Probability Interval Research Articles

Probability of Electron Ion 3Li^(2+) In Momentum Space At n ≤ 3

Lithium ions (3) is a particle that has a single electron and is formed because lithium metal loses 2 of its 3 valence electrons. Probability is the square modulus of a wave function. Electron probability is a description of the discovery of electrons in space. This study aims to determine the probability value of electron momentum of Lithium ion (3). This type of research is non-experimental research in the form of developing existing theories with theoretical research methods. When viewed from the main orbital quantum value, the greater the probability value or the easier the electron is found at a larger quantum number. When viewed from the level of the probability interval, the greater the probability interval, the greater the main quantum value. In accordance with the theory which states that the large main quantum number n and orbital quantum l result in the probability value of Lithium ions getting bigger.

A generalization of the Moore and Yang integral and interval probability density functions

ABSTRACT Based on an extension of Riemann sums, Moore and Yang have defined an integral notion for the context of continuous inclusion monotonic interval functions in which the limits of integration are real numbers. This integral notion generalizes the usual one for real-valued functions based on Riemann sums. In this paper we extend this approach by considering intervals as limits of integration and abolishing the inclusion monotonic restriction of the integrable interval functions. Also, such a new integration notion is used to define interval probability density functions and use it in interval probability distribution functions.

Optimal Histogram Filter

The article discusses a technique for constructing an optimal histogram filter and its modifications, taking into account a priori information about the expected probability distribution density. The main idea of constructing a histogram filter is to apply a special transformation that displays the profile of a section of any distribution law into a constant level of characteristic numbers equivalent to it. This transformation allows to determine the coefficients of the histogram filter. An estimate of the value of the number of data of a particular interval of the histogram is formed by the characteristic function of the filter containing real data and equivalent to the characteristic number. The convergence of the estimates obtained by the histogram filter to the true values of the interval probabilities is shown. Modifications of the optimal histogram filter that require less computational costs for their implementation are considered. The upper bounds of the qualitative characteristics of filters are obtained. It has been established that the optimal histogram filter, regardless of the type of distribution law, provides three times the best quality of identification (recognition) in comparison with the standard histogram estimate. The efficiency of the histogram filter is confirmed by simulations. The histogram filter is an easy-to-implement tool that can be easily integrated into any open distribution law identification (recognition) algorithm.

Verified propagation of imprecise probabilities in non-linear ODEs

We combine reachability analysis and probability bounds analysis, which allow for imprecisely known random variables (multivariate intervals or p-boxes) to be specified as the initial states of a dynamical system. In combination, the methods allow for the temporal evolution of p-boxes to be rigorously computed, and they give interval probabilities for formal verification problems, also called failure probability calculations in reliability analysis. The methodology places no constraints on the input probability distribution or p-box and can handle dependencies generally in the form of copulas. We also provide a consonant approximation method for multivariate p-boxes, which allows for the prediction sets of dynamical systems to be efficiently computed. The presented methodology is rigorous and automatically verified, as both the dynamics and uncertainties are represented and solved with guaranteed enclosures.

Short-term wind power forecasting by bidirectional attention mechanism LSTM and its probability interval prediction by sliding-window KDE

Deterministic point wind power forecasting (DP-WPF) and its probability interval prediction (PIP) are indispensable to short-term peak alleviation and frequency regulation in power systems with large-scale wind power injection. To improve short-term DP-WPF by long short-term memory (LSTM), a horizontal/vertical bidirectional feature attention (BFA) based LSTM model is proposed. More specifically, the BFA-LSTM model has three parts: first, multivariate time series are fed into LSTM to extract long-short-term temporal features; second, the LSTM outputs are processed horizontally as well as vertically for retrieving step-wise/multistep-wise temporal features, respectively, namely, in the bidirectional attention sense; third, both the horizontal and vertical attention weights are adaptively adjusted according to the feature importance in DP-WPF. Cases comparison shows that the suggested modeling is stably superior to most common counterparts. To address PIP by kernel density estimation (KDE), sliding-window KDE is leveraged for probability analysis. More precisely, probability density functions (PDF) and probability intervals are estimated with sliding-window samples, which are non-parametric operations and involve finitely many local samples. Superior performances of PIP by sliding-window KDE at different confidence levels indicate that the sliding-window PDF approach is highly effective in contrast to those with all samples.

