Beta Distribution Research Articles

Transformer-Based Parameter Estimation in Statistics

Parameter estimation is one of the most important tasks in statistics, and is key to helping people understand the distribution behind a sample of observations. Traditionally, parameter estimation is done either by closed-form solutions (e.g., maximum likelihood estimation for Gaussian distribution) or by iterative numerical methods such as the Newton–Raphson method when a closed-form solution does not exist (e.g., for Beta distribution). In this paper, we propose a transformer-based approach to parameter estimation. Compared with existing solutions, our approach does not require a closed-form solution or any mathematical derivations. It does not even require knowing the probability density function, which is needed by numerical methods. After the transformer model is trained, only a single inference is needed to estimate the parameters of the underlying distribution based on a sample of observations. In the empirical study, we compared our approach with maximum likelihood estimation on commonly used distributions such as normal distribution, exponential distribution and beta distribution. It is shown that our approach achieves similar or better accuracy as measured by mean-square-errors.

Coupled Confusion Correction: Learning from Crowds with Sparse Annotations

As the size of the datasets getting larger, accurately annotating such datasets is becoming more impractical due to the expensiveness on both time and economy. Therefore, crowd-sourcing has been widely adopted to alleviate the cost of collecting labels, which also inevitably introduces label noise and eventually degrades the performance of the model. To learn from crowd-sourcing annotations, modeling the expertise of each annotator is a common but challenging paradigm, because the annotations collected by crowd-sourcing are usually highly-sparse. To alleviate this problem, we propose Coupled Confusion Correction (CCC), where two models are simultaneously trained to correct the confusion matrices learned by each other. Via bi-level optimization, the confusion matrices learned by one model can be corrected by the distilled data from the other. Moreover, we cluster the ``annotator groups'' who share similar expertise so that their confusion matrices could be corrected together. In this way, the expertise of the annotators, especially of those who provide seldom labels, could be better captured. Remarkably, we point out that the annotation sparsity not only means the average number of labels is low, but also there are always some annotators who provide very few labels, which is neglected by previous works when constructing synthetic crowd-sourcing annotations. Based on that, we propose to use Beta distribution to control the generation of the crowd-sourcing labels so that the synthetic annotations could be more consistent with the real-world ones. Extensive experiments are conducted on two types of synthetic datasets and three real-world datasets, the results of which demonstrate that CCC significantly outperforms state-of-the-art approaches. Source codes are available at: https://github.com/Hansong-Zhang/CCC.

A Bayesian approach based on discounting factor for consistency assessment in multi-regional clinical trial

ABSTRACT Multi-regional clinical trial (MRCT) has become an increasing trend for its supporting simultaneous global drug development. After MRCT, consistency assessment needs to be conducted to evaluate regional efficacy. The weighted Z-test approach is a common consistency assessment approach in which the weighting parameter W does not have a good practical significance; the discounting factor approach improved from the weighted Z-test approach by converting the estimation of W in original weighted Z-test approach to the estimation of discounting factor D. However, the discounting factor approach is an approach of frequency statistics, in which D was fixed as a certain value; the variation of D was not considered, which may lead to un-reasonable results. In this paper, we proposed a Bayesian approach based on D to evaluate the treatment effect for the target region in MRCT, in which the variation of D was considered. Specifically, we first took D random instead of fixed as a certain value and specified a beta distribution for it. According to the results of simulation, we further adjusted the Bayesian approach. The application of the proposed approach was illustrated by Markov Chain Monte Carlo simulation.

On bivariate compound exponentiated survival function of the beta distribution: estimation and prediction

A bivariate distribution is a probability distribution that describes the joint behavior of two random variables. It provides information about the simultaneous variation of two variables, allowing us to analyze their relationship and dependencies. This article discussed the bivariate compound exponentiated survival function of the beta distribution. the joint cumulative distribution function and the joint probability density function were found in closed forms. Several characteristics of this distribution have been discussed. The maximum likelihood (ML) estimators of the parameters and two sample ML predictions of the future observations are derived. The Bayes estimators (BEs) of the parameters based on the squared error loss function and two sample Bayesian predictions of the future observations are presented. The performance of the proposed bivariate distributions is examined using a simulation study. Finally, two data sets are considered in the framework of the proposed distributions to demonstrate their flexibility for real-life applications.

