Discrete Probability Research Articles

Probabilistic cellular automata simulation of microstructure evolution: the role of model parameters on precision and uncertainty

Abstract Probabilistic cellular automata (PCA) is a widely used and cost-efficient method for simulating microstructural evolution. In this method, probabilistic state change rules determine the evolution of cell states at each time step. However, its stochastic nature introduces inherent uncertainty, leading to non-repeatable results. Most microstructural simulation studies assess the accuracy of simulations by comparing predicted results with experimental observations, neglecting uncertainties in mathematical models and algorithms. In this study, the precision and stochastic behavior of microstructure evolution in the PCA simulations were investigated. The probabilities of transformations of cell states at each time step were formulated, and discrete probability distribution functions (dPDF) were introduced to analyze the frequency distribution of simulation outcomes. The performance and consistency of these dPDFs were assessed by comparing statistical analyzes of PCA simulation results with dPDF predictions, revealing that the variance of simulation results is less than that of the binomial distribution function. Additionally, the effects of modeling parameters, such as model size, cellular resolution, and probability distribution of state changes in two- and three-dimensional PCA modeling, on the precision and reliability of simulation results were studied. In PCA models, simulation uncertainty inversely relates to the square root of model size. Furthermore, in 2D simulations, uncertainty is inversely proportional to the square root of the cellular resolution, while in 3D simulations, it is inversely proportional to the cellular resolution itself. These findings provide a simple and computationally efficient method for evaluating PCA simulation uncertainty and determining optimal simulation parameters, including model size, cellular resolution, and dimensionality.

Completions to discrete probability distributions in log-linear models

Completions to discrete probability distributions in log-linear models

On the degree of uniformity measure for probability distributions

Abstract A key challenge in studying probability distributions is quantifying the inherent inequality within them. Certain parts of the distribution have higher probabilities than others, and our goal is to measure this inequality using the concept of mathematical diversity, a novel approach to examining inequality. We introduce a new measure m D (P), called the degree of uniformity measure on a given probability space that generalizes the idea of the slope of secant of the slope of diversity curve. This measure generalizes the idea of degree of uniformity of a contiguous part (P = {k 1, k 2} in the discrete case or P = (a, b) in the continuous case) in a probability space related to a random variable X, to an arbitrary measurable part P. We also demonstrate the truly scale free and self-contained nature of the concept of degree of uniformity by relating the measure of two parts P 1 and P 2 from completely unrelated distributions with random variables X 1 and X 2 that have completely different scales of variation.

Reliability assessment of distribution networks with medium-voltage and low-voltage flexible interconnections considering graded transfer

Abstract For flexible interconnection, flexible interconnected devices (FID) are utilized to upgrade or build connection nodes, which can provide more ways for load recovery after faults, thereby improving the reliability level of the distribution network. Based on the characteristics of connection switch, medium-voltage (MV), and low-voltage (LV) flexible interconnection transfer methods, a reliability assessment method is proposed in this paper for medium-voltage and low-voltage flexible interconnection distribution networks (MLFIDN) considering graded transfer. First, a multi-state reliability model of the LV substation area and discrete probability distribution of DG output are established based on the system structure and composition. Then, the fault impact analysis based on graded transfer mode is proposed. A transfer recovery model that takes into account power balance, the node voltage, line current safety constraints, and flexible interconnection operation constraints is established. On this basis, a simulation method based on non-sequential Monte Carlo sampling is used for system reliability assessment. Finally, the effectiveness of the proposed method is verified by the IEEE-33 distribution network with MV and LV flexible interconnection reliability assessment. The reliability assessment results can quantify the improvement of flexible interconnection on the reliability of distribution networks.

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

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.

Mini-Workshop: Permutation Patterns

The study of permutation patterns has recently seen several surprising results, and the purpose of this mini-workshop was to bring together researchers from across the field to focus on four hot topics related to these recent developments. The topics covered the nature of generating functions that enumerate permutation classes, the structure of permutation classes and the impact this has on their growth rates, and the study of permutons, which lies at the interface of permutation patterns and discrete probability. The workshop offered an opportunity for knowledge exchange, but also time and space to initiate group collaborations on open problems related to these topics.

