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.

Investigation on the Effect of Probability Distribution on the Dynamic Response of Liquefiable Soils

This study investigated the effect of different probabilistic distributions (Lognormal, Gamma, and Beta) to characterize the spatial variability of shear modulus on the soil liquefiable response. The parameter sensitivity analysis included the coefficient of variation and scale of fluctuation of soil shear modulus. The results revealed that the distribution type had no significant influence on the liquefication zone. In particular, the estimation with Beta distribution is the worst scenario. It illuminated that the estimation with Beta distribution can provide a conservative design if site investigation is absent.

Evaluation of several initialization methods on arithmetic optimization algorithm performance

Arithmetic optimization algorithm (AOA) is one of the recently proposed population-based metaheuristic algorithms. The algorithmic design concept of the AOA is based on the distributive behavior of arithmetic operators, namely, multiplication (M), division (D), subtraction (S), and addition (A). Being a new metaheuristic algorithm, the need for a performance evaluation of AOA is significant to the global optimization research community and specifically to nature-inspired metaheuristic enthusiasts. This article aims to evaluate the influence of the algorithm control parameters, namely, population size and the number of iterations, on the performance of the newly proposed AOA. In addition, we also investigated and validated the influence of different initialization schemes available in the literature on the performance of the AOA. Experiments were conducted using different initialization scenarios and the first is where the population size is large and the number of iterations is low. The second scenario is when the number of iterations is high, and the population size is small. Finally, when the population size and the number of iterations are similar. The numerical results from the conducted experiments showed that AOA is sensitive to the population size and requires a large population size for optimal performance. Afterward, we initialized AOA with six initialization schemes, and their performances were tested on the classical functions and the functions defined in the CEC 2020 suite. The results were presented, and their implications were discussed. Our results showed that the performance of AOA could be influenced when the solution is initialized with schemes other than default random numbers. The Beta distribution outperformed the random number distribution in all cases for both the classical and CEC 2020 functions. The performance of uniform distribution, Rayleigh distribution, Latin hypercube sampling, and Sobol low discrepancy sequence are relatively competitive with the Random number. On the basis of our experiments’ results, we recommend that a solution size of 6,000, the number of iterations of 100, and initializing the solutions with Beta distribution will lead to AOA performing optimally for scenarios considered in our experiments.

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.

Multidimensional Poverty Modeling for Namibia Using the Beta Distribution

Despite global efforts in alleviating poverty, many people are still living in poverty. Different methods were employed to estimate poverty with many researchers moving from monetary to multidimensional poverty modeling approach. In Namibia, very few studies have been conducted to estimate poverty in a multidimensional sense. The 2015/2016 Namibia household income and expenditure survey dataset was employed to develop multidimensional poverty indices (MPIs) using beta distribution. We showed that the MPI is equivalent to the mean of the left truncated beta distribution. The results revealed that the northern regions of Namibia are the most affected by multidimensional poverty. The results from this study can be used to identify areas that are severely affected by poverty and consequently form a basis to develop appropriate measures intended to alleviate poverty.

Bayesian two-sided group chain sampling plan for beta binomial distribution under quality regions

PurposeThis paper introduces a Bayesian two-sided group chain sampling plan (BT-SGChSP) by using binomial distribution to estimate the average proportion of defectives. In this Bayesian approach, beta distribution is used as a suitable prior of binomial distribution. The proposed plan considers both consumer's and producer's risks. Currently, group chain sampling plans only consider the consumer's risk and do not account for the producer's risk. All existing plans are used to estimate only a single point, but this plan gives a quality region for the pre-specified values of different design parameters. In other words, instead of point wise description for the designing of sampling plan based on a range of quality by involving a novel approach called quality region.Design/methodology/approachThe methodology is based on five phases, which are (1) operating procedure, (2) derivation of the probability of lot acceptance, (3) constructing plans for given acceptable quality level (AQL) and limiting quality level (LQL), (4) construction of quality intervals for BT-SGChSP and (5) selection of the sampling plans.FindingsThe findings show that the operating characteristic (OC) curve of BT-SGChSP is more ideal than the existing Bayesian group chain sampling plan because the quality regions for BT-SGChSP give less proportion of defectives for same consumer's and producer's risks.Research limitations/implicationsThere are four limitations in this study: first is the use of binomial distribution when deriving the probability of lot acceptance. Alternatively, it can be derived by using distributions such as Poisson, weighted Poisson and weighted binomial. The second is that beta distribution is used as prior distribution. Otherwise, different prior distributions can be used like: Rayleigh, exponential and generalized exponential. The third is that we adopt mean as a quality parameter, whereas median and other quintiles can be used. Forth, this paper considers probabilistic quality region (PQR) and indifference quality region (IQR).Practical implicationsThe proposed plan is an alternative of traditional group chain sampling plans that are based on only current lot information. This plan considers current lot information with preceding and succeeding lot and also considers prior information of the product.Originality/valueThis paper first time uses a tight (three acceptance criteria) and introduces a BT-SGChSP to find quality regions for both producer's and consumer's risk.