Algorithmic and Software Program for Determining the Labor Intensity of One-Season Maintenance of Tractors

The paper describes a program for calculating the total specific labor intensity of tractor maintenance during their one-season use. The mathematical support of the program is represented by models that allow, based on data on the seasonal operation of tractors, to calculate the total specific labor intensity of maintenance. In the algorithmic support, the main stages of calculating the indicator under study are formulated. The user interface of the program is created in the form of a form using the Visual Basic for Applications programming language. In addition to the labor intensity itself, the program calculates statistical parameters, upper limits of seasonal operating time and probabilities of intervals. The proposed program can be used to calculate the total specific labor intensity of maintenance not only for tractors, but also for other machines used in agricultural production.

Robust actions for improving supply chain resilience and viability

It is vital for supply chains (SCs) to survive the dramatic and long-term impacts from severe disruptive events, such as COVID-19 pandemic. SC viability, an extension of SC resilience, is increasingly attracting attention from both academics and practitioners. To improve SC viability, the government can perform a series of costly interventions on SCs. Due to data scarcity on unpredictable disruptive events, especially under the pandemic, the information related to SC partners may not be accurately obtained. In this paper, we investigate a novel SC resilience and viability improving problem under severe disruptive events, in which only the probability intervals of SC partners’ states are known. The problem consists of the selection of appropriate intervention actions, respecting a limited capital budget. The objective is to minimize the worst-case disruption risk of the manufacturer. Specifically, Causal Bayesian Network (CBN) is applied to quantify the SC ripple effects; Do-calculus technique is used to measure the benefits of government intervention actions; and robust optimization is employed to minimize the disruption risk under the worst-case condition. For the problem, a new robust optimization model that combines the CBN and the Do-calculus is constructed. Based on analyses of problem features, an efficient problem-specific branch-and-bound (PS-BAB) algorithm is proposed to solve the problem exactly. Experimental results show the efficiency of our methodology and managerial insights are drawn.

A signomial programming-based approach for multi-echelon supply chain disruption risk assessment with robust dynamic Bayesian network

Disruption risk assessment is a primary and crucial step before taking measures to mitigate the negative impact of disruptions propagating along supply chains (SCs). Recently, robust dynamic Bayesian network (DBN) provides a valid tool for disruption risk estimation under the ripple effect in a data-scarce environment. However, existing literature has not considered such disruption risk assessment for multi-echelon SCs that are usually structurally complicated and thus vulnerable to disruptions with ripple effects. Motivated by this fact, we study the disruption risk assessment problem under the ripple effect for a multi-echelon SC with several suppliers and one manufacturer, in which only probability intervals of the suppliers’ states and those of the related disruption propagations are known. The aim is to acquire a robust risk estimation, measured by the worst-case total weighted probabilities for the manufacturer in the disrupted state over a time horizon. For the problem, a nonconvex nonlinear programming model is established to obtain the worst-case risk estimation. To efficiently solve the problem, a novel signomial programming (SP)-based approach is developed for finding near-optimal solutions. Numerical experiments on instances in the literature and our randomly generated instances are conducted to evaluate the efficiency of the proposed method. Besides, managerial insights are drawn.

Continuous time recurrent neural networks: Overview and benchmarking at forecasting blood glucose in the intensive care unit

