The field of uncertainty quantification is evolving rapidly because of increasing emphasis on models that require quantified uncertainties for large-scale applications, novel algorithm development, and new computational architectures that facilitate implementation of these algorithms. Uncertainty Quantification: Theory, Implementation, and Applications provides readers with the basic concepts, theory, and algorithms necessary to quantify input and response uncertainties for simulation models arising in a broad range of disciplines. The book begins with a detailed discussion of applications where uncertainty quantification is critical for both scientific understanding and policy. It then covers concepts from probability and statistics, parameter selection techniques, frequentist and Bayesian model calibration, propagation of uncertainties, quantification of model discrepancy, surrogate model construction, and local and global sensitivity analysis. The author maintains a complementary web page where readers can find data used in the exercises and other supplementary material. Uncertainty Quantification: Theory, Implementation, and Applications includes a large number of definitions and examples that use a suite of relatively simple models to illustrate concepts; numerous references to current and open research issues; and exercises that illustrate basic concepts and guide readers through the numerical implementation of algorithms for prototypical problems. It also features a wide range of applications, including weather and climate models, subsurface hydrology and geology models, nuclear power plant design, and models for biological phenomena, along with recent advances and topics that have appeared in the research literature within the last 15 years, including aspects of Bayesian model calibration, surrogate model development, parameter selection techniques, and global sensitivity analysis. Audience: The text is intended for advanced undergraduates, graduate students, and researchers in mathematics, statistics, operations research, computer science, biology, science, and engineering. It can be used as a textbook for one- or two-semester courses on uncertainty quantification or as a resource for researchers in a wide array of disciplines. A basic knowledge of probability, linear algebra, ordinary and partial differential equations, and introductory numerical analysis techniques is assumed. Contents: Chapter 1: Introduction; Chapter 2: Large-Scale Applications; Chapter 3: Prototypical Models; Chapter 4: Fundamentals of Probability, Random Processes, and Statistics; Chapter 5: Representation of Random Inputs; Chapter 6: Parameter Selection Techniques; Chapter 7: Frequentist Techniques for Parameter Estimation; Chapter 8: Bayesian Techniques for Parameter Estimation; Chapter 9: Uncertainty Propagation in Models; Chapter 10: Stochastic Spectral Methods; Chapter 11: Sparse Grid Quadrature and Interpolation Techniques; Chapter 12: Prediction in the Presence of Model Discrepancy; Chapter 13: Surrogate Models; Chapter 14: Local Sensitivity Analysis; Chapter 15: Global Sensitivity Analysis; Appendix A: Concepts from Functional Analysis; Bibliography; Index
This article introduces a novel approach to integrating correlation matrix information from training samples to construct a classification rule for testing samples. Traditional discriminant analysis methods that rely solely on mean vectors tend to perform poorly when the mean of the training samples is not indicative of the testing samples. To address this limitation, we propose a new discriminant analysis method called Correlation-matrix driven Discriminant Analysis (CorrDA). By considering the correlation matrices of different classes in the training samples, we can capture the unique patterns among the classes. CorrDA utilizes the Bayes classifier and mixture models to effectively incorporate the correlation matrix information derived from the training samples, thereby improving the discriminant analysis performance on the testing data. Through the analysis of COVID-19 datasets and extensive simulation studies, we provide empirical evidence demonstrating the superior performance of CorrDA.
Historically marked by cultural and socio-political marginalization, the Rif region of Morocco has seen intensified challenges
following the Hirak movement and the closing of Ceuta and Melilla, conditions which left the youth of the region lost between desperation and migration aspirations. Adopting the city of Nador as a case study, this research explores the perspectives of its youth on the socio-political and economic realities in the post-Hirak era and examines how these perceptions influence their migration aspirations and decisions. Semi-structured interviews with individuals originating from or currently
residing in Nador reveal the multifaceted drivers of migration, including economic hardship, limited opportunities, and the need to preserve personal dignity. The findings highlight tensions between the youth’s attachment to their homeland and the pursuit of better prospects abroad, underscoring the interplay of structural inequalities and individual agency. Situated within broader theoretical frameworks of migration and population studies, this study sheds light on the localized dynamics of marginalization and mobility, offering recommendations for addressing the root causes of youth migration and fostering development strategies tailored to the lived realities of communities in Nador and beyond.
