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In this article we propose a procedure for selecting a random size subset that contains all experimental treatments that are better than the standard. The comparison is made with regard to two binary endpoints. An experimental treatment is considered to be better than the standard if its two endpoints have successful rates higher than standard rates. We assume responses from different treatments are independent, however, responses from each treatment may be associated. We use the odds ratio as the association parameter and describe the bivariate binomial distribution involved in the computation of the probability of a correct selection for our procedure. We derive the design parameters for three different cases: independent endpoints, dependent endpoints with known association and dependent endpoints with unknown association, and provide tables including the minimum sample size needed for the probability requirements to be satisfied. [ABSTRACT FROM AUTHOR]

We study the problem of solvability of the equation ϕ (x) = ∫ Ω g (w) ϕ (f (x , ω)) P (d ω) + F (x) , where P is a probability measure on a σ-algebra of subsets of Ω, assuming Hölder continuity of F on the range of f. [ABSTRACT FROM AUTHOR]

Adverse Outcome Pathway (AOP) is a useful tool to glean mode of action (MOE) of a chemical. However, in order to use it for the purpose of risk assessment, an AOP needs to be quantified using in vitro or in vivo data. Majority of quantitative AOPs developed so far, were for single exposure to progressively higher doses. Limited attempts were made to include time in the modeling. Here as a proof-of concept, we developed a hypothetical AOP, and quantified it using a virtual dataset for six repeated exposures using a Bayesian Network Analysis (BN) framework. The virtual data was generated using realistic assumptions. Effects of each exposure were analyzed separately using a static BN model and analyzed in combination using a dynamic BN (DBN) model. Our work shows that the DBN model can be used to calculate the probability of adverse outcome when other upstream KEs were observed earlier. These probabilities can help in identification of early indicators of AO. In addition, we also developed a data driven AOP pruning technique using a lasso-based subset selection, and show that the causal structure of AOP is itself dynamic and changes over time. This proof-of-concept study revealed the possibility for expanding the applicability of the AOP framework to incorporate biological dynamism in toxicity appearance by repeated insults. [ABSTRACT FROM AUTHOR]

This article considers the goal of selecting the population with the largest mean among k normal populations when variances are not known. We propose a Stein-type two-sample procedure, denoted by R BCH , for selecting a nonempty random-size subset of size at most m ( 1 ≤ m ≤ k − 1 ) that contains the population associated with the largest mean, with a guaranteed minimum probability P * , whenever the distance between the largest mean and the second largest mean is at least δ * , where m, P * , and δ * are specified in advance of the experiment. The probability of a correct selection and the expected subset size of R BCH are derived. Critical values/procedure parameters that are required for certain k, m, P * , and δ * are obtained by solving simultaneous integral equations and are presented in tables. [ABSTRACT FROM AUTHOR]

This study aims to develop an efficient budget allocation procedure for the problem of selecting an optimal subset of designs from a finite number of alternative designs in stochastic environments. The optimal subset might contain more alternative designs beyond the Pareto optimal ones. In this study, we adopt the Pareto rank to measure the performance of each design and define the optimal subset. Our objective is to minimize the probability that the optimal subset is falsely selected within a fixed limited simulation budget. We propose an upper bound of the probability of false selection and derive an asymptotically optimal simulation budget allocation rule based on the large deviation theory. We also provide some useful insights into how the simulation budget can be allocated to identify the optimal subset. The proposed budget allocation algorithm is compared with existing methods through numerical experiments, and the results show the efficiency of our proposed algorithm. [ABSTRACT FROM AUTHOR]

Early detection of dreissenid mussels (Dreissena polymorpha and D. rostriformis bugensis) is crucial to mitigating the economic and environmental impacts of an infestation. Plankton tow sampling is a common method used for early detection of dreissenid mussels, but little is known about the sampling intensity required for a high probability of early detection using the method. We used implicit dynamic occupancy models to estimate plankton tow detection probabilities of dreissenid mussels from a long-term data set containing plankton tow samples collected across central and western United States. We fit models using a) the entire data set, including water bodies with unknown occupancy status in addition to heavily infested water bodies, b) a data subset that included water bodies with paired water temperature data, and c) a data subset that included water bodies with lower dreissenid densities. For the entire data set, we found that estimated detection probabilities varied by water body size and ranged from approximately 0.10 to 0.86. For the water temperature subset, we observed the same pattern between detection probability and water body size as we did for the full data but additionally found that the estimated detection probabilities were much higher when water temperatures were above 12 °C. For the lower dreissenid density subset, we found that the estimated probability of detecting dreissenid mussels with a single aggregated plankton tow sample was near zero. Given these estimates, we conclude that the number of aggregated plankton tow samples taken per water body in the data is far fewer than the number needed to ensure a high probability of detecting dreissenid mussels, especially if they are at low densities. We summarize the analyses with a discussion of plankton tow sampling protocol changes needed to improve estimates of dreissenid detection probabilities. [ABSTRACT FROM AUTHOR]

In this article, we study complete convergence and complete moment convergence for negatively dependent random variables under sub-linear expectations. The results obtained in sublinear expectation spaces extend the corresponding ones in probability space. [ABSTRACT FROM AUTHOR]

