The Probability Density Function Related to Shallow Cumulus Entrainment Rate and Its Influencing Factors in a Large-Eddy Simulation

The process of entrainment-mixing between cumulus clouds and the ambient air is important for the development of cumulus clouds.Accurately obtaining the entrainment rate(λ)is particularly important for its parameterization within the overall cumulus parameterization scheme.In this study,an improved bulk-plume method is proposed by solving the equations of two conserved variables simultaneously to calculateλof cumulus clouds in a large-eddy simulation.The results demonstrate that the improved bulk-plume method is more reliable than the traditional bulk-plume method,becauseλ,as calculated from the improved method,falls within the range ofλvalues obtained from the traditional method using different conserved variables.The probability density functions ofλfor all data,different times,and different heights can be well-fitted by a log-normal distribution,which supports the assumed stochastic entrainment process in previous studies.Further analysis demonstrate that the relationship betweenλand the vertical velocity is better than other thermodynamic/dynamical properties;thus,the vertical velocity is recommended as the primary influencing factor for the parameterization ofλin the future.The results of this study enhance the theoretical understanding ofλand its influencing factors and shed new light on the development ofλparameterization.

Probability Distribution Characteristics of Strong Nonlinear Waves Under Typhoon Conditions in the Northern South China Sea

The generation and propagation mechanism of strong nonlinear waves in the South China Sea is an essential research area. In this study, the third-generation wave model WAVEWATCH Ⅲ is employed to simulate wave fields under extreme sea states. The model, integrating the ST6 source term, is validated against observed data, demonstrating its credibility. The spatial distribution of the occurrence probability of strong nonlinear waves during typhoons is shown, and the waves in the straits and the northeastern part of the South China Sea show strong nonlinear characteristics. The high-order spectral model HOS-ocean is employed to simulate the random wave surface series beneath five different platform areas. The waves during the typhoon exhibit strong nonlinear characteristics, and freak waves exist. The space-varying probability model is established to describe the short-term probability distribution of nonlinear wave series. The exceedance probability distributions of the wave surface beneath different platform areas are compared and analyzed. The results show that with an increase in the platform area, the probability of a strong nonlinear wave beneath the platform increases.

Probability Laws Derived from the Gamma Function

Several densities or probability laws of continuous random variables derive from the Euler Gamma function. These laws form the basis of sampling theory, namely hypothesis testing and estimation. Namely the gamma, beta, and Student law, through the chi-square law and the normal law are all distributions resulting from applications of Euleur functions.

On the Application of Mixed Models of Probability and Convex Set for Time-Variant Reliability Analysis

In time-variant reliability problems,there are a lot of uncertain variables from different sources.Therefore,it is important to consider these uncertainties in engineering.In addition,time-variant reliability problems typically involve a complexmultilevel nested optimization problem,which can result in an enormous amount of computation.To this end,this paper studies the time-variant reliability evaluation of structures with stochastic and bounded uncertainties using a mixed probability and convex set model.In this method,the stochastic process of a limit-state function with mixed uncertain parameters is first discretized and then converted into a timeindependent reliability problem.Further,to solve the double nested optimization problem in hybrid reliability calculation,an efficient iterative scheme is designed in standard uncertainty space to determine the most probable point(MPP).The limit state function is linearized at these points,and an innovative random variable is defined to solve the equivalent static reliability analysis model.The effectiveness of the proposed method is verified by two benchmark numerical examples and a practical engineering problem.

Adaptive Update Distribution Estimation under Probability Byzantine Attack

The secure and normal operation of distributed networks is crucial for accurate parameter estimation.However,distributed networks are frequently susceptible to Byzantine attacks.Considering real-life scenarios,this paper investigates a probability Byzantine(PB)attack,utilizing a Bernoulli distribution to simulate the attack probability.Historically,additional detection mechanisms are used to mitigate such attacks,leading to increased energy consumption and burdens on distributed nodes,consequently diminishing operational efficiency.Differing from these approaches,an adaptive updating distributed estimation algorithm is proposed to mitigate the impact of PB attacks.In the proposed algorithm,a penalty strategy is initially incorporated during data updates to weaken the influence of the attack.Subsequently,an adaptive fusion weight is employed during data fusion to merge the estimations.Additionally,the reason why this penalty term weakens the attack has been analyzed,and the performance of the proposed algorithm is validated through simulation experiments.

