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Recurrence relations for integrals that involve the density of multivariate normal distributions are developed. These recursions allow fast computation of the moments of folded and truncated multivariate normal distributions. Besides being numerically efficient, the proposed recursions also allow us to obtain explicit expressions of low-order moments of folded and truncated multivariate normal distributions. Supplementary material for this article is available online.

Plane segmentation is a basic task in the automatic reconstruction of indoor and urban environments from unorganized point clouds acquired by laser scanners. As one of the most common plane-segmentation methods, standard Random Sample Consensus (RANSAC) is often used to continually detect planes one after another. However, it suffers from the spurious-plane problem when noise and outliers exist due to the uncertainty of randomly sampling the minimum subset with 3 points. An improved RANSAC method based on Normal Distribution Transformation (NDT) cells is proposed in this study to avoid spurious planes for 3D point-cloud plane segmentation. A planar NDT cell is selected as a minimal sample in each iteration to ensure the correctness of sampling on the same plane surface. The 3D NDT represents the point cloud with a set of NDT cells and models the observed points with a normal distribution within each cell. The geometric appearances of NDT cells are used to classify the NDT cells into planar and non-planar cells. The proposed method is verified on three indoor scenes. The experimental results show that the correctness exceeds 88.5% and the completeness exceeds 85.0%, which indicates that the proposed method identifies more reliable and accurate planes than standard RANSAC. It also executes faster. These results validate the suitability of the method.

Skew-symmetric families of distributions such as the skew-normal and skew-t represent supersets of the normal and t distributions, and they exhibit richer classes of extremal behaviour. By defining a non-stationary skew-normal process, which allows the easy handling of positive definite, non-stationary covariance functions, we derive a new family of max-stable processes – the extremal skew-t process. This process is a superset of non-stationary processes that include the stationary extremal-t processes. We provide the spectral representation and the resulting angular densities of the extremal skew-t process and illustrate its practical implementation.

Simulation from the truncated multivariate normal distribution in high dimensions is a recurrent problem in statistical computing and is typically only feasible by using approximate Markov chain Monte Carlo sampling. We propose a minimax tilting method for exact independently and identically distributed data simulation from the truncated multivariate normal distribution. The new methodology provides both a method for simulation and an efficient estimator to hitherto intractable Gaussian integrals. We prove that the estimator has a rare vanishing relative error asymptotic property. Numerical experiments suggest that the scheme proposed is accurate in a wide range of set-ups for which competing estimation schemes fail. We give an application to exact independently and identically distributed data simulation from the Bayesian posterior of the probit regression model.

In this study, we derive the characteristic function of the multivariate folded normal distribution, a distribution that arises when the magnitudes—but not the signs—of a normally distributed random vector are of interest. The folded normal distribution is widely applicable across various fields. Thus, obtaining an analytical expression for its characteristic function is pivotal in understanding its fundamental properties. Moreover, this allows one to facilitate numerical evaluations of complex distributions involving linear combinations of absolute values of dependent normal variables. The derivation is based on a novel expression of the moment generating function, formulated using the cumulative distribution function of the multivariate normal distribution. To validate our findings, we present two examples using our MATLAB implementation. We compare the characteristic function for the sum of the absolute values of elements of a multivariate normal vector with the simulated empirical counterpart. Additionally, we derive the second mixed moment of the bivariate folded normal distribution from the moment generating function, demonstrating its agreement with known theoretical expressions.

This study introduces the Multi-objective Generalized Normal Distribution Optimization (MOGNDO) algorithm, an advancement of the Generalized Normal Distribution Optimization (GNDO) algorithm, now adapted for multi-objective optimization tasks. The GNDO algorithm, previously known for its effectiveness in single-objective optimization, has been enhanced with two key features for multi-objective optimization. The first is the addition of an archival mechanism to store non-dominated Pareto optimal solutions, ensuring a detailed record of the best outcomes. The second enhancement is a new leader selection mechanism, designed to strategically identify and select the best solutions from the archive to guide the optimization process. This enhancement positions MOGNDO as a cutting-edge solution in multi-objective optimization, setting a new benchmark for evaluating its performance against leading algorithms in the field. The algorithm's effectiveness is rigorously tested across 35 varied case studies, encompassing both mathematical and engineering challenges, and benchmarked against prominent algorithms like MOPSO, MOGWO, MOHHO, MSSA, MOALO, MOMVO, and MOAOS. Utilizing metrics such as Generational Distance (GD), Inverted Generational Distance (IGD), and Maximum Spread (MS), the study underscores MOGNDO's ability to produce Pareto fronts of high quality, marked by exceptional precision and diversity. The results affirm MOGNDO's superior performance and versatility, not only in theoretical tests but also in addressing complex real-world engineering problems, showcasing its high convergence and coverage capabilities. The source codes of the MOGNDO algorithm are publicly available at  https://nimakhodadadi.com/algorithms-%2B-codes .