The critical problem of reliability design is how to obtain a more accurate failure probability with a smaller number of evaluations of actual complex and nonlinear performance function. To achieve this objective, an adaptive subset simulation method with a deep neural network (DNN) is proposed for accurate estimation of small failure probability. A determinate criterion for threshold values is developed, and the subset number is adaptively quantified according to the initial estimated value of small failure probability. Therefore, the estimation of small failure probability is converted to estimation problem of multiple large conditional probabilities. An adaptive deep neural network model is constructed in every subset to predict the conditional probability with a smaller number of evaluations of the actual performance function. Furthermore, the sampling points for the next subset can be adaptively selected according to the constructed DNN model, which can decrease the number of invalid sampling points and evaluations of actual performance function, then the computational efficiency for estimating the conditional probability in every subset is increased. The sampling points with high probability density functions are recalculated with actual performance function values to replace the predicted values of the DNN model, which can verify the accuracy of DNN model and increase the estimation accuracy of small failure probability. By analyzing a nonlinear problem, a multiple failure domain problem and two engineering examples, the effectiveness and accuracy of the proposed methodology for estimating small failure probability are verified. [ABSTRACT FROM AUTHOR]

Levin and Leu (2021) introduced some key inequalities that underlie the lower bound formula for the probability of lattice events when using adaptive members of the Levin-Robbins-Leu family of sequential subset selection procedures for binary outcomes. Here we provide a rigorous proof of the key inequality for each adaptive procedure in the special case of equal odds parameters. We also provide some further insight into why the key inequality holds for arbitrary odds parameters and we present a complete proof in that case for a simple yet non-trivial prototype example. Two errata in the abovementioned publication are also corrected herein. [ABSTRACT FROM AUTHOR]

Compositional data, which is data consisting of fractions or probabilities, is common in many fields including ecology, economics, physical science and political science. If these data would otherwise be normally distributed, their spread can be conveniently represented by a multivariate normal distribution truncated to the non-negative space under a unit simplex. Here this distribution is called the simplex-truncated multivariate normal distribution. For calculations on truncated distributions, it is often useful to obtain rapid estimates of their integral, mean and covariance; these quantities characterising the truncated distribution will generally possess different values to the corresponding non-truncated distribution. In this paper, three different approaches that can estimate the integral, mean and covariance of any simplex-truncated multivariate normal distribution are described and compared. These three approaches are (1) naive rejection sampling, (2) a method described by Gessner et al. that unifies subset simulation and the Holmes-Diaconis-Ross algorithm with an analytical version of elliptical slice sampling, and (3) a semi-analytical method that expresses the integral, mean and covariance in terms of integrals of hyperrectangularly-truncated multivariate normal distributions, the latter of which are readily computed in modern mathematical and statistical packages. Strong agreement is demonstrated between all three approaches, but the most computationally efficient approach depends strongly both on implementation details and the dimension of the simplex-truncated multivariate normal distribution. For computations in low-dimensional distributions, the semi-analytical method is fast and thus should be considered. As the dimension increases, the Gessner et al. method becomes the only practically efficient approach of the methods tested here. [ABSTRACT FROM AUTHOR]

Missing covariates is a common issue when fitting meta‐regression models. Standard practice for handling missing covariates tends to involve one of two approaches. In a complete‐case analysis, effect sizes for which relevant covariates are missing are omitted from model estimation. Alternatively, researchers have employed the so‐called "shifting units of analysis" wherein complete‐case analyses are conducted on only certain subsets of relevant covariates. In this article, we clarify conditions under which these approaches generate unbiased estimates of regression coefficients. We find that unbiased estimates are possible when the probability of observing a covariate is completely independent of effect sizes. When that does not hold, regression coefficient estimates may be biased. We study the potential magnitude of that bias assuming a log‐linear model of missingness and find that the bias can be substantial, as large as Cohen's d = 0.4–0.8 depending on the missingness mechanism. [ABSTRACT FROM AUTHOR]

This study introduces a novel point selection procedure for the Probability Density Evolution Method (PDEM) to estimate time-dependent reliability and failure probabilities in dynamic systems under first-passage failure conditions. The method integrates and modifies features of the Subset simulation procedure to adaptively generate dependent sample sets suitable for a reliability analysis by a direct probability integration approach levering PDEM. Performance function assessments are used as weighting factors in the Subset supported Point Selection (S-PS), enhancing the ultimate failure probability estimation accuracy. The presented approach effectively identifies samples in the failure region, particularly benefiting for dynamic systems under stochastic excitation, tested with random dimensions up to 60. It also offers a computationally efficient structural reliability estimation procedure by analyzing full time–history responses. The proposed method provides deeper insights into rare failure events and mechanisms through the visualization of intermediate results. This research presents an advanced framework for estimating structural reliability and understanding critical events in dynamic systems. [Display omitted] • Efficient estimation of small time-dependent failure probabilities for stochastic dynamic problems. • Insight on critical system behavior for structural dynamical first-passage problems. • Reliability analysis by utilizing performance-based information obtained by subset simulation in a direct probability integration approach. • Capturing the full response evolutionary probability density function for critical and rare events of structural dynamical systems. [ABSTRACT FROM AUTHOR]