ObjectiveBlood glucose measurements in the intensive care unit (ICU) are typically made at irregular intervals. This presents a challenge in choice of forecasting model. This article gives an overview of continuous time autoregressive recurrent neural networks (CTRNNs) and evaluates how they compare to autoregressive gradient boosted trees (GBT) in forecasting blood glucose in the ICU. MethodsContinuous time autoregressive recurrent neural networks (CTRNNs) are a deep learning model that account for irregular observations through incorporating continuous evolution of the hidden states between observations. This is achieved using a neural ordinary differential equation (ODE) or neural flow layer. In this manuscript, we give an overview of these models, including the varying architectures that have been proposed to account for issues such as ongoing medical interventions. Further, we demonstrate the application of these models to probabilistic forecasting of blood glucose in a critical care setting using electronic medical record and simulated data and compare with GBT and linear models. ResultsThe experiments confirm that addition of a neural ODE or neural flow layer generally improves the performance of autoregressive recurrent neural networks in the irregular measurement setting. However, several CTRNN architecture are outperformed by a GBT model (Catboost), with only a long short-term memory (LSTM) and neural ODE based architecture (ODE-LSTM) achieving comparable performance on probabilistic forecasting metrics such as the continuous ranked probability score (ODE-LSTM: 0.118 ± 0.001; Catboost: 0.118 ± 0.001), ignorance score (0.152 ± 0.008; 0.149 ± 0.002) and interval score (175 ± 1; 176 ± 1). ConclusionThe application of deep learning methods for forecasting in situations with irregularly measured time series such as blood glucose shows promise. However, appropriate benchmarking by methods such as GBT approaches (plus feature transformation) are key in highlighting whether novel methodologies are truly state of the art in tabular data settings.

A data-driven distributionally newsvendor problem for edge-cloud collaboration in intelligent manufacturing systems

In intelligent manufacturing systems, the industrial informatics has features of multi-source, multi-noise, and time series. It is difficult for small and medium enterprises (SMEs) to directly exploit the enormous amounts of data due to the limited budgets and computing capabilities. Edge intelligence is a key technique to power intelligent manufacturing systems and provide knowledge transferred from the cloud to SMEs at the edges. To address edge-cloud collaboration issue, we propose a refined data-driven distributionally robust newsvendor model based on φ-divergence measures and imprecise Dirichlet models (DRN-IDM). We construct new distributional uncertainty sets by effectively integrating local censored demand data and cloud knowledge, which helps SMEs to make intelligent production decisions and reduce significant decision deviations, even under a small censored data set. In particular, the novel demand uncertainty sets can depict the distance between distributions and probability intervals. Then, we transform the DRN-IDM model into a convex optimization model that is amenable to algorithmic implementation. Additionally, based on the coefficient of variation of limited historical data, we propose an adaptive demand information fusion procedure to achieve excellent synergy effect from cloud knowledge. We also validate the effectiveness of the DRN-IDM model and the practicability of adaptive procedure using extensive numerical studies with both simulated and real-life data. Furthermore, we measure the relative expected value of cloud knowledge and investigate the effect of censored demand samples. Our results verify the effectiveness condition of the DRN-IDM model and indicate that cloud knowledge can improve the precision and robustness of SMEs’ production decisions with small-scale censored data. Interestingly, the verified adaptive procedure can be applied in the learning criteria design of metaheuristics in intelligent manufacturing systems, and the reconstructed uncertainty set can narrow the search space to improve the convergence performance of algorithms.

Hybrid reliability analysis with incomplete interval data based on adaptive Kriging

Hybrid reliability analysis with mixed random and interval uncertainties is a significant challenge in the reliability assessment of engineering structures. The situation will be more intractable when involving incomplete interval data. To obtain reliable estimates of the failure probability limits, an effective parameter estimation method, integrating the quantile variant of the Expectation-Maximization algorithm and Kullback-Leibler divergence, is proposed to transform uncertain variables with incomplete data into random variables with distribution uncertainty. Then, an adaptive Kriging-assisted hybrid reliability analysis method is developed to ensure computational accuracy and efficiency. In this method, a candidate pool incorporating the distribution uncertainty is constructed and its size is adaptively reduced by removing the samples that violate the projection uniformity on input dimensions with exact distributions. Meanwhile, an improved U learning function and an error-based convergence criterion are defined to drive and stop the adaptive process. Then the failure probability limits are estimated by combining the refined Kriging model and Monte Carlo simulation. Four application examples are employed to verify the superiority of the proposed method. Comparison results show that the proposed method can significantly improve computational efficiency while ensuring the accuracy and reliability of the estimated failure probability interval under incomplete interval observations.

