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A particularly important substructure in modeling joint linear chance-constrained programs with random right-hand sides and finite sample space is the intersection of mixing sets with common binary variables (and possibly a knapsack constraint). In this paper, we first revisit basic mixing sets by establishing a strong and previously unrecognized connection to submodularity. In particular, we show that mixing inequalities with binary variables are nothing but the polymatroid inequalities associated with a specific submodular function. This submodularity viewpoint enables us to unify and extend existing results on valid inequalities and convex hulls of the intersection of multiple mixing sets with common binary variables. Then, we study such intersections under an additional linking constraint lower bounding a linear function of the continuous variables. This is motivated from the desire to exploit the information encoded in the knapsack constraint arising in joint linear CCPs via the quantile cuts. We propose a new class of valid inequalities and characterize when this new class along with the mixing inequalities are sufficient to describe the convex hull.

Image-space auxiliary features such as surface normal have significantly contributed to the recent success of Monte Carlo (MC) reconstruction networks. However, path-space features, another essential piece of light propagation, have not yet been sufficiently explored. Due to the curse of dimensionality, information flow between a regression loss and high-dimensional path-space features is sparse, leading to difficult training and inefficient usage of path-space features in a typical reconstruction framework. This paper introduces a contrastive manifold learning framework to utilize path-space features effectively. The proposed framework employs weakly-supervised learning that converts reference pixel colors to dense pseudo labels for light paths. A convolutional path-embedding network then induces a low-dimensional manifold of paths by iteratively clustering intra-class embeddings, while discriminating inter-class embeddings using gradient descent. The proposed framework facilitates path-space exploration of reconstruction networks by extracting low-dimensional yet meaningful embeddings within the features. We apply our framework to the recent image- and sample-space models and demonstrate considerable improvements, especially on the sample space. The source code is available at https://github.com/Mephisto405/WCMC.

This paper presents an accurate sampling method for suspension-bridge finite element (FE) model updating. The proposed method utilizes a coupled suspension-bridge modeling approach, which integrates a numerical shape-finding method with conventional FE analysis to efficiently achieve parametric modeling during the sampling process. This study also employs the Nelder-Mead algorithm (NMA) to ensure the efficiency and robustness of the numerical shape-finding calculation. First, the accuracy of the NMA-based coupled modeling approach is examined by numerical investigation of the baseline model of an actual suspension bridge. Then, the computational results of two groups of samples with different sample sizes are compared to test the efficiency of the proposed method. Finally, by relying on the structural health monitoring (SHM) system of this bridge, the measured natural frequencies over half a year are identified, and almost all of them are within the calculated sample space. Therefore, the feasibility of the proposed suspension-bridge FE model sampling method is validated by the uncertainties of the measured responses over half a year.

This paper proposes a new boundary uncertainty-based estimation method that has significantly higher accuracy, scalability, and applicability than our previously proposed boundary uncertainty estimation method. In our previous work, we introduced a new classifier evaluation metric that we termed “boundary uncertainty.” The name “boundary uncertainty” comes from evaluating the classifier based solely on measuring the equality between class posterior probabilities along the classifier boundary; satisfaction of such equality can be described as “uncertainty” along the classifier boundary. We also introduced a method to estimate this new evaluation metric. By focusing solely on the classifier boundary to evaluate its uncertainty, boundary uncertainty defines an easier estimation target that can be accurately estimated based directly on a finite training set without using a validation set. Regardless of the dataset, boundary uncertainty is defined between 0 and 1, where 1 indicates whether probability estimation for the Bayes error is achieved. We call our previous boundary uncertainty estimation method “Proposal 1” in order to contrast it with the new method introduced in this paper, which we call “Proposal 2.” Using Proposal 1, we performed successful classifier evaluation on real-world data and supported it with theoretical analysis. However, Proposal 1 suffered from accuracy, scalability, and applicability limitations owing to the difficulty of finding the location of a classifier boundary in a multidimensional sample space. The novelty of Proposal 2 is that it locally reformalizes boundary uncertainty in a single dimension that focuses on the classifier boundary. This convenient reduction with a focus toward the classifier boundary provides the new method’s significant improvements. In classifier evaluation experiments on Support Vector Machines (SVM) and MultiLayer Perceptron (MLP), we demonstrate that Proposal 2 offers a competitive classifier evaluation accuracy compared to a benchmark Cross Validation (CV) method as well as much higher scalability than both CV and Proposal 1.