We introduce and discuss some key inequalities that underlie the lower bound formula for the probability of lattice events in the Levin-Robbins-Leu family of sequential subset selection procedures for binary outcomes. The present work combines the notion of lattice events—as previously discussed for the nonadaptive member of the family—with the positive cumulative sum property for the adaptive members—as previously discussed for the special lattice event of correct selection, thereby extending the key inequality to its broadest scope. [ABSTRACT FROM AUTHOR]

Feature selection is a widely used dimension reduction technique to select feature subsets because of its interpretability. Many methods have been proposed and achieved good results, in which the relationships between adjacent data points are mainly concerned. But the possible associations between data pairs that are not adjacent are always neglected. Different from previous methods, we propose a novel and very simple approach for unsupervised feature selection, named MMFS (Multi-step Markov Probability Relationship for Feature Selection). The idea is using multi-step Markov transition probability to describe the relation between any data pair. Two ways from the positive and negative viewpoints are employed respectively to keep the data structure after feature selection. From the positive viewpoint, the maximum transition probability that can be reached in a certain number of steps is used to describe the relation between two points. Then, the features which can keep the compact data structure are selected. From the viewpoint of negative, the minimum transition probability that can be reached in a certain number of steps is used to describe the relation between two points. On the contrary, the features that least maintain the loose data structure are selected. The two ways can also be combined. Thus three algorithms are proposed. Our main contributions are a novel feature section approach which uses multi-step transition probability to characterize the data structure, and three algorithms proposed from the positive and negative aspects for keeping data structure and select the features to preserve such structure. The performance of our approach is compared with the state-of-the-art methods on eight real-world data sets, and the experimental results show that the proposed MMFS is effective in unsupervised feature selection. [ABSTRACT FROM AUTHOR]

Schedule management is an essential part of construction project management. In practical management affairs, many uncertainties may lead to potential project delays and make the schedule risky. To quantify such risk, the Probabilistic Critical Path Method (PCPM) is used to compute the overdue probability. Survey shows it could help project managers understand the schedule better. However, two critical factors limited the application of PCPM: computational efficiency and timeliness. To solve these constraints, we combined subset simulation and statistical learning to build a computationally efficient and dynamic simulation system. Numerical experiment shows that this method can effectively improve the computation efficiency without losing any accuracy and outperforms the other approaches with the same assumptions. Besides, we proposed a machine learning-based way to estimate task duration distributions in PCPM automatically. It collects real-time progress data through user interactions and learns the best PERT-Beta parameters based on these historical data. Our estimator provides our simulation system the ability to handle dynamic assessment without laborious human work. These improvements reduce the limitations of PCPM, making the application of PCPM in practical management affairs possible. [ABSTRACT FROM AUTHOR]

We exhibit some strong positivity properties of a certain function that implies a key inequality that in turn implies the lower bound formula for the probability of correct selection in the Levin-Robbins-Leu family of sequential subset selection procedures for binary outcomes. These properties provide a more direct and comprehensive demonstration of the key inequality than was discussed in Levin and Leu (2013a). [ABSTRACT FROM AUTHOR]

Abstract We consider random sampling in finitely generated shift-invariant spaces V (Φ) ⊂ L 2 (R n) generated by a vector Φ = (φ 1 , ... , φ r) ∈ (L 2 (R n)) r. Following the approach introduced by Bass and Gröchenig, we consider certain relatively compact subsets V R , δ (Φ) of such a space, defined in terms of a concentration inequality with respect to a cube with side lengths R. Under very mild assumptions on the generators, we show that for R sufficiently large, taking O (R n l o g (R)) many random samples (taken independently uniformly distributed within C R) yields a sampling set for V R , δ (Φ) with high probability. We give explicit estimates of all involved constants in terms of the generators φ 1 , ... , φ r. [ABSTRACT FROM AUTHOR]

Highlights • Relations between subset simulation (SuS) and SORM • Problems of SuS with invariance under reformulations. • Problems of SuS with extrapolation of limit state functions. • Problems of SuS with changing topological shapes of failure domains. • Problems of SuS with multiple design points. Abstract In the last fifteen years the subset sampling method has often been used in reliability problems as a tool for calculating small probabilities. This method is extrapolating from an initial Monte Carlo estimate, for which the probability content of a failure domain found by a suitable higher level of the original limit state function. Then iteratively conditional probabilities are estimated for failures domains decreasing to the original failure domain. However, there are implied premises, regarding the structure of the failure domains, which must be fulfilled for the method to work properly. The examples studied in this paper demonstrate that inaccurate results might be obtained if the said premises are not fulfilled. This demonstrates that there are limitations for the application of this method. [ABSTRACT FROM AUTHOR]

