Family Of Probability Distributions Research Articles

Strategic Model for Yellow Hydrogen Production Using the Metalog Family of Probability Distributions

Storing energy in hydrogen has been recognized by scientists as one of the most effective ways of storing energy for many reasons. The first of these reasons is the availability of technology for producing hydrogen from water using electrolytic methods. Another aspect is the availability of relatively cheap energy from renewable energy sources. Moreover, you can count on the availability of large amounts of this energy. The aim of this article is to support the decision-making processes related to the production of yellow hydrogen using a strategic model which exploits the metalog family of probability distributions. This model allows us to calculate, with accuracy regarding the probability distribution, the amount of energy produced by photovoltaic systems with a specific peak power. Using the model in question, it is possible to calculate the expected amount of electricity produced daily from the photovoltaic system and the corresponding amount of yellow hydrogen produced. Such a strategic model may be appropriate for renewable energy developers who build photovoltaic systems intended specifically for the production of yellow and green hydrogen. Based on our model, they can estimate the size of the photovoltaic system needed to produce the assumed hydrogen volume. The strategic model can also be adopted by producers of green and yellow hydrogen. Due to precise calculations, up to the probability distribution, the model allows us to calculate the probability of providing the required energy from a specific part of the energy mix.

A disaggregated integer L-shaped method for stochastic vehicle routing problems with monotonic recourse

This paper proposes a new integer L-shaped method for solving two-stage stochastic integer programs whose first-stage solutions can decompose into disjoint components, each one having a monotonic recourse function. In a minimization problem, the monotonicity property stipulates that the recourse cost of a component must always be higher or equal to that of any of its subcomponents. The method exploits new types of optimality cuts and lower bounding functionals that are valid under this property. The stochastic vehicle routing problem is particularly well suited to be solved by this approach, as its solutions can be decomposed into a set of routes. We consider the variant with stochastic demands in which the recourse policy consists of performing a return trip to the depot whenever a vehicle does not have sufficient capacity to accommodate a newly realized customer demand. This work shows that this policy can lead to a non-monotonic recourse function, but that the monotonicity holds when the customer demands are modeled by several commonly used families of probability distributions. We also present new problem-specific lower bounds on the recourse that strengthen the lower bounding functionals and significantly speed up the solving process. Computational experiments on instances from the literature show that the new approach achieves state-of-the-art results.

Topp-Leone Exponentiated Burr XII Distribution: Theory and Application to Real-Life Data Sets

This paper introduces a novel compound distribution, named the Topp Leone Exponentiated Burr XII (TLEBXII) Distribution, aimed at generating a more flexible family of probability distributions. By taking the Exponentiated Burr XII distribution as the base and the Topp Leone exponentiated G family as the generator, this new distribution seeks to enhance modeling capabilities in various statistical applications. We present general expressions for the density and distribution functions of the TLEBXII model and explore its fundamental properties, including survival and hazard rate functions, moments, moment-generating functions, and entropy. Additionally, we discuss the methodology for estimating the model parameters through maximum likelihood estimation. The utility and superior performance of the TLEBXII distribution are demonstrated through its application to two real datasets, where it shows significant improvements over existing models. This paper underscores the potential of the TLEBXII distribution in providing a robust tool for statistical modeling and analysis.

The Inverse of Exact Renormalization Group Flows as Statistical Inference.

We build on the view of the Exact Renormalization Group (ERG) as an instantiation of Optimal Transport described by a functional convection-diffusion equation. We provide a new information-theoretic perspective for understanding the ERG through the intermediary of Bayesian Statistical Inference. This connection is facilitated by the Dynamical Bayesian Inference scheme, which encodes Bayesian inference in the form of a one-parameter family of probability distributions solving an integro-differential equation derived from Bayes' law. In this note, we demonstrate how the Dynamical Bayesian Inference equation is, itself, equivalent to a diffusion equation, which we dub Bayesian Diffusion. By identifying the features that define Bayesian Diffusion and mapping them onto the features that define the ERG, we obtain a dictionary outlining how renormalization can be understood as the inverse of statistical inference.

