proposed to the distribution, which includes zero as a support point. The new distribution is valuable from a theoretical perspective since it is a member of the weighted negative binomial distribution family. In addition, the distribution differs from the NBD in the sense that the probability of lower counts is inflated. This study discusses the characteristics of the proposed distribution and its modified versions, such as moments, probability generating functions, likelihood stochastic ordering, log-concavity, and unimodality properties. Real-world data are used to evaluate the performance of the proposed models against other models. All computations shown in this paper were produced using the R programming language.

Serotypic and genotypic profile of dengue virus outbreaks in India between 2011 and 2017.

The epidemiology of dengue has been demonstrating significant changes in recent years, with rising incidence of infection in countries with known endemicity and occurrence of fresh outbreaks in previously unaffected territories. India, which has been a traditional hotspot dengue virus (DENV) transmission in the endemic south-east Asian region, has also been affected by the sweeping changes in dengue epidemiology. Two remarkable developments characterize the alterations witnessed by India 2011-2017. Firstly, all parts of the country have witnessed abrupt replacement of existing DENV lineages with emerging novel strains. Secondly, co-circulation of multiple serotypes of DENV have been reported from all across the country 2011-2017; thereby hinting at the transition of the country towards hyper-endemicity. Considering the potential clinical implications of such epidemiological transformation in terms increasing involvement of pediatric patients and growing predisposition to serious complications, the present review provides an update on the serotypic and genotypic profile of dengue outbreaks that have been witnessed by different zones of India between 2011 and 2017. Dividing the country into northern, southern, eastern, western, central and north-eastern zones, we describe discrete zone-specific distribution patterns of DENV serotypes and genotypes and observe simultaneous circulation of different DENV strains in different parts of the country. Random shifts in the genetic characteristics of the circulating strains and the widespread co- circulation of all four serotypes underscore the need for undertaking continuous and representative molecular surveillance of the circulating DENV strains across the country for prompt identification of emerging strains and novel mutants; gain insights into the formulation of Dengue vaccines and develop a clearer understanding of the molecular basis of immune evasion, disease epidemiology and pathogenesis.

Investigation of Precipitation Behavior of a Novel Ni-Fe-Based Superalloy during High-Temperature Aging Treatment.

The precipitation behavior of a novel Ni-Fe-based superalloy developed for advanced ultra-supercritical (A-USC) coal-fired power plant applications during high-temperature aging treatment was investigated. The results showed that the major precipitates in the novel alloy were randomly distributed MC carbides, M23C6 carbides at grain boundaries, and the γ'-Ni3 (Al, Ti) phase in grain interiors after aging. MC remained relatively stable during both short-term and long-term aging. M23C6 quickly precipitated and exhibited a discrete distribution at grain boundaries during short-term aging, and partly developed into continuous films during long-term aging. After uniform precipitation, the shape of γ' remained spherical, and the size kept increasing with aging time according to the Lifshitz-Slyozov-Wagner (LSW) model. The hardness of the novel alloy was mainly associated with the precipitation behavior of γ'; as γ' gradually precipitated, the hardness steadily increased; after complete precipitation, as the size of γ' increased, the hardness first increased and then decreased, reaching the peak hardness when the average radius of γ' achieved the critical size. In addition, the novel alloy exhibited abnormal coarsening behavior at grain boundaries during both short-term and long-term aging. The coarsened grain boundaries were actually precipitate-free zones (PFZs) and the coarsened and elongated rod-like particles inside were identified as γ' precipitates. The mechanism of strain-induced grain boundary migration and the discontinuous coarsening reaction is proposed for the formation of PFZs. Furthermore, PFZs were considered to be potential crack sources during the creep rupture test, leading to earlier failure of the material.

An extended mesoscaled M-K model for predicting discrete ultimate strains and evaluating the effect of normal stress on forming limits

Due to the size effect, the formability of metal sheets diminishes and the ultimate strain points exhibit a more discrete distribution. This poses a huge challenge to the process design and manufacture of microparts. This paper presents a modified mesoscopic scale forming limit model based on the M-K model. The model extends the forming limit curve (FLC) into forming limit bands to predict the discrete nature of the ultimate strain. In addition, a method for applying normal stresses to increase the forming limit of thin metal sheets is proposed. The effects of normal stresses on surface roughening and flow stresses in thin metal sheets were evaluated and quantified by uniaxial tensile experiments under normal stresses. The results show that the normal stress suppresses the increase of surface roughness and improves the work hardening rate. The effect of normal stresses on forming limits was obtained using a constructed model. The results show that the predictions of the proposed model are able to include almost all the ultimate strain points. The forming limit bands are more inclusive and safer than the single forming limit curve. In addition, the forming limits of thin plates under normal stresses are significantly increased, which is attributed to the suppression of surface roughening and the increase of strain hardening index. The forming limit bands proposed in this paper provide a more reliable form of forming limit assessment for thin metal plates. The proposed method of applying normal stresses provides a robust and inexpensive process route to address the challenge of poor formability of thin metal sheets.

Estimation and inference of multi-effect generalized geographically and temporally weighted regression models

Geographically and temporally weighted regression (GTWR) models have been an effective tool for exploring spatiotemporal heterogeneity of regression relationships. However, they cannot effectively model such response variables that follows discrete distributions. In this study, we first extend the distributions of response variables to one-parameter exponential family of distributions and formulate generalized geographically and temporally weighted regression (GGTWR) models with their unilaterally temporally weighted maximum likelihood estimation method. Furthermore, we propose so-called multi-effect GGTWR (MEGGTWR) models in which spatiotemporally varying, constant, temporally varying, and spatially varying coefficients may simultaneously be included to reflect different effects of explanatory variables. A coefficient-average-based estimation method is suggested to calibrate MEGGTWR models and a generalized likelihood ratio statistic based test is formulated to identify the types of coefficients. Simulation studies are then conducted to assess the performance of the proposed estimation and inference methods with the impact of multicollinearity among explanatory variables also examined. The results show that the estimation method for MEGGTWR models can accurately estimate various types of coefficients and the test method is of valid type I error and satisfactory power. Finally, the relationship between childhood hand, foot, and mouth disease cases and climate factors is analyzed by the proposed models with their estimation and inference methods and some interesting spatiotemporal patterns are uncovered.

Poole-Frenkel conduction in CdS single-layered and CdS/SnS2 heterojunction electrode system

In this communication, the prevalence of Poole-Frenkel conduction mechanism in two distinct semiconductor systems, CdS single-layered and CdS/SnS2 heterojunction electrode systems, is reported. X-ray diffraction (XRD) exhibits the formation of CdS quantum dots (QDs). A High resolution transmission electron microscopy (HRTEM) shows a discrete particle distribution of SnS2, tends to assemble into nanosheets. Poole-Frenkel conduction arises due to the trap distribution of CdS dots, modified by SnS2 sheets. Furthermore, the formation of heterojunctions with SnS2 shows promising enhancement in charge transport, characterized by reduced trap density and improved conductivity compared pristine CdS. The findings provide valuable insights into the fundamental charge transport processes in CdS/SnS2 system and offer potential avenues for optimizing the performance of electronic devices.