Beta functions adaptation on empirical damage for a fragility model of Italian buildings

In this study, heuristic vulnerability model is presented providing damage distribution as a function of ground motion severity, through the use of given distribution function.Firstly, observational damage distribution are obtained from data collected after the recent earthquakes occurred in Italy and made available by DPC through Da.D.O. platform. The latter collect information on the main characteristics about building position, metrical data (e.g., number of storeys, interstorey height, storey area and construction age), structural types, severity and extension of observed damage on different building components. Moreover, the ground motion characterization was described via the ShakeMap published by the National Institute of Geophysics and Volcanology.Then, the two statistical models mostly used in the literature, i.e. binomial and beta, are used to determine the one most suitable to reproduce the observational damage distributions by means of godness-of-fit test. An introductive section properly examine the pro and cons in their use (number of parameters, flexibility). The observation of mean value and variance couple for more than 2000 damage samples obtained based on the considered database is used to obtain a general relationship which allows to reduce the number of parameters required by beta distribution, preserving its renomate flexibility.This enables the formulation of a novel heuristic vulnerability model for derivation of fragility curves. The model relies on the knoledge of mean damage trend with intensity measure and employs a probability distribution function. In this study, specifically, binomial and beta distributions are utilized. The generality of the method is then customized to suit the characteristics of Italian residential building stock, leveraging the information gathered after the 2009 L'Aquila Earthquake. This database presented an unprecedented opportunity due to the extensive data available on building geometrical and typological characteristics, seismic vulnerability, to the high level of representativeness of samples at municipality level, and to the thorough representation of damage patterns observed on building components.The fragility curves obtained through the presented vulnerability model are compared with those empirically derived from the observational data and are used to fully derive seismic risk scenario after 2009 L'Aquila Earthquake, in terms of economic losses, number of long term unusable buildings, the number of collapse, the number of deaths and severely injuries. The ensuing consequences deriving from the adoption of different fragility curves, i.e. empirical or with beta and binomial vulnerability models are deeply investigated.

Uncertainty analysis of greenhouse gas emissions of monorail transit during the construction.

This paper examines the uncertainty of greenhouse gas (GHG) emissions during monorail construction. Firstly, a deterministic analysis is conducted. Subsequently, the obtained data are evaluated using the data quality indicator (DQI), and a Markov chain Monte Carlo (MCMC) simulation method is employed to assume different parameter distributions. The results of the deterministic calculation indicate that the calculated emissions per unit area of the station amount to 1.97 ton CO2e/m2, while the calculated emissions per unit section length reach 7.55 ton CO2e/m2. To simulate parameter distribution, we utilize a Beta distribution with good shape applicability. Furthermore, we establish scenarios involving system boundary reduction, low-emission factors, and reduced material and energy inputs in order to analyze scenario uncertainties. Regarding model uncertainty, this paper assumes that the material and energy quantity data conform to the normal, log-normal, uniform, and triangular distributions, respectively, subsequently analyzing the uncertainty distributions. This paper analyzes the GHG emission uncertainty evaluation of 16 monorail stations and sections during the construction period, which is divided into parameter, scenario, and model uncertainty. We provide a concrete framework for studying uncertainties related to GHG emissions at stations and sections during the monorail construction period. The scenario analysis results will help to make decisions about the choice of parameters, system boundaries, and other settings. It provides new guidance for emission reduction policies, such as reducing the use of steel-related products or using alternative environmentally friendly materials, considering emission reduction factors more comprehensively and setting emission reduction factors according to uniform distribution principle as far as possible.