Stochastic Model of Dengue: Analysing the Probability of Extinction and LLN

In this article, we develop and analyze a continuous-time Markov chain (CTMC) model to study the resurgence of dengue. We also explore the large population asymptotic behavior of probabilistic model of dengue using the law of large numbers (LLN). Initially, we calculate and estimate the probabilities of dengue extinction and major outbreak occurrence using multi-type Galton-Watson branching processes. Subsequently, we apply the LLN to examine the convergence of the stochastic model towards the deterministic model. Finally, theoretical numerical simulations are conducted exploration to validate our findings. Under identical conditions, our numerical results demonstrate that dengue could vanish in the stochastic model while persisting in the deterministic model. The highlighting of the law of large numbers through numerical simulations indicates from what population size a deterministic model should be considered preferable.

High-dimensional uncertainty quantification of projectile motion in the barrel of a truck-mounted howitzer based on probability density evolution method

This paper proposed an efficient research method for high-dimensional uncertainty quantification of projectile motion in the barrel of a truck-mounted howitzer.Firstly,the dynamic model of projectile motion is established considering the flexible deformation of the barrel and the interaction between the projectile and the barrel.Subsequently,the accuracy of the dynamic model is verified based on the external ballistic projectile attitude test platform.Furthermore,the probability density evolution method(PDEM)is developed to high-dimensional uncertainty quantification of projectile motion.The engineering example highlights the results of the proposed method are consistent with the results obtained by the Monte Carlo Simulation(MCS).Finally,the influence of parameter uncertainty on the projectile disturbance at muzzle under different working conditions is analyzed.The results show that the disturbance of the pitch angular,pitch angular velocity and pitch angular of velocity decreases with the increase of launching angle,and the random parameter ranges of both the projectile and coupling model have similar influence on the disturbance of projectile angular motion at muzzle.

Integrating host biological and ecological variables to predict probability of haemosporidian infection in raptors

Variations in host traits that influence their exposure and susceptibility may impact probability of vector-transmitted diseases.Therefore,identifying the predictors of infection probability is necessary to understand the risk of disease outbreaks during expanding environmental perturbation.Here,we conducted a large survey based on microscopic examination and molecular analysis of haemosporidian parasite infection in raptors rescued at the Beijing Raptor Rescue Centre.Combining these data with biological and ecological variables of the raptors,we determined predictors that affect the probability of haemosporidian infection using generalized linear mixed models and multimodel inference.Our results showed that infection probability exhibited considerable variation across host species in raptors,and body mass,sex,and evolutionary history played relatively weaker roles in driving infection probability.Instead,activity pattern,age,geographic range size,migration distance,and nest type were important predictors of the probability of haemosporidian infection,and the role of each predictor differed in the three main haemosporidian genera(Plasmodium,Haemoproteus,and Leucocytozoon).This macro-ecological analysis will add to our understanding of host traits that influence the probability of avian haemosporidian infection and will help inform risk of emerging diseases.

Secrecy Outage Probability Minimization in Wireless-Powered Communications Using an Improved Biogeography-Based Optimization-Inspired Recurrent Neural Network

This paper focuses on wireless-powered communication systems,which are increasingly relevant in the Internet of Things(IoT)due to their ability to extend the operational lifetime of devices with limited energy.The main contribution of the paper is a novel approach to minimize the secrecy outage probability(SOP)in these systems.Minimizing SOP is crucial for maintaining the confidentiality and integrity of data,especially in situations where the transmission of sensitive data is critical.Our proposed method harnesses the power of an improved biogeography-based optimization(IBBO)to effectively train a recurrent neural network(RNN).The proposed IBBO introduces an innovative migration model.The core advantage of IBBO lies in its adeptness at maintaining equilibrium between exploration and exploitation.This is accomplished by integrating tactics such as advancing towards a random habitat,adopting the crossover operator from genetic algorithms(GA),and utilizing the global best(Gbest)operator from particle swarm optimization(PSO)into the IBBO framework.The IBBO demonstrates its efficacy by enabling the RNN to optimize the system parameters,resulting in significant outage probability reduction.Through comprehensive simulations,we showcase the superiority of the IBBO-RNN over existing approaches,highlighting its capability to achieve remarkable gains in SOP minimization.This paper compares nine methods for predicting outage probability in wireless-powered communications.The IBBO-RNN achieved the highest accuracy rate of 98.92%,showing a significant performance improvement.In contrast,the standard RNN recorded lower accuracy rates of 91.27%.The IBBO-RNN maintains lower SOP values across the entire signal-to-noise ratio(SNR)spectrum tested,suggesting that the method is highly effective at optimizing system parameters for improved secrecy even at lower SNRs.