Analysis of single-cell RNA sequencing (scRNA-Seq) data often involves filtering out uninteresting or poorly measured genes and dimensionality reduction to reduce noise and simplify data visualization. However, techniques such as principal components analysis (PCA) fail to preserve non-negativity and sparsity structures present in the original matrices, and the coordinates of projected cells are not easily interpretable. Commonly used thresholding methods to filter genes avoid those pitfalls, but ignore collinearity and covariance in the original matrix. We show that a deterministic column subset selection (DCSS) method possesses many of the favorable properties of common thresholding methods and PCA, while avoiding pitfalls from both. We derive new spectral bounds for DCSS. We apply DCSS to two measures of gene expression from two scRNA-Seq experiments with different clustering workflows, and compare to three thresholding methods. In each case study, the clusters based on the small subset of the complete gene expression profile selected by DCSS are similar to clusters produced from the full set. The resulting clusters are informative for cell type. [ABSTRACT FROM AUTHOR]

We consider a class of chance-constrained combinatorial optimization problems, which we refer to as probabilistic partial set-covering (PPSC) problems. Given a prespecified risk level ϵ ∊ [0; 1], the PPSC problem aims to find the minimum cost selection of subsets of items such that a target number of items is covered with probability at least 1 - ϵ. We show that PPSC admits an efficient probability oracle that computes the coverage probability exactly, under certain distributions of the random variables representing the coverage relation. Using this oracle, we give a compact mixed-integer program that solves the PPSC problem for a special case. For large-scale instances for which an exact oracle-based method exhibits slow convergence, we propose a samplingbased approach that exploits the special structure of PPSC. The oracle can be used as a tool for checking and fixing the feasibility of the solution given by this approach. In particular, we introduce a new class of facet-defining inequalities for a submodular substructure of PPSC, and show that a sampling-based algorithm coupled with the probability oracle effectively provides high-quality feasible solutions to the large-scale test instances. [ABSTRACT FROM AUTHOR]

Highlights • Reliability-based alarm limits for civil engineering structures. • Algorithm that combines Subset simulation MCMC with the finite element method. • Discussion on the applicability of the algorithm to different types of structures. Abstract Civil engineering structures are commonly monitored to assess their structural behaviour, using alarm thresholds to indicate when contingency actions are needed to improve safety. However, there is a need for guidelines on how to establish thresholds that ensure sufficient safety. This paper therefore proposes a general computational algorithm for establishment of reliability-based alarm thresholds for civil engineering structures. The algorithm is based on Subset simulation with independent-component Markov chain Monte Carlo simulation and applicable with both analytical structural models and finite element models. The reliability-based alarm thresholds can straightforwardly be used in the monitoring plans that are developed in the design phase of a construction project, in particular for sequentially loaded structures such as staged construction of embankments. With the reliability-based alarm thresholds, contingency actions will only be implemented when they are needed to satisfy the target probability of failure. [ABSTRACT FROM AUTHOR]

• A wrapper method of feature selection considering both relevance and redundancy is proposed for similar-day choosing. • A novel similar-day choosing method is used to excavate the uncertainties in residential loads. • An efficient and accurate probability density forecasting framework is proposed for residential loads. • Superior performances in both point forecasting and probabilistic forecasting are obtained. Probabilistic load forecasting can provide much richer information about uncertainties in residential loads than point load forecasting, and is gaining increasing attention in building efficient and economical demand response programs. However, obtaining satisfactory forecasting results for residential loads is challenging due to their intrinsic nonlinearity and volatility. Furthermore, existing probabilistic forecasting methods are limited in their ability to directly generate probability density forecasting in an efficient and accurate way. To address these issues, a new probability density forecasting framework for residential loads is proposed in this paper. Firstly, a similar-day choosing method is proposed by combining weighted external features and internal load curves to build a similar-day dataset. Then, point forecasting is achieved by training an exclusive predictor on the corresponding similar-day dataset for each forecasting target. Finally, probability density forecasting is realized by conducting kernel density estimation on the similar-day dataset and calibrating it according to the point forecasting result. Comparisons with other state-of-the-art methods carried out on the HUE dataset demonstrate the efficiency and superiority of the proposed framework in both point and probability density forecasting. [ABSTRACT FROM AUTHOR]

Abstract In aerospace engineering, the reliability analysis technique attracts increasing attention from many structure designers. Compared with the conventional reliability analysis, the probability and fuzzy state assumption (profust) reliability analysis is proposed as a more comprehensive and objective theory to evaluate the structural safety. Because of great computational burden of the existing methods in estimating profust failure probability, an efficient numerical algorithm is developed in this paper. The proposed method is based on an equivalent transformation of the profust failure probability, then the profust failure probability can be rewritten as the form of a series of conventional failure probabilities which have similar constructions. The subset simulation method is employed to compute all the conventional failure probabilities by using a set of samples repeatedly. Next, this method is applied to estimate the profust failure probability of a wing box structure. The calculation results indicate that the proposed method can reduce the computational cost dramatically with acceptable precision. [ABSTRACT FROM AUTHOR]