Ramalingam Shanmugam,1 Karan P Singh2 1School of Health Administration, Texas State University, San Marcos, TX, 78666, USA; 2Department of Epidemiology and Biostatistics, School of Community and Rural Health, The University of Texas Health Sciences Center at Tyler, Tyler, TX, 75708, USACorrespondence: Karan P SinghDepartment of Epidemiology and Biostatistics, School of Community and Rural Health, The University of Texas Health Sciences Center at Tyler, Tyler, TX, 75708, USAEmail karan.singh@uthct.eduBackground: Challenges to manage, mitigate, or prevent the COVID-19’s pandemics are felt by medical, healthcare professionals and governing agencies. Health researchers conduct survey among the citizens to capture their opinion on COVID-19. In such surveys like in Hanafiah and Wan (2020), structural-zero (different from sampling zero) category occurs as they question about perception, knowledge, and communication regarding COVID-19.Materials: The data were collected in a survey conducted among Malaysians by Hanafiah and Wan regarding COVID-19. The survey focused on people’s response about the public communication, knowledge, and perception.Methods: One of the four question categories in the survey is mutually exclusive with the other three questions. Consequently, there will be no entry in that category. Such group is called structurally zero category in the literature. The literature never probed the migrative split to other categories of the unknown proportion belonging to the structural zero category. In this article, the probability-based new and innovative method configures what proportion in that mutually exclusive category and it is the essence of our method.Results: The mutually exclusive nature of subquestions manufactured structural zero in their data. A careful analysis of the data has created so far unknown probability concepts in the literature, which we named as “Exodus probabilities” in this article. Its discovery and utility are illustrated and elaborated with application in COVID-19. This methodology is also useful in applications in engineering, epidemiology, marketing, communication networking, etc.Conclusion: What is quite novel about the discovery of the exodus probability in this article is the evolution of the concepts from the structural-zero category. In such situation, when a category is eliminated, the proportions of the sample might have uncommunicatively transited to other viable categories and our research question is all about configuring their proportions. This is an innovative approach.Keywords: public knowledge, perception, communication, sample space, dependent probabilities, Venn layout, odds, odds ratio, survey outcomes

This study introduces a novel probability concept, random probability density function (PDF), as an efficient alternative of the random sampling for probability/reliability analysis of multivariate problems. To this end, a solution is proposed for drawing 1-D random PDFs according to the joint PDF of random variables. It is shown that using this approach, few random PDFs could be used instead of millions of random samples for covering the sample space in probabilistic analysis. By drawing PDFs instead of samples, the expected value of multivariable functions can be represented as the expected value of several one-variable probability integrals and therefore, any capable 1-D integration technique can be henceforth used for accurate statistical moment estimation and reliability analysis of multivariable problems. The law of large numbers proves the correctness of the proposed approach and therefore, achieving correct solutions of complex probability and reliability problems will not be hereafter restricted to only using crude Monte Carlo simulation. Besides, a novel probability density simulator is proposed that can attain probability density and moments of multivariate functions in an efficient manner different from existing approaches. Accurate and efficient probability/reliability analyses of complex problems with small failure probabilities confirm that the proposed method can be a reliable and efficient alternative of the crude Monte Carlo method, especially for extremely rare-events and time-consuming problems. For such problems, the suggested approach can serve as a benchmark against which new methods can be compared.