Beta regression for double‐bounded response with correlated high‐dimensional covariates

Continuous responses measured on a standard unit interval are ubiquitous in many scientific disciplines. Statistical models built upon a normal error structure do not generally work because they can produce biassed estimates or result in predictions outside either bound. In real‐life applications, data are often high‐dimensional, correlated and consist of a mixture of various data types. Little literature is available to address the unique data challenge. We propose a semiparametric approach to analyse the association between a double‐bounded response and high‐dimensional correlated covariates of mixed types. The proposed method makes full use of all available data through one or several linear combinations of the covariates without losing information from the data. The only assumption we make is that the response variable follows a Beta distribution; no additional assumption is required. The resulting estimators are consistent and efficient. We illustrate the proposed method in simulation studies and demonstrate it in a real‐life data application. The semiparametric approach contributes to the sufficient dimension reduction literature for its novelty in investigating double‐bounded response which is absent in the current literature. This work also provides a new tool for data practitioners to analyse the association between a popular unit interval response and mixed types of high‐dimensional correlated covariates.

Facilitating clinical use of the Amsterdam Instrumental Activities of Daily Living Questionnaire: Normative data and a diagnostic cutoff value.

The Amsterdam Instrumental Activities of Daily Living Questionnaire (A-IADL-Q) is well validated and commonly used to assess difficulties in everyday functioning regarding dementia. To facilitate interpretation and clinical implementation across different European countries, we aim to provide normative data and a diagnostic cutoff for dementia. Cross-sectional data from Dutch Brain Research Registry (N = 1,064; mean (M) age = 62 ± 11 year; 69.5% female), European Medial Information Framework-Alzheimer's Disease 90 + (N = 63; Mage = 92 ± 2 year; 52.4% female), and European Prevention of Alzheimer's Dementia Longitudinal Cohort Study (N = 247; Mage = 63 ± 7 year; 72.1% female) were used. The generalized additive models for location, scale, and shape framework were used to obtain normative values (Z-scores). The beta distribution was applied, and combinations of age, sex, and educational attainment were modeled. The optimal cutoff for dementia was calculated using area under receiver operating curves (AUC-ROC) and Youden Index, using data from Amsterdam Dementia Cohort (N = 2,511, Mage = 64 ± 8 year, 44.4% female). The best normative model accounted for a cubic-like decrease of IADL performance with age that was more pronounced in low compared to medium/high educational attainment. The cutoff for dementia was 1.85 standard deviation below the population mean (AUC = 0.97; 95% CI [0.97-0.98]). We provide regression-based norms for A-IADL-Q and a diagnostic cutoff for dementia, which help improve clinical assessment of IADL performance across European countries.

Beta-informativeness-diffusion multilayer graph embedding for brain network analysis.

Brain network analysis provides essential insights into the diagnosis of brain disease. Integrating multiple neuroimaging modalities has been demonstrated to be more effective than using a single modality for brain network analysis. However, a majority of existing brain network analysis methods based on multiple modalities often overlook both complementary information and unique characteristics from various modalities. To tackle this issue, we propose the Beta-Informativeness-Diffusion Multilayer Graph Embedding (BID-MGE) method. The proposed method seamlessly integrates structural connectivity (SC) and functional connectivity (FC) to learn more comprehensive information for diagnosing neuropsychiatric disorders. Specifically, a novel beta distribution mapping function (beta mapping) is utilized to increase vital information and weaken insignificant connections. The refined information helps the diffusion process concentrate on crucial brain regions to capture more discriminative features. To maximize the preservation of the unique characteristics of each modality, we design an optimal scale multilayer brain network, the inter-layer connections of which depend on node informativeness. Then, a multilayer informativeness diffusion is proposed to capture complementary information and unique characteristics from various modalities and generate node representations by incorporating the features of each node with those of their connected nodes. Finally, the node representations are reconfigured using principal component analysis (PCA), and cosine distances are calculated with reference to multiple templates for statistical analysis and classification. We implement the proposed method for brain network analysis of neuropsychiatric disorders. The results indicate that our method effectively identifies crucial brain regions associated with diseases, providing valuable insights into the pathology of the disease, and surpasses other advanced methods in classification performance.