A novel compound G family with properties, copulas, modelling the asymmetric and positive-skewed engineering and medical data

It is essential to use compound families in statistical modeling for asymmetric-positive-skewed reliability and relief in real-life data for a number of reasons, including the capacity to capture underlying patterns and characteristics in the data more accurately. The current work introduces a novel compound and continuous G family of probability distributions characterized by two parameters, denoted as the compounded two-parameters general Rayleigh geometric family of probabilistic distributions. The research involves rigorous calculations to ascertain and analyze the relevant mathematical properties of this newly proposed family. We present aging notions under the new family, providing corresponding density, hazard, and survival functions. Notably, a specific emphasis is placed on the examination of the reciprocal-Weibull baseline model, both from a mathematical and statistical perspective, in a dedicated section of the study. To generate various multivariate and bivariate type distributions, the copula method is employed. These newly derived models hold significant utility for effectively modeling the multivariate and bivariate data. The study showcases the wide applicability in different sides under real-life datasets. The presented applications serve as concrete demonstrations of the family's effectiveness and versatility in addressing real-world scenarios.

A novel flexible exponent power-X family of distributions with applications to COVID-19 mortality rate in Mexico and Canada.

This paper aims to introduce a novel family of probability distributions by the well-known method of the T-X family of distributions. The proposed family is called a "Novel Generalized Exponent Power X Family" of distributions. A three-parameters special sub-model of the proposed method is derived and named a "Novel Generalized Exponent Power Weibull" distribution (NGEP-Wei for short). For the proposed family, some statistical properties are derived including the hazard rate function, moments, moment generating function, order statistics, residual life, and reverse residual life. The well-known method of estimation, the maximum likelihood estimation method is used for estimating the model parameters. Besides, a comprehensive Monte Carlo simulation study is conducted to assess the efficacy of this estimation method. Finally, the model selection criterion such as Akaike information criterion (AINC), the correct information criterion (CINC), the Bayesian information criterion (BINC), the Hannan-Quinn information criterion (HQINC), the Cramer-von-Misses (CRMI), and the ANDA (Anderson-Darling) are used for comparison purpose. The comparison of the NGEP-Wei with other rival distributions is made by Two COVID-19 data sets. In terms of performance, we show that the proposed method outperforms the other competing methods included in this study.

A New Modified-X family of distributions with applications in modeling biomedical data

In the present article, we derive a new Modified-X (NM-X) family of probability distributions by using a popular approach known as the T-X method. For the NM-X family, some of the main distributional (or statistical) properties such as quantile function, identifiability property, order statistics, residual life and reverse residual life, moments, and moments generating function are determined. A well-known method of estimation such as the Maximum Likelihood estimation method is used for estimating the model parameters of the introduced family of distributions. A Monte Carlo simulation method is incorporated to judge the performance of the estimators with different selected values of parameters. To check the tails behaviors of the proposed model, the risk measurements such as value-at-risk (VaR), tailed-value-at-risk (TVaR), tailed-variance (TV), and tailed-variance-premium (TVP) are investigated. For illustrative purposes, a special sub-model of the proposed class is derived and named a New Modified Weibull distribution (abbreviated as the NM-Weib). Furthermore, four applications from the field of medicine are analyzed to demonstrate the application, and effectiveness of the NM-Weib distribution. By adopting different diagnostic criteria, the NM-Weib distribution is compared with Weibull (Weib) distribution, Alpha Power Transformed-Weib (APTra-Weib) distribution, new Exponent-Weib (NExpo-Weib) distribution, Flexible Reduced Logarithmic-Weib (FRLog-Weib) distribution, Marshal Olkin-Weib (MO-Weib) distribution, and Kumaraswamy-Weib (K-Weib) distribution. Finally, based on the diagnostic criteria, it is observed that the NM-Weib distribution provides a suitable and best fit for each considered biomedical data set.