In this paper, we propose a Bayesian PG-INLA algorithm which is tailored to both one-dimensional and multidimensional 2-PL IRT models. The proposed PG-INLA algorithm utilizes a computationally efficient data augmentation strategy via the Pólya-Gamma variables, which can avoid low computational efficiency of traditioanl Bayesian MCMC algorithms for IRT models with a logistic link function. Meanwhile, combined with the advanced and fast INLA algorithm, the PG-INLA algorithm is both accurate and computationally efficient. We provide details on the derivation of posterior and conditional distributions of IRT models, the method of introducing the Pólya-Gamma variable into Gibbs sampling, and the implementation of the PG-INLA algorithm for both one-dimensional and multidimensional cases. Through simulation studies and an application to the data analysis of the IPIP-NEO personality inventory, we assess the performance of the PG-INLA algorithm. Extensions of the proposed PG-INLA algorithm to other IRT models are also discussed.
Generative Artificial Intelligence is emerging as an important technology, promising to be transformative in many areas. At the same time, generative AI techniques are based on sampling from probabilistic models, and by default, they come with no guarantees about correctness, safety, fairness, or other properties. Statistical methods offer a promising potential approach to improve the reliability of generative AI techniques. In addition, statistical methods are also promising for improving the quality and efficiency of AI evaluation, as well as for designing interventions and experiments in AI. In this paper, we review some of the existing work on these topics, explaining both the general statistical techniques used, as well as their applications to generative AI. We also discuss limitations and potential future directions.
The maxima and the minima of a randomly stopped sample of a random variable, $X$, together with two newly defined random variables that make $X$ into the maxima or minima of a randomly stopped sample of them, can be used to define statistical model transformation mechanisms. These transformations can be used to define models for extreme value data that are not grounded on large sample theory. The relationship between the stopping model and characteristics of the corresponding model transformations obtained is investigated. In particular, one looks into which stopping models make these model transformations into model extensions, and which stopping models lead to statistically stable extensions in the sense that using the model extension a second time leaves the extended model unchanged. The stopping models under which the extensions based on randomly stopped maxima and their inverses coincide with the extensions based on randomly stopped minima and their inverses are also characterized. The advantages of using models obtained through these model extension mechanisms instead of resorting to extreme value models grounded on asymptotic arguments is illustrated by way of examples.
This paper is concerned with the concepts of upper and lower almost α-*-continuous multifunctions. Moreover, several characterizations of upper and lower almost α-*-continuous multifunctions are investigated. In particular, the relationships between α-*-continuity and almost α-*-continuity are established.
Candace Savonen, Carrie Wright, Ava M. Hoffman
et al.
Data science and informatics tools are developing at a blistering rate, but their users often lack the educational background or resources to efficiently apply the methods to their research. Training resources and vignettes that accompany these tools often deprecate because their maintenance is not prioritized by funding, giving teams little time to devote to such endeavors. Our group has developed Open-source Tools for Training Resources (OTTR) to offer greater efficiency and flexibility for creating and maintaining these training resources. OTTR empowers creators to customize their work and allows for a simple workflow to publish using multiple platforms. OTTR allows content creators to publish training material to multiple massive online learner communities using familiar rendering mechanics. OTTR allows the incorporation of pedagogical practices like formative and summative assessments in the form of multiple choice questions and fill in the blank problems that are automatically graded. No local installation of any software is required to begin creating content with OTTR. Thus far, 15 training courses have been created with OTTR repository template. By using the OTTR system, the maintenance workload for updating these courses across platforms has been drastically reduced. For more information about OTTR and how to get started, go to ottrproject.org. Supplementary materials for this article are available online.
Probabilities. Mathematical statistics, Special aspects of education
With the development of modern science and technology, more and more high-dimensional data appear in the application fields. Since the high dimension can potentially increase the complexity of the covariance structure, comparing the covariance matrices among populations is strongly motivated in high-dimensional data analysis. In this article, we consider the proportionality test of two high-dimensional covariance matrices, where the data dimension is potentially much larger than the sample sizes, or even larger than the squares of the sample sizes. We devise a novel high-dimensional spatial rank test that has much-improved power than many existing popular tests, especially for the data generated from some heavy-tailed distributions. The asymptotic normality of the proposed test statistics is established under the family of elliptically symmetric distributions, which is a more general distribution family than the normal distribution family, including numerous commonly used heavy-tailed distributions. Extensive numerical experiments demonstrate the superiority of the proposed test in terms of both empirical size and power. Then, a real data analysis demonstrates the practicability of the proposed test for high-dimensional gene expression data.