Cumulative Histograms under Uncertainty: An Application to Dose–Volume Histograms in Radiotherapy Treatment Planning

In radiotherapy treatment planning, the absorbed doses are subject to executional and preparational errors, which propagate to plan quality metrics. Accurately quantifying these uncertainties is imperative for improved treatment outcomes. One approach, analytical probabilistic modeling (APM), presents a highly computationally efficient method. This study evaluates the empirical distribution of dose–volume histogram points (a typical plan metric) derived from Monte Carlo sampling to quantify the accuracy of modeling uncertainties under different distribution assumptions, including Gaussian, log-normal, four-parameter beta, gamma, and Gumbel distributions. Since APM necessitates the bivariate cumulative distribution functions, this investigation also delves into approximations using a Gaussian or an Ali–Mikhail–Haq Copula. The evaluations are performed in a one-dimensional simulated geometry and on patient data for a lung case. Our findings suggest that employing a beta distribution offers improved modeling accuracy compared to a normal distribution. Moreover, the multivariate Gaussian model outperforms the Copula models in patient data. This investigation highlights the significance of appropriate statistical distribution selection in advancing the accuracy of uncertainty modeling in radiotherapy treatment planning, extending an understanding of the analytical probabilistic modeling capacities in this crucial medical domain.

Gbdmr: identifying differentially methylated CpG regions in the human genome via generalized beta regressions

BackgroundDNA methylation is a biochemical process in which a methyl group is added to the cytosine-phosphate-guanine (CpG) site on DNA molecules without altering the DNA sequence. Multiple CpG sites in a certain genome region can be differentially methylated across phenotypes. Identifying these differentially methylated CpG regions (DMRs) associated with the phenotypes contributes to disease prediction and precision medicine development.ResultsWe propose a novel DMR detection algorithm, gbdmr. In contrast to existing methods under a linear regression framework, gbdmr assumes that DNA methylation levels follow a generalized beta distribution. We compare gbdmr to alternative approaches via simulations and real data analyses, including dmrff, a new DMR detection approach that shows promising performance among competitors, and the traditional EWAS that focuses on single CpG sites. Our simulations demonstrate that gbdmr is superior to the other two when the correlation between neighboring CpG sites is strong, while dmrff shows a higher power when the correlation is weak. We provide an explanation of these phenomena from a theoretical perspective. We further applied the three methods to multiple real DNA methylation datasets. One is from a birth cohort study undertaken on the Isle of Wight, United Kingdom, and the other two are from the Gene Expression Omnibus database repository. Overall, gbdmr identifies more DMR CpGs linked to phenotypes than dmrff, and the simulated results support the findings.ConclusionsGbdmr is an innovative method for detecting DMRs based on generalized beta regression. It demonstrated notable advantages over dmrff and traditional EWAS, particularly when adjacent CpGs exhibited moderate to strong correlations. Our real data analyses and simulated findings highlight the reliability of gbdmr as a robust DMR detection tool. The gbdmr approach is accessible and implemented by R on GitHub: https://github.com/chengzhouwu/gbdmr.

Employing zero-inflated beta distribution in an exposure-response analysis of TYK2/JAK1 inhibitor brepocitinib in patients with plaque psoriasis.

Brepocitinib is an oral selective dual TYK2/JAK1 inhibitor and based on its cytokine inhibition profile is expected to provide therapeutic benefit in the treatment of plaque psoriasis. Efficacy data from a completed Phase 2a study in patients with moderate-to-severe plaque psoriasis were utilized to develop a population exposure-response model that can be employed to inform dose selection decisions for further clinical development. A modeling approach that employs the zero-inflated beta distribution was used to account for the bounded nature and distributional characteristics of the Psoriasis Area and Severity Index (PASI) score data. The developed exposure-response model provided an adequate description of the observed PASI scores across all the treatment arms tested and across both the induction and maintenance dosing periods of the study. In addition, the developed model exhibited a good predictive capacity with regard to the derived responder metrics (e.g., 75%/90%/100% improvement in PASI score [PASI75/90/100]). Clinical trial simulations indicated that the induction/maintenance dosing paradigm explored in this study does not offer any advantages from an efficacy perspective and that doses of 10, 30, and 60mg once-daily may be suitable candidates for clinical evaluation in subsequent Phase 2b studies.