Generating the International Knee Documentation Committee Score using PROMIS Computer Adaptive Testing with Multivariable Predictive Models: Reduce Survey Burden with Comparable Data (102)

Objectives: Patient reported outcomes (PROs) serve as a means of measuring improvement and quality of care. Legacy PROs rely on a list of questions that have had to demonstrate accuracy, responsiveness, and validity in testing for intended measurements. While certain legacy PROs such as the International Knee Documentation Committee (IKDC) survey have demonstrated these properties well, a lengthy PRO creates a time burden on patients, making patient adherence and completion a challenge. In recent years, PROs such as the Patient Reported Outcomes Measurement Information System (PROMIS) Physical Function (PF) and Pain Interference (PI) surveys have been developed which leverage computer adaptive testing that produce equivalent accuracy, responsiveness, and validity of legacy PROs, but use only 4-12 questions per survey. This results in significant reduction in time to complete. As these new PROs are now being adopted, the ability to compare outcomes to prior studies that relied on legacy PROs is necessary. While prior studies have examined correlation between legacy PROs and PROMIS computer adaptive tests, no studies to date have developed effective prediction models utilizing PROMIS surveys to create an IKDC index score. The objective of this study was to develop a predictive model utilizing PROMIS PF and PI to effectively recreate IKDC survey scores. Methods: The Military Orthopaedics Tracking Injuries and Outcomes Network (MOTION) database is a prospectively collected repository of patient reported outcomes and intraoperative variables. As part of inclusion in MOTION, research patients who underwent knee surgery were asked to complete the IKDC as well as the PROMIS PF and PROMIS PI at varying time points. This cohort of patients that completed both IKDC and PROMIS scores were included in the present analysis. Nonlinear multivariable predictive models using both Gaussian and beta distributions were created to establish an IKDC index score, which was then validated using leave-one-out techniques and minimal clinically important difference (MCID) analysis. Results: A total of 1,011 knee patients (Age: 37.7±14.4 years; 30% Female) completed the IKDC, PROMIS PF, and PROMIS PI providing 1,618 complete observations. The algorithms for both the Gaussian and beta distribution were strongly validated to predict the IKDC (Table 1). The MCID for IKDC was 27.0 (95% confidence intervals: 15.0-39.7) whereas the IKDC-index MCIDs for the Gaussian and beta distribution models were both 13.3 (95% confidence intervals 2.7-27.9), suggesting that the derived IKDC-index is effective and can reliably re-create IKDC scores. The surface plots of this nonlinear multivariable model also confirm the necessity that nonlinear prediction is necessary for effective modeling of legacy PRO scores (Figures 1 & 2). Conclusions: PROMIS PF and PI predictive models are able to approximate the IKDC score within 9.3-10.0 points. Given the 27.0 point minimally clinically important difference for the IKDC survey in this cohort, the results of this study can be used to compare PROMIS PF and PI scores to prior IKDC data by creating an IKDC index score. Moreover, repeated use of the IKDC-index from PROMIS PF and PI allows for a lower MCID than using the conventional IKDC survey. PROMIS PF and PI scores can be substituted in both clinical and research settings to reduce patient time burden, increase completion rates, and still create data that can effectively be compared with studies utilizing legacy IKDC scores.

Random Variables and Probability

The chapter “Random Variables and Probability” serves as both a review and a reference on probability. The random variable is the core concept in understanding probability, parameter estimation, and model selection. This chapter reviews the basic idea of a random variable and discusses the two main kinds of random variables: discrete random variables and continuous random variables. It covers the distinction between discrete and continuous random variables and outlines the most common probability mass or density functions used in ecology. Advanced sections cover distributions such as the gamma distribution, Student’s t-distribution, the beta distribution, the beta-binomial distribution, and zero-inflated models.

A CASE STUDY ON ANIMAL BEHAVIOR ANALYSIS USING GAMLSS

Animal behavior studies usually produce large amounts of data and a wide variety of data structures, including nonlinear relationships, interaction effects, nonconstant variance, correlated measures, overdispersion, and zero inflation, among others. We aimed to explore here the potential of generalized additive models for location, scale and shape (GAMLSS) in analyzing data from animal behavior studies. Data from 20 Romane ewes from two genetic lineages submitted to brushing by a familiar observer were analyzed. Behavioral responses through ear posture changes, a count random variable, and the proportion of time to perform the horizontal ear posture, a continuous random variable on the interval (0,1), with non-null probabilities in zero and one, were analyzed. The Poisson, negative binomial, and their zero-inflated and zero-adjusted extensions models were considered for the count data, whereas the beta distribution and its inflated versions were evaluated for the proportions. Random effects were also included to consider the multilevel structure of the experiment. The zero adjusted negative binomial model has better fitted the count data, whereas the inflated beta distribution performed the best for the proportions. Both models allowed us to properly assess the effects of social separation, brushing, and genetic lineages on sheep behavioral. We may conclude that GAMLSS is a flexible framework to analyze animal behavior data.