It is crucial to take into account the dynamics of the species while investigating how a species may survive in an environment. A species can be classified as either semelparous or iteroparous depending on how it reproduces. In this article, we present a model, which consists of two semelparous species by considering two age classes. We specifically discuss the effects of density-dependent in the interaction between the two semelparaous species and examine the equilibria of the system in the absence and presence of harvesting in the system. Then, the local stability of the equilibria is also investigated. A modified Leslie matrix population model with the addition of density-dependent in the equation is used. The model is analyzed in the presence and absence of competition between these species. We assume that density-dependent only occurred in the first age class of both species and that harvesting only occurred in the second age class of both species. Then, we assume that competition only occurs in the first age class in both species in the form of interspecific and intraspecific competition. This assumption is intended to simplify the complexity of the problem in the model. Our results show that there are three equilibria in the model without competition and four equilibria in the model with the competition. Hence, the presence of competition has influenced the number of equilibria. We also investigate the relation between the stability of the equilibria with the net reproduction rate of the system. Furthermore, we found the condition for the local stability of the co-existence equilibrium point, which is related to the degree of interspecific and intraspecific competition. This theory may be applied to investigate the dynamics of natural resources, whether in the absence of human exploitation and in the presence of various strategies in managing the exploitation of the resources, such as in fisheries industries.
In this paper, we propose a generalization of the so-called truncated inverse Weibull-generated family of distributions by the use of the power transform, adding a new shape parameter. We motivate this generalization by presenting theoretical and practical gains, both consequences of new flexible symmetric/asymmetric properties in a wide sense. Our main mathematical results are about stochastic ordering, uni/multimodality analysis, series expansions of crucial probability functions, probability weighted moments, raw and central moments, order statistics, and the maximum likelihood method. The special member of the family defined with the inverse Weibull distribution as baseline is highlighted. It constitutes a new four-parameter lifetime distribution which brightensby the multitude of different shapes of the corresponding probability density and hazard rate functions. Then, we use it for modelling purposes. In particular, a complete numerical study is performed, showing the efficiency of the corresponding maximum likelihood estimates by simulation work, and fitting three practical data sets, with fair comparison to six notable models of the literature.
The paper assumes a Bayesian estimate of the telecommunication systems availability ratio. Downtime and uptime are described by gamma distributions with positive integer parameters. Estimates of the distribution parameters are obtained using the maximum likelihood method. For the set samples, the values of the desired probability distribution densities are found and an expression for estimating the availability ratio is derived. Numerical estimates for the standard and assumed estimates are given. For a system with two states, a Bayesian estimate of the availability function with consideration of downtime and serviceable condition takes into account the features of backup equipment and the effect of its failure defined by performance reliability and features that ensure the reliability of information signals. The proposed Bayesian approach has the following advantages: it is possible to conduct quantitative estimates with lack of sufficient statistics on functional use indicators; it takes into account all destabilizing factors of various nature; the presence of a lower mean square error compared to traditional methods. To implement the proposed approach that estimates the availability ratio, confidence probabilities are introduced relative to the indicator of failure flows and equipment recovery. The parameters of the a priori information can be determined by different methods or on the basis of sufficient statistical data. To illustrate the discussed calculation algorithm, a digital data transmission system of a standard satellite navigation system consisting of terminal, radio equipment, and a transponder is considered. To estimate the required values, we used data on interruptions in the operation of equipment due to its malfunction during a conditional year. The frequency of downtime caused by signal propagation conditions and equipment failures was evaluated. It was shown that the gamma distribution is suitable for describing the frequency distribution of downtime. The frequency distribution of the cyclicity coefficient with the condition of the selected time interval was also taken into account. Sample mathematical expectations and mean square deviations of the downtime coefficient were found. As a result, the numerical example shows the correctness of using the Bayesian estimate of weighted equipment preparedness.
Titin Siswantining, Muhammad Ihsan, Saskya Mary Soemartojo
et al.
Missing values are a problem that is often encountered in various fields and must be addressed to obtain good statistical inference such as parameter estimation. Missing values can be found in any type of data, included count data that has Poisson distributed. One solution to overcome that problem is applying multiple imputation techniques. The multiple imputation technique for the case of count data consists of three main stages, namely the imputation, the analysis, and pooling parameter. The use of the normal distribution refers to the sampling distribution using the central limit theorem for discrete distributions. This study is also equipped with numerical simulations which aim to compare accuracy based on the resulting bias value. Based on the study, the solutions proposed to overcome the missing values in the count data yield satisfactory results. This is indicated by the size of the bias parameter estimate is small. But the bias value tends to increase with increasing percentage of observation of missing values and when the parameter values are small.