Decision-making for stochastic multi-objective dispatch of integrated electrical and heating systems

In the realm of modern energy systems, addressing the challenges of enhancing flexibility and efficiency under uncertain conditions is of paramount importance. This paper explores the stochastic multi-objective optimal multi-energy flow problem within the context of integrated electrical and heating systems (IEHS). First, the electrical network, the heating network, and the energy hubs were modeled in a completely linearized form. The linear weighted sum method with variable weights was used to transform the multi-objective problem into a single-objective problem and generate a large number of Pareto-optimal solutions. Second, the input stochastic variables were divided into multi-interval scenarios by employing the Cartesian product. For each interval scenario, the interval satisfaction degree level was proposed to convert the constraints involving interval numbers into deterministic ones. Third, a multiple attributes decision analysis (MADA) approach was proposed based on evidential reasoning theory. Six evaluation attributes, namely, the power purchase cost and pollution gas emissions of IEHS, the sum of power loss and sum of voltage deviation in the electrical system, the sum of temperature drop in the heating system, and the interval probability value of the multi-interval scenarios, were considered to rank the Pareto-optimal solutions collected from the multi-interval scenarios and determine the final dispatch solution (called the Utopia solution). Numerical simulations demonstrated that the Utopia solution can comprehensively evaluate various attributes, making it the most suitable option for meeting the operational requirements of IEHS.

АНАЛИЗ НАДЕЖНОСТИ ЭЛЕМЕНТОВ СТРОИТЕЛЬНЫХ КОНСТРУКЦИЙ С ЯДЕРНОЙ ОЦЕНКОЙ РАСПРЕДЕЛЕНИЯ ВЕРОЯТНОСТЕЙ

The article describes an approach to structural reliability analysis, in which distribution functions of random variables are formed on the basis of nonparametric statistics, in particular, by kernel density estimation. This approach avoids hypotheses about the choice of the distribution function of a random variable and the estimation of its parameters. The proposed approach acquires particular relevance in practical tasks of probabilistic reliability assessment during the inspection of buildings and structures when it is not possible to obtain a large amount of statistical data about the structural material or soil properties. An example of the formation of a p-box is shown on the example of probability distribution functions obtained on the basis of kernel estimation. Increasing the level of reliability in conditions of data uncertainty can be achieved by increasing the cross-sections / reinforcement of structures or by increasing the quantity and quality of statistical data, which will make the interval of the non-failure probability more informative.

Nonparametric Bootstrap Confidence Intervals for the Population Mean of a Zero-Truncated Poisson-Lindley Distribution and Their Application

Recently, the zero-truncated Poisson-Lindley distribution has been proposed for studying count data containing non-zero values. However, the nonparametric bootstrap confidence interval estimation of the population mean has not yet been studied. In this study, confidence interval estimation based on percentile, simple, and biased-corrected bootstrap methods was compared in terms of coverage probability and average interval length via Monte Carlo simulation. The true values of parameter () were set as 0.25, 0.5, 1, 1.5, and 2, and the population means are approximate 7.7586, 4.0909, 2.4000, 1.8817, and 1.6364, respectively. The bootstrap samples (=1,000) of size were generated from the original sample, and each simulation was repeated 1,000 times. The results indicate that attaining the nominal confidence level using the bootstrap confidence intervals was impossible for small sample sizes regardless of the other settings. Moreover, when the sample size was large, the performance of the nonparametric bootstrap confidence intervals was not substantially different. Overall, the bias-corrected bootstrap confidence interval outperformed the others, even for small sample sizes. Last, the nonparametric bootstrap confidence intervals were used to calculate the confidence interval for the population mean of the zero-truncated Poisson-Lindley distribution via two numerical examples, the results of which match those from the simulation study.

LITESPARK-024: A randomized phase 1/2 study of belzutifan with or without palbociclib for previously treated advanced renal cell carcinoma

