Mapping of Beta Distribution for the Study of Dispersed Materials

In the study of polydisperse materials, most of the experimental particle size distributions were obtained on bounded intervals. In these cases, it is also desirable to use bounded models with different shapes to simulate the results of studying polydisperse and powder materials. The beta distribution is often used to approximate results due to the fact that this distribution contains many forms for displaying realizations on a limited interval. With the development of computer technology, there has been an increased interest in the use of beta distribution in the modern practice of analyzing results. Meanwhile, there remains a limitation in the use of the beta distribution that is associated with the choice of distribution shape. The possibilities of using known shape measures for mapping beta distribution in this paper is discusses. On the example of the space of shape measure of kurtosis and skewness, the limited use of only probabilistic measures of shapes is illustrated. It is proposed to use the entropy coefficients as an additional informational parameter of the beta distribution shape. On the base of a features comparison of the entropy coefficients for biased and unbiased beta distributions, recommendations for their application are given. By using the example of beta distributions mapping in the space of asymmetry and the entropy coefficient, it is shown that the synergistic combination of probabilistic and informational measures of the shape allows expanding the possibilities of estimating the shape parameters beta distributions. Two methods to display the positions of realizations of beta distributions is proposed. There are trajectories on a constant ratio of shape and realizations position curve on equal values of one parameter. In particular, the features of the choice of beta distributions with negative skewness are discussed.

Shape measures of generalized beta distributions

This paper presents shape measures for generalized beta distributions that unit many subfamilies of distributions. For the study of complex systems, the information entropy of the whole family of the generalized beta distribution is obtained. The paper uses the interval of entropy uncertainty as an estimate of the entropy uncertainty for probable models, which are given in units of an observable random variable. The entropy uncertainty interval was used to construct the entropy coefficient of unbiased subfamilies of the generalized beta distribution. Particular entropy coefficients are given for frequently used subfamilies of beta distribution, that greatly facilitates the use of coefficients as independent information measures in determining the shape of models. The paper contains the most general formulas for probabilistic measures of the distributions shape also.

Probabilistic measures of HIV-1 transmission in different HIV-1 key population groups of Larkana, Pakistan

Objective: To estimate the probability of human immunodeficiency virus (HIV)-1 transmission from different key HIV population groups using probabilistic modelling. Methods: This study was conducted in December 2020. A probabilistic model was used to estimate the probability of HIV-1 transmission from different key HIV population groups in Larkana. Our model was run on three probabilistic assumptions: 1) each replication gave two conceivable results: ‘true’ or ‘false’; 2) the chance of giving a ‘true’ result is the same for each replication; and 3) the replications are independent - ‘true’ in one will not impact the likelihood of ‘true’ in another. Results: The results estimated the probability of HIV transmission in key HIV population groups in Larkana to range between 0.42–0.54 per trial, where the highest probability of transmission was predicted for men who have sex with men (MSM; 0.54 per trial), followed by transgender (TG; 0.46 per trial) and people who inject drugs (PWID; 0.457 per trial). Conclusion: Our results suggest that there is a high likelihood of HIV transmission by key population groups in Larkana, such as MSM, TG, and PWID. Mathematic models, such as one proposed in our study can aid the HIV and acquired immunodeficiency syndrome (AIDS) control programmes in evaluating and optimising the strategies in controlling transmission of HIV from the key population groups. Continuous...

Communicative efficiency and differential case marking: a reverse-engineering approach

The use of differential case marking of A and P has been explained in terms of efficiency (economy) and markedness. The present study tests predictions based on these accounts, using conditional probabilities of a particular feature given the syntactic role (cue availability), and conditional probabilities of a particular syntactic role given the feature in question (cue reliability). Cue availability serves as a measure of markedness, whereas cue reliability is central for the efficiency account. Similar to reverse engineering, we determine which of the probabilistic measures could have been responsible for the recurrent cross-linguistic patterns described in the literature. The probabilities are estimated from spontaneous informal dialogues in English and Russian (Indo-European), Lao (Tai-Kadai), N||ng (Tuu) and Ruuli (Bantu). The analyses, which involve a series of mixed-effects Poisson models, clearly demonstrate that cue reliability matches the observed cross-linguistic patterns better than cue availability. Thus, the results support the efficiency account of differential marking.

Total Repair Cost Simulation Considering Multiple Probabilistic Measures and Service Life

In this study, the total maintenance cost for public houses in South Korea was analyzed, and the effect of each repair process on the total maintenance cost was evaluated with probabilistic and deterministic methods. In the probabilistic method, quality of repair materials and construction skills were considered in the variability of extended service life through repair, while the deterministic method considered it by simple summation of repair step. The repair cost was analyzed considering the coefficient of variation (COV) of extended service life, so the reasonable total maintenance cost was able to be evaluated. Since the results through the probabilistic method provided a continuous cost line, a reasonable repair strategy was carried out by simply changing the intended service life of the structure. The repair cost was additionally analyzed with constant COV (0.15) of each repair process for considering various situations. The analysis results with a COV of 0.15 exhibited a slightly higher maintenance cost than those with current COV. The total maintenance costs can be adjusted if the initial repair timing is extended to the largest possible extent for the highest-repair-cost process since the total repair cost is dominated by the process with the highest repair cost.

Wind Power Forecasting

The wind power generation depends on wind speed and its derivatives like: wind speed and direction. With consideration of stochastic nature of wind power, this work addresses three main issues: first, it discusses the state of art of energy forecasting with emphasis on wind power forecasting. It provides an overview of different variables on which wind power generation depends and explains various key features regarding the design framework of forecasting models. Second, it performs an assessment, detailed comparison and evaluation of the forecasting performance of various types of models; and third, evaluates the uncertainty of expected outcomes with the help of probabilistic measures.

The Informative Power of all Possible Linear Functionals and the Mean-Square Error in the Diccretization of Solutions of the Diriclet Problem for the Laplace Equation in the Circle

The Dirichlet problem for the Laplace equation in the case of a circle belongs to the classical ones and in various aspects has been the subject of study in various fields of mathematics. Among them are such topics as - "Boundary properties of analytic functions", in the study of which powerful methods of function theories were created and honed, - The Banach problem on the existence of a basis for a class of functions consisting of continuous in a closed circle and analytic in, - Numerical methods, since this problem as a mathematical model describes many real processes. In this article, we consider the discretization problem of solutions of the Dirichlet problem for the Laplace equation in a circle from finite numerical information obtained from the boundary function as a result of applying all possible linear functionals. The optimal order of discretization error is found and the corresponding optimal operator of discretization is constructed. The problem of constructing probabilistic measures on functional classes is also considered. Probabilistic measures on the Korobov 𝐸𝑟 (0, 2𝜋) and Nikolsky 𝐻𝑟 2 (0, 2𝜋) classes are introduced. Two-sided estimates of the mean-square error of discretization the solution of the problem by operator (𝑇𝑁 𝑓) (𝛼, 𝜃) are established.