A Sparse Smoothing Newton Method for Solving Discrete Optimal Transport Problems

The discrete optimal transport (OT) problem, which offers an effective computational tool for comparing two discrete probability distributions, has recently attracted much attention and played essential roles in many modern applications. This paper proposes to solve the discrete OT problem by applying a squared smoothing Newton method via the Huber smoothing function for solving the corresponding KKT system directly. The proposed algorithm admits appealing convergence properties and is able to take advantage of the solution sparsity to greatly reduce computational costs. Moreover, the algorithm can be extended to solve problems with similar structures, including the Wasserstein barycenter (WB) problem with fixed supports. To verify the practical performance of the proposed method, we conduct extensive numerical experiments to solve a large set of discrete OT and WB benchmark problems. Our numerical results show that the proposed method is efficient compared to state-of-the-art linear programming (LP) solvers. Moreover, the proposed method consumes less memory than existing LP solvers, which demonstrates the potential usage of our algorithm for solving large-scale OT and WB problems.

Generalised Poisson-new linear-exponential distribution

It is a compound discrete probability distribution which has two parameters and whose particular case is Poisson-New Linear-Exponential Distribution (PNLED) [1]. The statistical characteristics required for this distribution such as probability mass function (pmf), statistical moments, estimation of parameters and goodness of fit have been derived and explained nicely. To test theoretical reliability of this distribution, goodness of fit has been applied to some over-dispersed data which were used by other researchers and further, we found that this distribution looks more appropriate for statistical modelling than PNLED, generalised Poisson-Lindley distribution (GPLD) [1, 2] and Poisson-Lindley distribution (PLD) [3].

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.

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.

Ultrasound normalized cumulative residual entropy imaging: Theory, methodology, and application

Background and Objective:Ultrasound information entropy imaging is an emerging quantitative ultrasound technique for characterizing local tissue scatterer concentrations and arrangements. However, the commonly used ultrasound Shannon entropy imaging based on histogram-derived discrete probability estimation suffers from the drawbacks of histogram settings dependence and unknown estimator performance. In this paper, we introduced the information-theoretic cumulative residual entropy (CRE) defined in a continuous distribution of cumulative distribution functions as a new entropy measure of ultrasound backscatter envelope uncertainty or complexity, and proposed ultrasound CRE imaging for tissue characterization. Methods:We theoretically analyzed the CRE for Rayleigh and Nakagami distributions and proposed a normalized CRE for characterizing scatterer distribution patterns. We proposed a method based on an empirical cumulative distribution function estimator and a trapezoidal numerical integration for estimating the normalized CRE from ultrasound backscatter envelope signals. We presented an ultrasound normalized CRE imaging scheme based on the normalized CRE estimator and the parallel computation technique. We also conducted theoretical analysis of the differential entropy which is an extension of the Shannon entropy to a continuous distribution, and introduced a method for ultrasound differential entropy estimation and imaging. Monte-Carlo simulation experiments were performed to evaluate the estimation accuracy of the normalized CRE and differential entropy estimators. Phantom simulation and clinical experiments were conducted to evaluate the performance of the proposed normalized CRE imaging in characterizing scatterer concentrations and hepatic steatosis (n = 204), respectively. Results:The theoretical normalized CRE for the Rayleigh distribution was π/4, corresponding to the case where there were ≥10 randomly distributed scatterers within the resolution cell of an ultrasound transducer. The theoretical normalized CRE for the Nakagami distribution decreased as the Nakagami parameter m increased, corresponding to that the ultrasound backscattered statistics varied from pre-Rayleigh to Rayleigh and to post-Rayleigh distributions. Monte-Carlo simulation experiments showed that the proposed normalized CRE and differential entropy estimators can produce a satisfying estimation accuracy even when the size of the test samples is small. Phantom simulation experiments showed that the proposed normalized CRE and differential entropy imaging can characterize scatterer concentrations. Clinical experiments showed that the proposed ultrasound normalized CRE imaging is capable to quantitatively characterize hepatic steatosis, outperforming ultrasound differential entropy imaging and being comparable to ultrasound Shannon entropy and Nakagami imaging. Conclusion:This study sheds light on the theory and methodology of ultrasound normalized CRE. The proposed ultrasound normalized CRE can serve as a new, flexible quantitative ultrasound envelope statistics parameter. The proposed ultrasound normalized CRE imaging may find applications in quantified characterization of biological tissues. Our code will be made available publicly at https://github.com/zhouzhuhuang.

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.

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.

The Wasserstein Metric between a Discrete Probability Measure and a Continuous One

This paper examines the Wasserstein metric between the empirical probability measure of n discrete random variables and a continuous uniform measure in the d-dimensional ball, providing an asymptotic estimation of their expectations as n approaches infinity. Furthermore, we investigate this problem within a mixed process framework, where n discrete random variables are generated by the Poisson process.