Strategic Model for Charging a Fleet of Electric Vehicles with Energy from Renewable Energy Sources

The ever-growing number of electric vehicles requires increasing amounts of energy to charge their traction batteries. Electric vehicles are the most ecological when the energy for charging them comes from renewable energy sources. Obtaining electricity from renewable sources such as photovoltaic systems is also a way to reduce the operating costs of an electric vehicle. However, to produce cheap electricity from renewable energy sources, you first need to invest in the construction of a photovoltaic system. The article presents a strategic model for charging a fleet of electric vehicles with energy from photovoltaic systems. The model is useful for sizing a planned photovoltaic system to the energy needs of a vehicle fleet. It uses the Metalog family of probability distributions to determine the probability of producing a given amount of energy needed to power electric vehicle chargers. Using the model, it is possible to determine the percentage of energy from photovoltaic systems in the total energy needed to charge a vehicle fleet. The research was carried out on real data from an operating photovoltaic system with a peak power of 50 kWp. The approach presented in the strategic model takes into account the geographical and climatic context related to the location of the photovoltaic system. The model can be used for various renewable energy sources and different sizes of vehicle fleets with different electricity demands to charge their batteries. The presented model can be used to manage the energy produced both at the design stage of the photovoltaic system and during its operation.

Accounting for hydroclimatic properties in flood frequency analysis procedures

Abstract. Flood hazard is typically evaluated by computing extreme flood probabilities from a flood frequency distribution following nationally defined procedures in which observed peak flow series are fit to a parametric probability distribution. These procedures, also known as flood frequency analysis, typically recommend only one probability distribution family for all watersheds within a country or region. However, large uncertainties associated with extreme flood probability estimates (>50-year flood or Q50) can be further biased when fit to an inappropriate distribution model because of differences in the tails between distribution families. Here, we demonstrate that hydroclimatic parameters can aid in the selection of a parametric flood frequency distribution. We use L-moment diagrams to visually show the fit of gaged annual maxima series across the United States, grouped by their Köppen climate classification and the precipitation intensities of the basin, to a general extreme value (GEV), log normal 3 (LN3), and Pearson 3 (P3) distribution. Our results show that in real space basic hydroclimatic properties of a basin exert a significant influence on the statistical distribution of the annual maxima. The best-fitted family distribution shifts from a GEV towards an LN3 distribution across a gradient from colder and wetter climates (Köppen group D, continental climates) towards more arid climates (Köppen group B, dry climates). Due to the diversity of hydrologic processes and flood-generating mechanisms among watersheds within large countries like the United States, we recommend that the selection of distribution model be guided by the hydroclimatic properties of the basin rather than relying on a single national distribution model.

Parametric Information Geometry with the Package Geomstats

We introduce the information geometry module of the Python package Geomstats. The module first implements Fisher–Rao Riemannian manifolds of widely used parametric families of probability distributions, such as normal, gamma, beta, Dirichlet distributions, and more. The module further gives the Fisher–Rao Riemannian geometry of any parametric family of distributions of interest, given a parameterized probability density function as input. The implemented Riemannian geometry tools allow users to compare, average, interpolate between distributions inside a given family. Importantly, such capabilities open the door to statistics and machine learning on probability distributions. We present the object-oriented implementation of the module along with illustrative examples and show how it can be used to perform learning on manifolds of parametric probability distributions.

A New Approach of Presenting Lindley Expontiated Gumbel Distribution with Application to Environmental Data

The Lindley exponentiated Gumbel (LEGu) Distribution, a new family of probability distribution developed for modelling environmental data is introduced in this research paper. The investigation defines the new distribution's basic statistical properties, such as its shape, density function, hazard rate function, moment generation function, and maximum likelihood estimates of its model parameters. The study shows that, in comparison to other baseline distributions and comparators, increasing the number of parameters in the distribution improves its robustness and adaptability, making it more flexible in accommodating different types of data. The model is now more adaptable and can handle a wider range of data sets, making it a useful and effective tool for environmental modelling. The creation of the Lindley exponentiated Gumbel Distribution and the statistical qualities that go along with it bring up new possibilities for the analysis of environmental data, making a significant contribution to the larger initiatives in climate research and decision-making. The results of this study have the potential to advance our comprehension of environmental mechanisms and raise the precision of climate assessments and projections.