A self-organization formation configuration based assignment probability and collision detection

The formation control of multiple unmanned aerial vehicles(multi-UAVs)has always been a research hotspot.Based on the straight line trajectory,a multi-UAVs target point assignment algorithm based on the assignment probability is proposed to achieve the shortest overall formation path of multi-UAVs with low complexity and reduce the energy consumption.In order to avoid the collision between UAVs in the formation process,the concept of safety ball is introduced,and the collision detection based on continuous motion of two time slots and the lane occupation detection after motion is proposed to avoid collision between UAVs.Based on the idea of game theory,a method of UAV motion form setting based on the maximization of interests is proposed,including the maximization of self-interest and the maximization of formation interest is proposed,so that multi-UAVs can complete the formation task quickly and reasonably with the linear trajectory assigned in advance.Finally,through simulation verification,the multi-UAVs target assignment algorithm based on the assignment probability proposed in this paper can effectively reduce the total path length,and the UAV motion selection method based on the maximization interests can effectively complete the task formation.

Time series prediction of reservoir bank landslide failure probability considering the spatial variability of soil properties

Historically,landslides have been the primary type of geological disaster worldwide.Generally,the stability of reservoir banks is primarily affected by rainfall and reservoir water level fluctuations.Moreover,the stability of reservoir banks changes with the long-term dynamics of external disastercausing factors.Thus,assessing the time-varying reliability of reservoir landslides remains a challenge.In this paper,a machine learning(ML)based approach is proposed to analyze the long-term reliability of reservoir bank landslides in spatially variable soils through time series prediction.This study systematically investigated the prediction performances of three ML algorithms,i.e.multilayer perceptron(MLP),convolutional neural network(CNN),and long short-term memory(LSTM).Additionally,the effects of the data quantity and data ratio on the predictive power of deep learning models are considered.The results show that all three ML models can accurately depict the changes in the time-varying failure probability of reservoir landslides.The CNN model outperforms both the MLP and LSTM models in predicting the failure probability.Furthermore,selecting the right data ratio can improve the prediction accuracy of the failure probability obtained by ML models.

Prediction and Output Estimation of Pattern Moving in Non-Newtonian Mechanical Systems Based on Probability Density Evolution

A prediction framework based on the evolution of pattern motion probability density is proposed for the output prediction and estimation problem of non-Newtonian mechanical systems,assuming that the system satisfies the generalized Lipschitz condition.As a complex nonlinear system primarily governed by statistical laws rather than Newtonian mechanics,the output of non-Newtonian mechanics systems is difficult to describe through deterministic variables such as state variables,which poses difficulties in predicting and estimating the system’s output.In this article,the temporal variation of the system is described by constructing pattern category variables,which are non-deterministic variables.Since pattern category variables have statistical attributes but not operational attributes,operational attributes are assigned to them by posterior probability density,and a method for analyzing their motion laws using probability density evolution is proposed.Furthermore,a data-driven form of pattern motion probabilistic density evolution prediction method is designed by combining pseudo partial derivative(PPD),achieving prediction of the probability density satisfying the system’s output uncertainty.Based on this,the final prediction estimation of the system’s output value is realized by minimum variance unbiased estimation.Finally,a corresponding PPD estimation algorithm is designed using an extended state observer(ESO)to estimate the parameters to be estimated in the proposed prediction method.The effectiveness of the parameter estimation algorithm and prediction method is demonstrated through theoretical analysis,and the accuracy of the algorithm is verified by two numerical simulation examples.

Variations in Probability Assessment With Atypical Scenarios

Probability assessment in some scenarios may involve unusual aspects such as requiring certain values for some events and extremely high or low probabilities in other cases.