Abstract Background Immunotherapy is a standard-of-care first-line treatment for advanced clear cell renal cell carcinoma (ccRCC). However, many patients will develop resistance to first-line therapy, and effective second- and later-line treatment options are therefore needed. The von Hippel-Lindau (VHL) gene is inactivated in approximately 90% of patients with RCC and results in the constitutive activation of hypoxia-inducible factor 2α (HIF-2α), a key oncogenic driver in RCC. Belzutifan, a first-in-class HIF-2α inhibitor, has demonstrated antitumor activity with manageable safety in previously treated patients with advanced ccRCC. The cyclin-dependent kinase (CDK) pathway is also associated with poor clinical outcomes in ccRCC. In RCC cell lines, the CDK 4/6 inhibitor palbociclib inhibited cell growth. CDK 4/6 inhibition has shown synergistic antiproliferative effects with HIF-2α inhibition in HIF-2α–dependent VHL -/- RCC cell lines. Palbociclib could therefore potentially enhance the efficacy of belzutifan as combination therapy for previously treated pts with advanced RCC. Methods LITESPARK-024 is a 2-part, open-label, multicenter, phase 1/2 randomized study (NCT05468697). Part 1 is intended to establish the recommended phase 2 dose (RP2D) of belzutifan plus palbociclib using a modified toxicity probability interval design. After the RP2D is established, part 2 will directly compare the safety and efficacy of belzutifan monotherapy with belzutifan + palbociclib in patients with advanced ccRCC. In both parts, patients with measurable disease per RECIST v1.1, a Karnofsky Performance Status score of ≥70%, and histologically confirmed unresectable stage IV RCC with a clear cell component and disease progressing on or after receiving at least 2 systemic treatments (both an anti–PD-1/L1 monoclonal antibody and a VEGF receptor–targeted tyrosine kinase inhibitor, in sequence or in combination) will be enrolled. Up to 30 patients will be enrolled into 3 dose groups in part 1 and will receive belzutifan 120 mg once daily plus palbociclib (75, 100, or 125 mg) daily for 21 consecutive days followed by 7 days off. In part 2, approximately 150 patients will be randomly assigned 2:1 to receive belzutifan 120 mg once daily plus palbociclib RP2D (21 consecutive days/7 days off) or belzutifan 120 mg once daily. Patients will be stratified by International metastatic RCC Database Consortium risk (0 vs 1-2 vs 3-6) and sarcomatoid histology (yes vs no) at randomization in part 2. The primary end point for part 1 is to assess dose-limiting toxicities and adverse events and to determine the RP2D of belzutifan plus palbociclib. The primary end point for part 2 is objective response rate per RECIST v1.1 by investigator assessment. Secondary end points for part 2 are clinical benefit rate, duration of response, and progression-free survival per RECIST v1.1 by investigator assessment; overall survival, and safety and tolerability. Enrollment began in July 2022. ©2023 American Society of Clinical Oncology, Inc. Reused with permission. This abstract was accepted and previously presented at the 2023 ASCO Genitourinary Cancers Symposium. All rights reserved. CDMRP DOD Funding: no

Robust target area search using sets of probabilities

Target area search in the standard Bayesian information-theoretic formulation consists of a repetitive cycle of sensing, recursive Bayesian estimation and motion control. The paper formulates the problem of target area search in the framework of imprecise probability theory using probability sets. The rationale is that the measurement model parameters, such as the probability of detection or the probability of false alarm, are rarely known as precise values. Instead, we adopt probability intervals to express beliefs. The paper formulates the Bayes-like state estimation equations as well as the reward function for motion control. The reward function is based on an uncertainty measure which takes into account both randomness and epistemic uncertainty.

Applying rainfall threshold estimates and frequency ratio model for landslide hazard assessment in the coastal mountain setting of South Asia

Landslides pose a serious risk to life and property in the mountainous regions around the globe. Understanding the interplay of landslide conditioning and triggering factors is essential for lessening the impacts caused by the hazard. Cox's Bazar — a coastal mountainous district in Bangladesh is recurrently affected by rainfall-triggered landslides. Based on analysis of 14 experiential landslides and combination of gauged and satellite rainfall estimates for the period from 2003 to 2019, the present study determines three landslide-triggering rainfall thresholds for the Cox's Bazar District (CBD): 1. Intensity-Duration (ID) threshold derived in this study revealed that any rainfall event with an intensity of ≥4.04 ​mm/h if prolonging for ≥12h can cause slope failures; 2. Event-Duration (ED) threshold suggested that a normalized cumulative event rainfall (EMAP) of 0.15 for one day is expected to trigger landslides; and 3. threshold calculated using randomly chosen antecedent rainfall expressed best distinction on 30-day rainfall and the equation of the threshold came out as Rth ​= ​64–0.02 Ra30. The recurrence probability of the derived antecedent rainfall threshold and likely landslides was determined through the Poisson distribution. Moreover, we assess the landslide susceptibility of the district with a coupled use of Frequency Ratio (FR) statistical measure and Geographic Information System (GIS). Considering the combined role of selected conditioning factors, the landslide susceptibility status of the CBD was quantified and classified into probability intervals. The accuracy of the susceptibility maps was assessed through the Relative Landslide Density Index (R-Index) that used a field landslide inventory, comprising well distributed 891 events. Moreover, gridded population data was superimposed on the derived susceptibility maps to understand the risk levels of people. The derivation of landslide-triggering rainfall thresholds and spatial susceptibility assessment has been useful to propose a low-cost Landslide Early Warning System (LEWS) which can contribute in alleviating the adverse effects of landslide hazard in the CBD.