A New Sine-Based Distributional Method with Symmetrical and Asymmetrical Natures: Control Chart with Industrial Implication

Control charts are widely used in quality control and industrial sectors. Because of their important role, researchers are focusing on the development of new control charts. According to our study, there is no significant amount of published work on control charts using trigonometrically generated distribution methods. In this paper, we contribute to this interesting research gap by developing a new control chart using a sine-based distributional method. The proposed distributional method (or family of probability distributions) may be called a new modified sine-G family of distributions. Based on the new modified sine-G method, a novel modification of the Weibull distribution, namely, a new modified sine-Weibull distribution, is introduced. The new modified sine-Weibull distribution is flexible enough to capture symmetrical and asymmetrical behaviors of its density function. An industrial application is considered to show the importance and implacability of the proposed distribution in quality control. Based on the proposed model, an attribute control chart is developed under a truncated life test. The control chart limits (ARLs) are also computed for the proposed model. The ARLs of the proposed control chart are compared with the attribute control chart of the Weibull distribution. The results show that the developed chart is more efficient than the existing attribute control chart for the Weibull distribution.

Sine Type II Topp-Leone G Family of Probability Distribution: Mathematical Properties and Application

In this paper we introduced a new trigonometric family of continuous distributions called the Sine Type II Topp-Leone-G family of distributions. The new proposed family is more flexible than some of the existing families. The models of the family have many applications in physics, engineering, and medicine. Some statistical properties of the Sine Type II Topp-Leone-G family of distributions such as moments, moment generating function, quantile function and order statistics are derived. Maximum likelihood estimation method was used to estimate parameters of the model. Two real life datasets were analyzed to show the flexibility of the proposed model and the results of the analysis showed that, the proposed model was more efficient and best fit the data sets than its competitors.

Selection of the photovoltaic system power for the electric vehicle

The article presents a method of a photovoltaic system power selecting for proper electric vehicle. At the beginning, the characteristics of the electric vehicle itself and its traction batteries were made in order to determine the method of its charging. In the presented studies, measurement data from the operation of real photovoltaic systems was recorded and processed. The algorithm used includes examining the energy consumption profile of the owner's residential building. The combined demand for electricity of the electric vehicle and the building made it possible to determine the planned photovoltaic system. The authors presented three possibilities of its location. It can be installed on the roof of the building, on the ground next to the building or on the carport under which an electric vehicle can be parked. Finally, the Metalog family of probability distributions was used to analytically validate the power choice of the photovoltaic system. The authors have developed an algorithm using human and artificial intelligence that helps to properly select the power of the photovoltaic system for the vehicle.

A Method of Assessing the Selection of Carport Power for an Electric Vehicle Using the Metalog Probability Distribution Family

This article presents a method for assessing the selection of carport power for an electric vehicle using the Metalog probability distribution family. Carports are used to generate electricity and provide shade for vehicles parked underneath them. On the roof of the carport, there is a photovoltaic system consisting of photovoltaic panels and an inverter. An inverter with Internet of Things functions generates data packets which describe the operation of the entire system at certain intervals and sends them via wireless transmission to a cloud server. The transmitted data can be processed offline and used to determine the charging capacity of individual electric vehicles. This article presents the use of the Metalog family of distributions to predict the production of electricity by a photovoltaic carport with the accuracy of the probability distribution. Based on the calculations, an electric vehicle was selected that can be charged from the carport.

A Novel Odd Beta Prime-Logistic Distribution: Desirable Mathematical Properties and Applications to Engineering and Environmental Data