Coronaviruses are enveloped RNA viruses from the Coronaviridae family affecting neurological, gastrointestinal, hepatic and respiratory systems. In late 2019 a new member of this family belonging to the Betacoronavirus genera (referred to as COVID-19) originated and spread quickly across the world calling for strict containment plans and policies. In most countries in the world, the outbreak of the disease has been serious and the number of confirmed COVID-19 cases has increased daily, while, fortunately the recovered COVID-19 cases have also increased. Clearly, forecasting the “confirmed” and “recovered” COVID-19 cases helps planning to control the disease and plan for utilization of health care resources. Time series models based on statistical methodology are useful to model time-indexed data and for forecasting. Autoregressive time series models based on two-piece scale mixture normal distributions, called TP–SMN–AR models, is a flexible family of models involving many classical symmetric/asymmetric and light/heavy tailed autoregressive models. In this paper, we use this family of models to analyze the real world time series data of confirmed and recovered COVID-19 cases.

INTRODUCTION This study introduces a language‐independent, acoustic‐based method to identify the bimodal pauses in connected speech related to Alzheimer's disease (AD) through a log‐normal distribution, aiming to explore pausing behavior as a digital marker of cognitive decline. METHODS We fitted a bimodal log‐normal distribution to 4473 pauses automatically extracted through acoustic analysis. We compared linear and logarithmic pause indices between cognitive groups and explored their neurocognitive correlates. RESULTS We empirically revealed a dual‐mode pause distribution, customizing a threshold of ≈ 180 ms to differentiate short and long pauses. This bimodal distribution effectively distinguished cognitive groups, driven by variations in the central tendency of long pauses. Both pause types were elevated in individuals with mild cognitive impairment and correlated with tau and amyloid levels. DISCUSSION Bimodal pause distribution shows promise as a sensitive speech‐based indicator of cognitive decline, linking closely to AD biomarkers. We introduce a refined, unbiased, language‐independent framework for broader application across diverse populations. Highlights Pausing in connected speech was investigated as a digital marker of cognitive decline. Bimodal log‐normal pause distribution distinguishes between cognitive groups. Short (80–180 ms) and long (> 180 ms) pauses correlate with tau and amyloid.

The problem regarding the optimal integration of efficient reactive power compensation in radial and meshed distribution networks using fixed-step capacitor banks and distribution static compensators (D-STATCOMs) is addressed in this research paper by proposing a master–slave optimization methodology. Radial and meshed distribution topologies are considered for the grid structure while including variable active and reactive demand curves. An economic analysis is performed, considering the net present value of the optimization plan, as well as the costs of energy losses and the capacitor banks’ acquisition, installation, and operation. In the case of the D-STATCOMs, an annualized costs analysis is presented. In the master stage, the discrete version of the generalized normal distribution optimization (GNDO) algorithm selects the nodes and the sizes of the capacitor banks. In the slave stage, the successive approximations power flow approach is implemented. Numerical results in the IEEE 33-bus grid (with both radial and meshed topologies) and the IEEE 85-bus grid (with a radial configuration) demonstrated the proposed master–slave optimization’s effectiveness in minimizing the project’s expected net present value for a planning period of five years. Moreover, a simulation in the IEEE 69-bus grid under peak operation conditions showed that the GNDO approach is an excellent optimization technique to solve the studied problem when compared to combinatorial and exact optimization methods. In addition, numerical validations considering D-STATCOMs in the IEEE 85-bus grid confirmed the effectiveness and robustness of the GNDO approach in addressing problems associated with optimal reactive power compensation in medium-voltage distribution systems.

Multiple researchers have proposed skew random fields derived from multivariate skew distributions, yet the consistency of these fields has been questioned. Mahmoudian (2018) and Saber et al. (2018) have put forth alternative suggestions to address these concerns. In our study, we identify that the random fields outlined by Mahmoudian (2018) continue to demonstrate consistency issues, suggesting a flaw in their definition. Finally we propose a skew random field and apply it to spatial prediction.