Dynamic modeling and uncertainty quantification of district heating systems considering renewable energy access

The district heating system (DHS) is one of the essential carriers for coordinating renewable energy with fossil energy and achieving flexible energy consumption. Considering the uncertainty of both the renewable energy units' power and the consumers' aggregate loads impact the transport process of the DHS, the thermal characteristics of the DHS transportation and the node temperature response under the source-load uncertainty need to be quantified and analyzed. This paper proposed a method to calculate the nodes' temperature dynamic response of the DHS under multiple source-load uncertainty scenarios. The prediction and combination methods of supply and demand probability intervals of source and load nodes under varying credibility levels were investigated. Additionally, the effects of source-load uncertainty boundary conditions on the DHS transport process were analyzed. The temperature response of the nodes was calculated based on the established dynamic DHS model under multiple source-load uncertainty scenarios. A DHS in Beijing was selected for case validation and analysis, and it has 90 nodes and 109 pipes. The simulation results showed that the relative average error of node flow rate in the dynamic DHS model was 3.02%, and the average difference in node temperature was 1 °C. The probability distribution semi-analytical method and the Bayesian credible interval calculation method can accurately describe the uncertainty on both source and load sides. The models and algorithms proposed in this paper could effectively quantify the nodes' dynamic temperature response under the source-load uncertainty.

Classical and Bayesian Inference for the Length Biased Weighted Lomax Distribution under Progressive Censoring Scheme

In this study, the distribution’s reliability and hazard functions, as well as the population parameters, are estimated for the length biased weighted Lomax (LBWLo) based on progressively Type II censored samples. The maximum likelihood and Bayesian methods are implanted to get the proposed estimators. Gamma and Jeffery's priors serve as informative and non-informative priors, respectively, from which the posterior distribution of the LBWLo distribution is constructed. To obtain the Bayesian estimates, the Metropolis-Hasting (MH) algorithm is also used. We obtain asymptotic confidence intervals based on the Fisher information matrix. Using the sample produced by the MH method, we construct the intervals with the highest posterior densities. A numerical simulation research is done to evaluate the effectiveness of the approaches. Through Monte Carlo simulation, we compare the proposed estimators in terms of mean squared error. It is possible to get coverage probability and average interval lengths of 95% .The study's findings supported the idea that, in the majority of cases, Bayes estimates with an informative prior are more appropriate than other estimates.

A Metaheuristic Approach to Parameter Estimation of a Flexible Parametric Mixture Cure Rate Model with Interval-Censored Data

A flexible parametric mixture cure model, called bi-lognormal cure rate model or simply BLN model, is defined and studied. The BLN model can be effectively used to analyze survival dataset in the presence of long-term survivors, especially when the dataset presents the underlying phenomenon of latent competing risks or when there is evidence that a bimodal hazard function is appropriated to described it, which are advantages over other cure rate models found in the literature. We discuss the maximum likelihood estimation for the model parameters considering interval-censored data through the differential evolution algorithm that is a nature-inspired computing metaheuristic used for global optimization of functions defined in multidimensional spaces. This approach is also used because the likelihood function of the model is multimodal and the direct application of gradient methods in this case is not ideal, since such methods are local search methods with a high chance of getting stuck at a local maximum when the starting point is chosen outside the basin of attraction of a global maximum. In addition, a simulation study was implemented to compare the performance of differential evolution algorithm with the performance of the Newton-Raphson algorithm in terms of bias, root mean square error, and the coverage probability of the asymptotic confidence intervals for the parameters. Finally, an application of the BLN model to real data is presented to illustrate that it can provide a better fit than other mixture cure rate models.