In parametric statistical modeling, it is important to construct new extensions of existing probability distributions (PDs) that can make modeling data more flexible and help stakeholders make better decisions. In the present study, a new family of probability distributions (FPDs) called the odd beta prime generalized (OBP-G) FPDs is proposed to improve the traditional PDs. A new PD called the odd beta prime-logistic (OBP-logistic) distribution has been developed based on the developed OBP-G FPDs. Some desirable mathematical properties of the proposed OBP-logistic distribution, including the moments, moment-generating function, information-generating function, quantile function, stress–strength, order statistics, and entropies, are studied and derived. The proposed OBP-logistic distribution’s parameters are determined by adopting the maximum likelihood estimation (MLE) method. The applicability of the new PD was demonstrated by employing three data sets and these were compared by the known extended logistic distributions, such as the gamma generalized logistic distribution, new modified exponential logistic distribution, gamma-logistic distribution, exponential modified Weibull logistic distribution, exponentiated Weibull logistic distribution, and transmuted Weibull logistic distribution. The findings reveal that the studied distribution provides better results than the competing PDs. The empirical results showed that the new OBP-logistic distribution performs better than the other PDs based on several statistical metrics. We hoped that the newly constructed OBP-logistic distribution would be an alternative to other well-known extended logistic distributions for the statistical modeling of symmetric and skewed data sets.

Efficient Comparison Of Independence Structures Of Log-Linear Models

Abstract Log-linear models are a family of probability distributions which capture relationships between variables. They have been proven useful in a wide variety of fields such as epidemiology, economics and sociology. The interest in using these models is that they are able to capture context-specific independencies, relationships that provide richer structure to the model. Many approaches exist for automatic learning of the independence structure of log-linear models from data. The methods for evaluating these approaches, however, are limited, and are mostly based on indirect measures of the complete density of the probability distribution. Such computation requires additional learning of the numerical parameters of the distribution, which introduces distortions when used for comparing structures. This work addresses this issue by presenting the first measure for the direct and efficient comparison of independence structures of log-linear models. Our method relies only on the independence structure of the models, which is useful when the interest lies in obtaining knowledge from said structure, or when comparing the performance of structure learning algorithms, among other possible uses. We present proof that the measure is a metric, and a method for its computation that is efficient in the number of variables of the domain.

A Goodness-of-Fit Test Based on Fuzzy Random Variables

During the last decades, several methods have been proposed for Kolmogorov−Smirnov one-sample test based on fuzzy random variables to describe the impression of classical random variables. However, such techniques do not discuss the modeling of imprecise observations and simulation of such data from the distribution of a fuzzy random variable. Moreover, such methods rely on a fuzzy cumulative distribution function with known parameters. In this paper, however, a modified Kolmogorov−Smirnov one-sample test is introduced based on a novel notion of fuzzy random variables which comes down to model fuzziness and randomness in the distribution of population in a frequently used family of probability distributions called location and scale distribution functions. A method of moment estimator was also utilized to estimate the location and scale parameters. Then, a notion of non-fuzzy Kolmogorov−Smirnov one-sample test was developed based on fuzzy hypotheses. Monte Carlo simulation was also employed to evaluate the critical value corresponding to a significance level and the performance of the test using power studies. Comparing the observed test statistics and the given fuzzy significance level, a classical procedure was finally used to accept or reject the null fuzzy hypothesis. Two numerical examples including a simulation study and an applied example were provided to clarify the discussions in this paper. The proposed method was also compared with some existing methods. The goodness-of-fit results demonstrated that the proposed Kolmogorov−Smirnov provides an efficient tool to handle statistical inference fuzzy observations.

Discrete mixture representations of spherical distributions

We obtain discrete mixture representations for parametric families of probability distributions on Euclidean spheres, such as the von Mises–Fisher, the Watson and the angular Gaussian families. In addition to several special results we present a general approach to isotropic distribution families that is based on density expansions in terms of special surface harmonics. We discuss the connections to stochastic processes on spheres, in particular random walks, discrete mixture representations derived from spherical diffusions, and the use of Markov representations for the mixing base to obtain representations for families of spherical distributions.

Invariant measures for stochastic delay nonlocal lattice dynamical systems

This paper deals with the dynamics of the stochastic delay lattice systems with fractional discrete Laplacian driven by nonlinear noise. The existence of a invariant measure for the stochastic delay nonlocal lattice systems is established under certain conditions. The idea of the uniform estimates on the tails of the solutions, the technique of diadic division and the Arzela-Ascoli theorem are employed to show the tightness of a family of probability distributions of the solutions.