Statistical model selection is a great challenge when the number of accessible measurements is much smaller than the dimension of the parameter space. We study the problem of model selection in the context of subset selection for high-dimensional linear regressions. Accordingly, we propose a new model selection criterion with the Fisher information that leads to the selection of a parsimonious model from all the combinatorial models up to some maximum level of sparsity. We analyze the performance of our criterion as the number of measurements grows to infinity, as well as when the noise variance tends to zero. In each case, we prove that our proposed criterion gives the true model with a probability approaching one. Additionally, we devise a computationally affordable algorithm to conduct model selection with the proposed criterion in practice. Interestingly, as a side product, our algorithm can provide the ideal regularization parameter for the Lasso estimator such that Lasso selects the true variables. Finally, numerical simulations are included to support our theoretical findings. [ABSTRACT FROM AUTHOR]

In this study, we present a method for improving the quality of automatic single fallen tree stem segmentation in ALS data by applying a specialized constrained conditional random field (CRF). The entire processing pipeline is composed of two steps. First, short stem segments of equal length are detected and a subset of them is selected for further processing, while in the second step the chosen segments are merged to form entire trees. The first step is accomplished using the specialized CRF defined on the space of segment labelings, capable of finding segment candidates which are easier to merge subsequently. To achieve this, the CRF considers not only the features of every candidate individually, but incorporates pairwise spatial interactions between adjacent segments into the model. In particular, pairwise interactions include a collinearity/angular deviation probability which is learned from training data as well as the ratio of spatial overlap, whereas unary potentials encode a learned probabilistic model of the laser point distribution around each segment. Each of these components enters the CRF energy with its own balance factor. To process previously unseen data, we first calculate the subset of segments for merging on a grid of balance factors by minimizing the CRF energy. Then, we perform the merging and rank the balance configurations according to the quality of their resulting merged trees, obtained from a learned tree appearance model. The final result is derived from the top-ranked configuration. We tested our approach on 5 plots from the Bavarian Forest National Park using reference data acquired in a field inventory. Compared to our previous segment selection method without pairwise interactions, an increase in detection correctness and completeness of up to 7 and 9 percentage points, respectively, was observed. [ABSTRACT FROM AUTHOR]

In this paper we introduce the notions of double left stabilizer of X and double left stabilizer of X with respect to Y , for nonempty subsets X and Y of BL-algebra A and we study some properties of them. After that we state and prove some theorems which determine the relationship between these notions and other types of filters in BL-algebras. Finally we introduce the set N(F), for every filter F of A. Also we prove A is an MV -algebra iff N(F) = N(A) = {1} iff D(Xl) = Xl iff D((X,{1})l) = Xl, for each nonempty subset X and every proper filter F of A. [ABSTRACT FROM AUTHOR]

It follows from the famous result of Cook about the NP-completeness of the Boolean satisfiability problem that there is no polynomial algorithm for this problem if P ≠ NP. In this paper, we prove that the Boolean satisfiability problem remains computationally hard on polynomial strongly generic subsets of formulas provided P ≠ NP and P = BPP. Boolean formulas are represented in the natural way by labeled binary trees. [ABSTRACT FROM AUTHOR]

The aim of this text is to provide an elementary and self-contained exposition of Gromov's argument on topological overlap (the presentation is based on Gromov's work, as well as two follow-up papers of Matoušek and Wagner, and of Dotterrer, Kaufman and Wagner). We also discuss a simple generalization in which the vertices are weighted according to some probability distribution. This allows to use von Neumann's minimax theorem to deduce a dual statement. [ABSTRACT FROM AUTHOR]

Considerk( ⩾ 2) normal populations whose means are all known or unknown and whose variances are unknown. Let σ2[1]⩽ ⋅⋅⋅ ⩽ σ[k]2denote the ordered variances. Our goal is to select a non empty subset of thekpopulations whose size is at mostm(1 ⩽m⩽k− 1) so that the population associated with the smallest variance (called the best population) is included in the selected subset with a guaranteed minimum probabilityP whenever σ2[2]/σ[1]2⩾ δ > 1, whereP and δ are specified in advance of the experiment. Based on samples of sizenfrom each of the populations, we propose and investigate a procedure calledRBCP. We also derive some asymptotic results for our procedure. Some comparisons with an earlier available procedure are presented in terms of the average subset sizes for selected slippage configurations based on simulations. The results are illustrated by an example. [ABSTRACT FROM AUTHOR]

Let S be a subset of $$\mathbb {R}^d$$ with finite positive Lebesgue measure. The Beer index of convexity $${\text {b}}(S)$$ of S is the probability that two points of S chosen uniformly independently at random see each other in S. The convexity ratio $${\text {c}}(S)$$ of S is the Lebesgue measure of the largest convex subset of S divided by the Lebesgue measure of S. We investigate the relationship between these two natural measures of convexity. We show that every set $$S\subseteq \mathbb {R}^2$$ with simply connected components satisfies $${\text {b}}(S)\leqslant \alpha {\text {c}}(S)$$ for an absolute constant $$\alpha $$ , provided $${\text {b}}(S)$$ is defined. This implies an affirmative answer to the conjecture of Cabello et al. that this estimate holds for simple polygons. We also consider higher-order generalizations of $${\text {b}}(S)$$ . For $$1\leqslant k\leqslant d$$ , the k-index of convexity $${\text {b}}_k(S)$$ of a set $$S\subseteq \mathbb {R}^d$$ is the probability that the convex hull of a $$(k+1)$$ -tuple of points chosen uniformly independently at random from S is contained in S. We show that for every $$d\geqslant 2$$ there is a constant $$\beta (d)>0$$ such that every set $$S\subseteq \mathbb {R}^d$$ satisfies $${\text {b}}_d(S)\leqslant \beta {\text {c}}(S)$$ , provided $${\text {b}}_d(S)$$ exists. We provide an almost matching lower bound by showing that there is a constant $$\gamma (d)>0$$ such that for every $$\varepsilon \in (0,1)$$ there is a set $$S\subseteq \mathbb {R}^d$$ of Lebesgue measure 1 satisfying $${\text {c}}(S)\leqslant \varepsilon $$ and $${\text {b}}_d(S)\geqslant \gamma \frac{\varepsilon }{\log _2{1/\varepsilon }}\geqslant \gamma \frac{{\text {c}}(S)}{\log _2{1/{\text {c}}(S)}}$$ . [ABSTRACT FROM AUTHOR]