Detection of abdominopelvic lymph nodes in multi-parametric MRI

Reliable localization of lymph nodes (LNs) in multi-parametric MRI (mpMRI) studies plays a major role in the assessment of lymphadenopathy and staging of metastatic disease. Radiologists routinely measure the nodal size in order to distinguish benign from malignant nodes, which require subsequent cancer staging. However, identification of lymph nodes is a cumbersome task due to their myriad appearances in mpMRI studies. Multiple sequences are acquired in mpMRI studies, including T2 fat suppressed (T2FS) and diffusion weighted imaging (DWI) sequences among others; consequently, the sizing of LNs is rendered challenging due to the variety of signal intensities in these sequences. Furthermore, radiologists can miss potentially metastatic LNs during a busy clinical day. To lighten these imaging and workflow challenges, we propose a computer-aided detection (CAD) pipeline to detect both benign and malignant LNs in the body for their subsequent measurement. We employed the recently proposed Dynamic Head (DyHead) neural network to detect LNs in mpMRI studies that were acquired using a variety of scanners and exam protocols. The T2FS and DWI series were co-registered, and a selective augmentation technique called Intra-Label LISA (ILL) was used to blend the two volumes with the interpolation factor drawn from a Beta distribution. In this way, ILL diversified the samples that the model encountered during the training phase, while the requirement for both sequences to be present at test time was nullified. Our results showed a mean average precision (mAP) of 53.5% and a sensitivity of ∼78% with ILL at 4 FP/vol. This corresponded to an improvement of ≥10% in mAP and ≥12% in sensitivity at 4FP (p ¡ 0.05) respectively over current LN detection approaches evaluated on the same dataset. We also established the out-of-distribution robustness of the DyHead model by training it on data acquired by a Siemens Aera scanner and testing it on data from the Siemens Verio, Siemens Biograph mMR, and Philips Achieva scanners. Our pilot work represents an important first step towards automated detection, segmentation, and classification of lymph nodes in mpMRI.

Construction of the beta distributions using the random permutation divisors

A subset of cycles comprising a permutation σ in the symmetric group Sn, n ∈ N, is called a divisor of σ. Then the partial sums over divisors with sizes up to un, 0 ≤ u ≤ 1, of values of a nonnegative multiplicative function on a random permutation define a stochastic process with nondecreasing trajectories. When normalized the latter is just a random distribution function supported by the unit interval. We establish that its expectations under various weighted probability measures defined on the subsets of Sn are quasihypergeometric distribution functions. Their limits as n -> 1 cover the class of two-parameter beta distributions. It is shown that, under appropriate conditions, the convergence rate is of the negative power of n order.

Probability distribution of write failure in a memory cell array consisting of magnetic tunnel junction elements with distributed write error rates

Write failure (WF) is a major reliability issue for applications of magnetoresistive random access memory (MRAM), and much effort has been devoted to reducing the write error rate (WER), which is the probability of write failures of a memory cell. Recently, it was shown that the WER of MRAM obeys a skewed probability distribution even though the variation in material parameters obeys a normal distribution. However, little is known about the effect of WER distribution on WF in a memory cell array. Here, we study WF in a memory cell array consisting of magnetic tunnel junction elements with distributed WERs based on numerical simulations. We simulated Bernoulli trials of writing, assuming that the WER obeys a beta distribution. The results show that for typical writing patterns, WF in a memory cell array obeys a binomial distribution, with the mean of the WER as the probability of success. The statistical properties of WF in a memory cell array are not affected by the variance and skewness of the WER. The results provide a basic understanding of the statistical properties of WF in a memory cell array and will be useful for the development of computing systems that exploit erroneous memories.

Quantitative Methods for Evaluating Long-Term Career Attitudes of the Subjects of Modern Society («Model of the Life Trajectory»)

The article is devoted to the problems of ideas of the subjects of modern society about the long-term prospects for the development of their life path (the so-called “life trajectories”). Particular attention is paid to methods for formalizing such representations based on the approximation of life trajectories using curves from the family of beta distribution density functions. The parameters of the beta-density function can be used as the basis for the formal typification of subjects within society. Their changes can be used as indicators of structural changes in society. First of all, in terms of shifts in ideas about the «ideal» (reference) configurations of the life path. In part, these conclusions can be incorporated into procedures for identifying generations.