Generating Chaos in Dynamical Systems: Applications, Symmetry Results, and Stimulating Examples

In this paper, we present a new class of extended oscillators in light of chaos theory. It is based on dynamical complex systems built on the concept of self-describing with a stopping criterion process. We offer an effective studying approach with a specific focus on learning, provoking students’ thinking through the triad of enigmatics–creativity–acmeology. Dynamic processes are the basis of mathematical modeling; thus, we can reach the goal of the above-mentioned triad by the proposed differential systems. The results we derive strongly confirm the presence of symmetry in the outcomes of the proposed models. We suggest a stochastic approach to structuring the proposed dynamical systems by modeling the coefficients that drive them by some discrete probability distribution that exhibits symmetry or asymmetry. We propose specific tools for researching the behavior of these systems.

Subdivision scheme for discrete probability measure-valued data

This paper deals with the approximation of discrete probability measure-valued data by a new subdivision scheme. Its construction relies on a coupling between linear subdivision and optimal transport. A mathematical analysis is performed to study its convergence. Two test cases are finally described to emphasize its capability: the first one is related to point cloud interpolation while the second one is a first attempt in the framework of image approximation.

Yet Another Attempt to Classify Positive Univariate Probability Distributions

We propose an original classification of several discrete and continuous probability distributions. We establish links between these distributions, in particular the little known relationship between the negative hypergeometric distribution and the beta distribution. These relations allow us to propose a structure of relations which is summarized in graphic form.Our classification emphasises the analogy between certain discrete and continuous distributions. This analogy makes it possible to establish relations between the theory of point processes and the theory of survey sampling. It also makes it possible to envisage the use of link functions that are little used in generalised regression.

Construction and application of a risk index of Echinococcus infection based on the classification of echinococcosis lesions

To investigate the feasibility of constructing the risk index of Echinococcus infection based on the classification of echinococcosis lesions, so as to provide insights into the management of echinococcosis. The imaging data of echinococcosis cases were collected from epidemiological surveys of echinococcosis in China from 2012 to 2016, and the detection of incident echinococcosis cases was captured from the annual echinococcosis prevention and control reports across provinces (autonomous regions) and Xinjiang Production and Construction Corps in China from 2017 to 2022. After echinococcosis lesions were classified, a risk index of Echinococcus infection was constructed based on the principle of discrete distribution marginal probability and multi-group classification data tests. The correlation between the risk index of Echinococcus infection and the detection of incident echinococcosis cases was evaluated in the provinces (autonomous regions and corps) from 2017 to 2022, and the correlations between the short and medium-term risk indices and between the medium and long-term risk indices of Echinococcus infection were examined using a univariate linear regression model. A total of 4 014 echinococcosis cases in China from 2012 to 2016 were included in this study. The short-, medium- and long-term risk indices of E. granulosus infection varied in echinococcosis-endemic provinces (autonomous regions and corps) of China (χ2 = 4.12 to 708.65, all P values < 0.05), with high short- (0.058), medium- (0.137) and long-term risk indices (0.104) in Tibet Autonomous Region, and the short-, medium- and long-term risk indices of E. multilocularis infection varied in echinococcosis-endemic provinces (autonomous regions and corps) of China (χ2 = 6.74 to 122.60, all P values < 0.05), with a high short-term risk index in Sichuan Province (0.016) and high medium- (0.009) and long-term risk indices in Qinghai Province (0.018). There were no significant correlations between the risk index of E. granulosus infection and the detection of incident cystic echinococcosis cases during the study period (t = -0.518 to 2.265, all P values > 0.05), and strong correlations were found between the risk indices of E. multilocularis infection and the detection of incident alveolar echinococcosis cases (including mixed type) in 2018, 2020, 2021, 2022, during the period from 2017 through 2020, from 2017 through 2021, from 2017 through 2022 (all r values > 0.7, t = 2.521 to 3.692, all P values < 0.05). Linear regression models were established between the risk index of E. multilocular infection and the detection of alveolar echinococcosis cases (including mixed type), and the models were all statistically significant (b = 0.214 to 2.168, t = 2.458 to 3.692, F = 6.044 to 13.629, all P values < 0.05). The regression coefficients for the correlations between the medium- and short-term, and between the long- and medium-term risk indices of E. granulosus infection were 2.339 and 0.765, and the regression coefficients for the correlations between the medium- and short-term, and between the long- and medium-term risk indices of E. multilocular infection were 0.280 and 1.842, with statistical significance seen in both the regression coefficients and regression models (t = 16.479 to 197.304, F = 271.570 to 38 928.860, all P values < 0.05). The risk index of Echinococcus infection has been successfully established based on the classification of echinococcosis lesions, which may provide insights into the prevention and control, prediction, diagnosis and treatment, and classified management of echinococcosis.