After more than a century of study, we do not yet fully understand how shapes and patterns are encoded and identified. Greater progress might result from quantifying stimulus information, thus allowing manipulation of the degree to which a shape or pattern can elicit recognition. The present work used discrete dot patterns that are seen as letters of the alphabet. By adjusting the density of the dots in each pattern, one can determine the probability that it will be recognized. The experiments displayed low-density dot patterns to human respondents, assessing the interval across which non-redundant information provided by two compatible subsets would combine to elicit recognition. This provided a measure of the time required for decay of information persistence. Viewed in the context of prior work, the evidence indicates that the retina mediates initial visibility of the stimulus trace, but the longer-duration persistence required for memory retrieval is mediated by visual cortex. [ABSTRACT FROM AUTHOR]

Let G = (V,E) be a graph, k is a positive integer and let S = (U,X) be a semigraph. A subset D ⊆ V is called a k-strong(weak) dominating set of the graph G if every vertex v ϵ V - D is strongly(weakly) dominated by at least k vertices of D. The k-strong(weak)domination number γsk(γwk) is the minimum cardinality of k-strong!weak) dominating set of G. A subset D ⊆ U is called k-strong(weak) adjacent dominating set of semigraph S if every vertex u ∊ U -D is strongly(weakly) adjacent dominated by at least k vertices of D. The minimum cardinality of a k- strong adjacent dominating set, a k- weak adjacent dominating set are respectively denoted as γs,ak(S) and γw,ak(S). In this paper, we determine γsk and γWk for standard graphs. Also, we establish bounds on the above domination parameters for graphs and semigraphs. We prove that the k-strong(weak) domination problem is NP-complete for general graphs. [ABSTRACT FROM AUTHOR]

In this article, the problem of selecting an optimal subset from a finite set of simulated designs is considered. Given the total simulation budget constraint, the selection problem aims to maximize the Probability of Correct Selection (PCS) of the topmdesigns. To simplify the complexity of the PCS, an approximated probability measure is developed and an asymptotically optimal solution of the resulting problem is derived. A subset selection procedure, which is easy to implement in practice, is then designed. More important, we provide some useful insights on characterizing an efficient subset selection rule and howit can be achieved by adjusting the simulation budgets allocated to all of the designs. [ABSTRACT FROM AUTHOR]

The Levin-Robbins-Leu (LRL) family of sequential subset selectionprocedures admits of a simple formula that provides a lower bound for the probability of various types of subset selection events. Here we demonstrate that a corresponding lower bound formula holds forlattice eventswhen using the nonadaptive family member with binary outcomes. Lattice events are more general than the type of events that we previously considered in demonstrating that the nonadaptive LRL procedure selects “acceptable” subsets with an arbitrarily large prespecified probability irrespective of the true population parameters. Interestingly, the proof of the lower bound formula for lattice events that we present here simplifies the methods previously used and sheds additional light on why the lower bound formula holds. As for acceptable subset selection, we conjecture that the other LRL family members, which allow for adaptive elimination of inferior candidates and/or recruitment of superior candidates, obey the same lower bound formulas for lattice events. [ABSTRACT FROM AUTHOR]

The random--or orderly--seating of passengers on a bus is used to motivate several questions about cycles of permutations. These in turn motivates the investigation of bijections between special subsets of permutations. The goal, of course, is to give simple explanations of surprising facts. [ABSTRACT FROM AUTHOR]

Direct Monte Carlo simulation for the reliability analysis of slope stability with spatially variable soil properties suffers from a serious lack of efficiency when the probability of failure, p f , is low (e.g., p f < 10 - 6 ). Based on the multiple response surfaces and Subset simulation, this paper proposes an efficient approach for the estimating of small probabilities of slope failure in spatially variable soils. An improved Cholesky decomposition technique is presented for the simulation of the globally non-stationary random fields of spatially variable soil properties in the multiple soil layers. Two slope examples are investigated to demonstrate the effectiveness of the proposed method. The efficiency of the proposed approach for parametric sensitivity analysis is also highlighted. [ABSTRACT FROM AUTHOR]

