Discrete Probability Distributions Research Articles

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

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.

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.

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.

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.

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.

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.

Estimating Skellam distribution and regression parameters in Stata

The Skellam distribution is a discrete probability distribution related to the difference between two independent Poisson-distributed random variables. It has been used in a variety of contexts, including sports or supply and demand imbalances in shared transportation. Stata does not support the Skellam distribution or Skellam regression. We present a command, skellamreg, to estimate the parameters of a Skellam distribution and Skellam regression model using Mata’s optimize function.

A Quantum Algorithm Framework for Discrete Probability Distributions With Applications to Rényi Entropy Estimation

A Quantum Algorithm Framework for Discrete Probability Distributions With Applications to Rényi Entropy Estimation

Spectral decomposition of [formula omitted] and Poincaré inequality on a compact interval — Application to kernel quadrature

Motivated by uncertainty quantification of complex systems, we aim at finding quadrature formulas of the form ∫abf(x)dμ(x)=∑i=1nwif(xi) where f belongs to H1(μ). Here, μ belongs to a class of continuous probability distributions on [a,b]⊂R and ∑i=1nwiδxi is a discrete probability distribution on [a,b]. We show that H1(μ) is a reproducing kernel Hilbert space with a continuous kernel K, which allows to reformulate the quadrature question as a kernel (or Bayesian) quadrature problem. Although K has not an easy closed form in general, we establish a correspondence between its spectral decomposition and the one associated to Poincaré inequalities, whose common eigenfunctions form a T-system (Karlin and Studden, 1966). The quadrature problem can then be solved in the finite-dimensional proxy space spanned by the first eigenfunctions. The solution is given by a generalized Gaussian quadrature, which we call Poincaré quadrature.We derive several results for the Poincaré quadrature weights and the associated worst-case error. When μ is the uniform distribution, the results are explicit: the Poincaré quadrature is equivalent to the midpoint (rectangle) quadrature rule. Its nodes coincide with the zeros of an eigenfunction and the worst-case error scales as b−a23n−1 for large n. By comparison with known results for H1(0,1), this shows that the Poincaré quadrature is asymptotically optimal. For a general μ, we provide an efficient numerical procedure, based on finite elements and linear programming. Numerical experiments provide useful insights: nodes are nearly evenly spaced, weights are close to the probability density at nodes, and the worst-case error is approximately O(n−1) for large n.

A Rényi-type quasimetric with random interference detection

This paper introduces a new dissimilarity measure between two discrete and finite probability distributions. The followed approach is grounded jointly on mixtures of probability distributions and an optimization procedure. We discuss the clear interpretation of the constitutive elements of the measure under an information-theoretical perspective by also highlighting its connections with the Rényi divergence of infinite order. Moreover, we show how the measure describes the inefficiency in assuming that a given probability distribution coincides with a benchmark one by giving formal writing of the random interference between the considered probability distributions. We explore the properties of the considered tool, which are in line with those defining the concept of quasimetric—i.e. a divergence for which the triangular inequality is satisfied. As a possible usage of the introduced device, an application to rare events is illustrated. This application shows that our measure may be suitable in cases where the accuracy of the small probabilities is a relevant matter.

Poisson-New Quadratic-Exponential Distribution

This proposed distribution is a discrete compound probability distribution with only one parameter. To get this distribution, Poisson distribution has been mixed with the New Quadratic-Exponential distribution of Sah (2022). Hence, it is named as “Poisson-New Quadratic-Exponential Exponential Distribution (PNLED)”. The important statistical characteristics needed to check the validity of this distribution have been derived and clearly explained. To check the validity of the theoretical works of this distribution, while using goodness of fit on some over-dispersed count data, what we have been found that this distribution seems a better alternative of Poisson-Lindley distribution (PLD) of Sankaran (1970), Poisson Mishra distribution (PMD) of Sah (2017) and Poisson-Modified Mishra distribution (PMMD) of Sah and Sahani (2023).

Particle filtering supported probability density estimation of mobility patterns

This paper presents a methodology that aims to enhance the accuracy of probability density estimation in mobility pattern analysis by integrating prior knowledge of system dynamics and contextual information into the particle filter algorithm. The quality of the data used for density estimation is often inadequate due to measurement noise, which significantly influences the distribution of the measurement data. Thus, it is crucial to augment the information content of the input data by incorporating additional sources of information beyond the measured position data. These other sources can include the dynamic model of movement and the spatial model of the environment, which influences motion patterns. To effectively combine the information provided by positional measurements with system and environment models, the particle filter algorithm is employed, which generates discrete probability distributions. By subjecting these discrete distributions to exploratory techniques, it becomes possible to extract more certain information compared to using raw measurement data alone. Consequently, this study proposes a methodology in which probability density estimation is not solely based on raw positional data but rather on probability-weighted samples generated through the particle filter. This approach yields more compact and precise modeling distributions. Specifically, the method is applied to process position measurement data using a nonparametric density estimator known as kernel density estimation. The proposed methodology is thoroughly tested and validated using information-theoretic and probability metrics. The applicability of the methodology is demonstrated through a practical example of mobility pattern analysis based on forklift data in a warehouse environment.

Premium Linear-Exponential Mixture of Poisson Distribution

The proposed distribution is a discrete probability distribution with only one parameter and it has been derived by compounding the Poisson distribution with the Premium Linear-exponential distribution (PLED) of Sah (2022). So, it is named as Premium Linear-Exponential Mixture of Poisson Distribution (PLEMPD). The required statistical characteristics have been derived and explained to give the theoretical concept of this distribution. The main purpose of proposing this distribution is to provide a better goodness of fit for over-dispersed count data than the previously derived one parameter continuous mixtures of Poisson distribution of same nature by others. Taking some secondary over-dispersed count data and using the theoretical concept of this distribution, it has been observed that this distribution is more suitable for statistical modeling of over-dispersed count data than Poisson-Lindley distribution (PLD) of Sankaran (1970) and Poisson-Mishra distribution (PMD) of Sah (2017).

이산확률분포의 표본 분위수 계산 방법에 관한 연구

In this paper, we studied the methods of computing sample quantiles for the case of discrete probability distributions. There are two main methods which are introduced in statistics textbooks. We compared these two methods by simulation from practical and educational necessity. We considered some cases of Poisson, binomial, geometric, negative binomial, and discrete uniform distributions by setting up the parameters, and for each distribution we performed 10,000 times of simulation of drawing random samples of size 20 and 50. At each time of simulation, we computed the difference between the population quantile and the sample quantile obtained by each method. We compared the two methods by using two criteria: one is the mean square error over 10,000 times of simulation, and the other is the frequency of obtaining closer sample quantile to the population quantile than the competing method. We also obtained the estimated probabilities of exact estimation of population quantiles for the two methods, and performed the tests of homogeneity on the probability distribution of under-estimation, exact estimation, and over-estimation of the population quantiles.

A Dirichlet process mixture regression model for the analysis of competing risk events

We develop a regression model for the analysis of competing risk events. The joint distribution of the time to these events is flexibly characterized by a random effect which follows a discrete probability distribution drawn from a Dirichlet Process, explaining their variability. This entails an additional layer of flexibility of this joint model, whose inference is robust with respect to the misspecification of the distribution of the random effects. The model is analysed in a fully Bayesian setting, yielding a flexible Dirichlet Process Mixture model for the joint distribution of the time to events. An efficient MCMC sampler is developed for inference. The modelling approach is applied to the empirical analysis of the surrending risk in a US life insurance portfolio previously analysed by Milhaud and Dutang (2018). The approach yields an improved predictive performance of the surrending rates.