Parametric Quantile Beta Regression Model

SummaryIn this paper, we develop a fully parametric quantile regression model based on the generalised three‐parameter beta (GB3) distribution. Beta regression models are primarily used to model rates and proportions. However, these models are usually specified in terms of a conditional mean. Therefore, they may be inadequate if the observed response variable follows an asymmetrical distribution. In addition, beta regression models do not consider the effect of the covariates across the spectrum of the dependent variable, which is possible through the conditional quantile approach. In order to introduce the proposed GB3 regression model, we first reparameterise the GB3 distribution by inserting a quantile parameter, and then we develop the new proposed quantile model. We also propose a simple interpretation of the predictor–response relationship in terms of percentage increases/decreases of the quantile. A Monte Carlo study is carried out for evaluating the performance of the maximum likelihood estimates and the choice of the link functions. Finally, a real COVID‐19 dataset from Chile is analysed and discussed to illustrate the proposed approach.

Comparison of state-of-knowledge correlation effect associated with lognormal, beta, and gamma uncertainty distributions

In the probabilistic safety assessment (PSA) of nuclear power plants, the state-of-knowledge correlation (SOKC) is required to be considered when estimating the risk measures and their uncertainties. Specifically, the SOKC must be taken into account when the failure data for similar components are aggregated to produce the failure rate or failure probability for the components and their uncertainty distributions. The objective of this study is to mathematically prove what has been numerically observed in the existing studies: The SOKC effect is more significant if the uncertainty is modeled with a lognormal distribution than when it is modeled with a gamma or beta distribution when the parameters of the two distributions are estimated from the same failure data by using the method of moments. An example based on the frequency calculation for interfacing system loss of coolant accidents is provided to demonstrate the proof. The mathematical proof provides a relationship between the raw moments of lognormal, gamma, and beta distributions, which is expected to be valuable in probability and statistics theories.

Recent Advances in Group-Based Trajectory Modeling for Clinical Research.

Group-based trajectory modeling (GBTM) identifies groups of individuals following similar trajectories of one or more repeated measures. The categorical nature of GBTM is particularly well suited to clinical psychology and medicine, where patients are often classified into discrete diagnostic categories. This review highlights recent advances in GBTM and key capabilities that remain underappreciated in clinical research. These include accounting for nonrandom subject attrition, joint trajectory and multitrajectory modeling, the addition of the beta distribution to modeling options, associating trajectories with future outcomes, and estimating the probability of future outcomes. Also discussed is an approach to selecting the number of trajectory groups.

Are There Dragon Kings in the Stock Market?

In this study, we undertake a systematic study of historic market volatility spanning roughly five preceding decades. We focus specifically on the time series of the realized volatility (RV) of the S&P500 index and its distribution function. As expected, the largest values of RV coincide with the largest economic upheavals of the period: Savings and Loan Crisis, Tech Bubble, Financial Crisis and Covid Pandemic. We address the question of whether these values belong to one of the three categories: Black Swans (BS), that is, they lie on scale-free, power-law tails of the distribution; Dragon Kings (DK), defined as statistically significant upward deviations from BS; or Negative Dragons Kings (nDK), defined as statistically significant downward deviations from BS. In analyzing the tails of the distribution with RV>40, we observe the appearance of “potential” DK, which eventually terminate in an abrupt plunge to nDK. This phenomenon becomes more pronounced with the increase in the number of days over which the average RV is calculated—here from daily, n=1, to “monthly”, n=21. We fit the entire distribution with a modified Generalized Beta (mGB) distribution function, which terminates at a finite value of the variable but exhibits a long power-law stretch prior to that, as well as a Generalized Beta Prime (GB2) distribution function, which has a power-law tail. We also fit the tails directly with a straight line on a log-log scale. In order to ascertain BS, DK or nDK behavior, all fits include their confidence intervals and p-values are evaluated for the data points to check whether they can come from the respective distributions.