One of the main challenges facing wireless sensor networks (WSNs) is the limited power resources available at small sensor nodes. It is therefore desired to reduce the power consumption of sensors while keeping the distortion between the source and its estimate at the fusion centre (FC) below a specific threshold. In this paper, we analyze a subset selection strategy to reduce the average transmission power of the WSN. We consider a two-hop network and assume the channels between the source and the relay sensors to be time-varying fading channels, modeled as Gilbert-Elliott channels. We show that when these channels are known at the FC, a subset of sensors can be selected by the FC to minimize transmit power while satisfying the distortion criterion. Through analysis, we derive the probability distribution of the size of this subset. We also consider practical aspects of implementing the proposed scheme, including channel estimation at relays. Through simulations, we compare the performance of the proposed scheme with schemes appearing in the literature. Simulation results confirm that for a certain range of end-to-end bit-error rates (BERs), the proposed scheme succeeds to achieve a superior power reduction compared to other schemes. [ABSTRACT FROM PUBLISHER]

Estimates of quantities characterizing the complexity of the family of convex subsets of the d-dimensional cube [1, n] as n→∞ are given. The geometric properties of spaces with norm generated by the generalized majorant of partial sums are studied. [ABSTRACT FROM AUTHOR]

Let b ≥ 2 be a positive integer. Let D be a finite subset of Z and { n k } k = 1 ∞ ⊆ N be a sequence of strictly increasing numbers. A Moran measure μ b , D , { n k } is a Borel probability measure generated by the Moran iterated function system (Moran IFS) { f k , d ( x ) = b n k − 1 − n k ( x + d ) : d ∈ D , k ∈ N , n 0 = 0 } . In this paper we study one of the basic problems in Fourier analysis associated with μ b , D , { n k } . More precisely, we give some conditions under which the measure μ b , D , { n k } is a spectral measure, i.e., there exists a discrete subset Λ ⊆ R such that E ( Λ ) = { e 2 π i λ x : λ ∈ Λ } is an orthonormal basis for L 2 ( μ b , D , { n k } ) . [ABSTRACT FROM AUTHOR]

The operating characteristics (OCs) of a subset ranking and selection procedure are derived for the hybrid randomized response model developed by Jia and McDonald (2009). The OCs include the probability of a correctP(CS), the individual selection probability γi, and the expected subset sizeE(S), under the slippage configuration or the equi-spaced configuration. An example comparing failure rates of contraceptive methods is used to illustrate the use of these new results. [ABSTRACT FROM PUBLISHER]

The paper deals with the well known set packing problem and its special case, when the number of subsets is maximized. It is assumed that some of the problem coefficients are realizations of mutually independent random variables. Average case (i.e. asymptotical probabilistic) properties of selected problem characteristics are investigated for the variety of possible instances of the problem. The important results of the paper are: • Behavior of the optimal solution values of the set packing problem is presented for the special asymptotic case, where mutual asymptotical relation between m (number of elements of the packed set) and n (number of sets provided) is playing an essential role. • Probability of reaching feasible solution is reasonably high (i.e. ⩾ 2/e, 2/e ≈ 0.736); moreover, it may be set arbitrarily close to 1 (e.g. 0.999), although the deterioration in the quality of approximation of the behavior of the optimal solution values may be substantial. • Some relations between the general case of the set packing problem and its maximization for the special case are investigated. [ABSTRACT FROM AUTHOR]

One of the main problems with designing a production line is to find the optimal number of buffers between workstations in order to maximizes the throughput. This problem known as buffer allocation problem. Previous work in this problem focus on selecting a single buffer profile that has the maximum throughput. The objective in this paper would be to selecting from a large number of alternatives, the best subset of buffer profiles where its throughput are at its maximum. The ordinal optimization with optimal computing budget allocation approaches will be used to isolating the best subset of buffer profile, where its throughput is maximum, from the set of all alternatives. Numerical results show that the proposed algorithm finds the best subset of the puffer allocation with high probability and small replications numbers of samples. [ABSTRACT FROM AUTHOR]

Stochastic Shortest Path Problems (SSPs) are a common representation for probabilistic planning problems. Two approaches can be used to solve SSPs: (i) consider all probabilistically reachable states and (ii) plan only for a subset of these reachable states. Closed policies, the solutions obtained in the former approach, require significant computational effort, and they do not require replanning, i.e., the planner is never re-invoked. The second approach, employed by replanners, computes open policies, i.e., policies for a subset of the probabilistically reachable states. Therefore, when a state is reached in which the open policy is not defined, the replanner is reinvoked to compute a new open policy. In this article, we introduce a special case of SSPs, the depth-based short-sighted SSPs, in which every state has a nonzero probability of being reached using at most t actions. We also introduce the novel algorithm Short-Sighted Probabilistic Planner (SSiPP), which solves SSPs through depth-based short-sighted SSPs and guarantees that at least t actions can be executed without replanning. Therefore, SSiPP can compute both open and closed policies: as t increases, the returned policy approaches the behavior of a closed policy, and for t large enough, the returned policy is closed. Moreover, we present two extensions to SSiPP: Labeled-SSiPP and SSiPP-FF. The former extension incorporates a labeling mechanism to avoid revisiting states that have already converged. The latter extension combines SSiPP and determinizations to improve the performance of SSiPP in problems without dead ends. We also performed an extensive empirical evaluation of SSiPP and its extensions in several problems against state-of-the-art planners. The results show that (i) Labeled-SSiPP outperforms SSiPP and the considered planners in the task of finding the optimal solution when the problems have a low percentage of relevant states; and (ii) SSiPP-FF outperforms SSiPP in the task of quickly finding suboptimal solutions to problems without dead ends while performing similarly in problems with dead ends. [ABSTRACT FROM AUTHOR]

We consider a class of packing problems with uncertain data, which we refer to as the chance-constrained binary packing problem. In this problem, a subset of items is selected that maximizes the total profit so that a generic packing constraint is satisfied with high probability. Interesting special cases of our problem include chance-constrained knapsack and set packing problems with random coefficients. We propose a problem formulation in its original space based on the so-called probabilistic covers. We focus our solution approaches on the special case in which the uncertainty is represented by a finite number of scenarios. In this case, the problem can be formulated as an integer program by introducing a binary decision variable to represent feasibility of each scenario. We derive a computationally efficient coefficient strengthening procedure for this formulation, and demonstrate how the scenario variables can be efficiently projected out of the linear programming relaxation. We also study how methods for lifting deterministic cover inequalities can be leveraged to perform approximate lifting of probabilistic cover inequalities. We conduct an extensive computational study to illustrate the potential benefits of our proposed techniques on various problem classes. [ABSTRACT FROM AUTHOR]

Considerk( ⩾ 2) normal populations with unknown means μ1, …, μk, and a common known variance σ2. Let μ[1]⩽ ⋅⋅⋅ ⩽ μ[k]denote the ordered μi.The populations associated with thet(1 ⩽t⩽k− 1) largest means are called the t best populations. Hsu and Panchapakesan (2004) proposed and investigated a procedureRHPfor selecting a non empty subset of the k populations whose size is at mostm(1 ⩽m⩽k−t) so that at least one of the t best populations is included in the selected subset with a minimum guaranteed probability P* whenever μ[k − t + 1]− μ[k − t]⩾ δ*, whereP* and δ* are specified in advance of the experiment. This probability requirement is known as the indifference-zone probability requirement. In the present article, we investigate the same procedureRHPfor the same goal as before but whenk−t

A new approximate Bayesian computation (ABC) algorithm for Bayesian updating of model parameters is proposed in this paper, which combines the ABC principles with the technique of subset simulation for efficient rare-event simulation, first developed in S. K. Au and J. L. Beck [Probabilistic Engrg. Mech., 16 (2001), pp. 263-277]. It has been named ABC-SubSim. The idea is to choose the nested decreasing sequence of regions in subset simulation as the regions that correspond to increasingly closer approximations of the actual data vector in observation space. The efficiency of the algorithm is demonstrated in two examples that illustrate some of the challenges faced in real-world applications of ABC. We show that the proposed algorithm outperforms other recent sequential ABC algorithms in terms of computational efficiency while achieving the same, or better, measure of accuracy in the posterior distribution. We also show that ABC-SubSim readily provides an estimate of the evidence (marginal likelihood) for posterior model class assessment, as a by-product. [ABSTRACT FROM AUTHOR]

We show, through local estimates and simulation, that if one constrains simple graphs by their densities ϵ of edges and τ of triangles, then asymptotically (in the number of vertices) for over 95% of the possible range of those densities there is a well-defined typical graph, and it has a very simple structure: the vertices are decomposed into two subsets V1 and V2 of fixed relative size c and 1 − c, and there are well-defined probabilities of edges, gjk, between vj ∈ Vj, and vk ∈ Vk. Furthermore the four parameters c, g11, g22 and g12 are smooth functions of (ϵ, τ) except at two smooth ‘phase transition’ curves. [ABSTRACT FROM AUTHOR]

For low failure probability prediction, subset simulation can reduce the number of simulations significantly compared to the traditional MCS method for a target prediction error limit. To further reduce the computational effort for cases where the performance function evaluation is tedious and time-consuming, the performance function is approximated by a sequentially updated (instead of a global) Kriging model. For this purpose, an active learning technique with a new learning and stopping criterion is employed to efficiently select points to train the computationally cheaper Kriging model at each simulation level, which is used to estimate the intermediate threshold and generate a new simulation sample. The updated Kriging model at the final subset simulation level is used to compute the conditional failure probability. The failure probability is estimated based on an initial simulation sample size N , and an updated N is computed and employed to obtain the final failure probability within a desired bound on the variability. The efficiency (in terms of the number of expensive evaluations using the actual performance function) and prediction error (represented by the mean square error (MSE)) of the proposed method are benchmarked using several examples. The method is shown to be more efficient (using lesser expensive evaluations) with smaller MSE for problems having low failure probabilities compared with selected existing methods. • New active learning technique is employed to facilitate Kriging model updating at each simulation level. • Kriging model is used to determine intermediate threshold for generation of new simulation sample at low cost. • Simulation sample size is estimated based on desired limit in variability of failure probability. • Method yields better results than those from method of Ling et al. (2019) in terms of efficiency and prediction error. [ABSTRACT FROM AUTHOR]