Adjuvant chemotherapy for isolated resectable colorectal lung metastasis: A retrospective study using inverse probability treatment weighting propensity analysis

BACKGROUND The benefit of adjuvant chemotherapy(ACT)for patients with no evidence of disease after pulmonary metastasis resection(PM)from colorectal cancer(CRC)remains controversial.AIM To assess the efficacy of ACT in patients after PM resection for CRC.METHODS This study included 96 patients who underwent pulmonary metastasectomy for CRC at a single institution between April 2008 and July 2023.The primary end-point was overall survival(OS);secondary endpoints included cancer-specific survival(CSS)and disease-free survival(DFS).An inverse probability of treat-ment-weighting(IPTW)analysis was conducted to address indication bias.Sur-vival outcomes compared using Kaplan-Meier curves,log-rank test,Cox regre-ssion and confirmed by propensity score-matching(PSM).RESULTS With a median follow-up of 27.5 months(range,18.3-50.4 months),the 5-year OS,CSS and DFS were 72.0%,74.4%and 51.3%,respectively.ACT had no significant effect on OS after PM resection from CRC[original cohort:P=0.08;IPTW:P=0.15].No differences were observed for CSS(P=0.12)and DFS(P=0.68)between the ACT and non-ACT groups.Multivariate analysis showed no association of ACT with better survival,while sublobar resection(HR=0.45;95%CI:0.20-1.00,P=0.049)and longer disease-free interval(HR=0.45;95%CI:0.20-0.98,P=0.044)were associated with improved survival.CONCLUSION ACT does not improve survival after PM resection for CRC.Further well-designed randomized controlled trials are needed to determine the optimal ACT regimen and duration.

Calculation of Two-Tailed Exact Probability in the Wald-Wolfowitz One-Sample Runs Test

The objectives of this paper are to demonstrate the algorithms employed by three statistical software programs (R, Real Statistics using Excel, and SPSS) for calculating the exact two-tailed probability of the Wald-Wolfowitz one-sample runs test for randomness, to present a novel approach for computing this probability, and to compare the four procedures by generating samples of 10 and 11 data points, varying the parameters n0 (number of zeros) and n1 (number of ones), as well as the number of runs. Fifty-nine samples are created to replicate the behavior of the distribution of the number of runs with 10 and 11 data points. The exact two-tailed probabilities for the four procedures were compared using Friedman’s test. Given the significant difference in central tendency, post-hoc comparisons were conducted using Conover’s test with Benjamini-Yekutielli correction. It is concluded that the procedures of Real Statistics using Excel and R exhibit some inadequacies in the calculation of the exact two-tailed probability, whereas the new proposal and the SPSS procedure are deemed more suitable. The proposed robust algorithm has a more transparent rationale than the SPSS one, albeit being somewhat more conservative. We recommend its implementation for this test and its application to others, such as the binomial and sign test.

Flashover Probability of Wind Turbine Blade and Impact of Strong Electromagnetic Pulse from Lightning Strikes on Wind Turbine Safety

This paper systematically studies the flashover probability of wind turbine blade lightning arrester and the impact of strong electromagnetic pulses on the local and surrounding wind turbines during lightning strikes.The research results indicate that the flashover probability of direct lightning strikes by the wind turbine blade lightning arrester is almost negligible,and the strong electromagnetic pulse of wind turbine blade during lightning strikes has a serious impact on the electronic equipment of the machine,while the impact on the surrounding wind turbine is relatively small.At the same time,the calculation formula for the reflection of lightning current on the carbon brush between the wind turbine hub and the engine compartment during the flashing of the wind turbine blades is provided,and the calculation method for calculating the spatial gradient distribution of electromagnetic field intensity using Biot-Savart Law theorem is applied.The limitations of using wind turbine blades for lightning protection are pointed out,and a technical route for achieving wind turbine lightning safety is proposed,which can be used as a reference for wind turbine lightning protection technicians.

Integration of Ideological and Political Education into the Probability Theory and Mathematical Statistics Course:A Teaching Design Based on Estimation and Hypothesis Testing

With the rapid development of higher education in China,colleges and universities are facing new challenges and impacts in talent training.Probability Theory and Mathematical Statistics is one of the important courses in higher education for science and engineering majors and economics and management majors.Its critical role in cultivating students’thinking skills and improving their problem-solving skills is self-evident.Course ideological and political education construction is an important link in college talent training work.Combining ideological and political education with course teaching can help students establish correct value concepts and play a certain role in improving their comprehensive ability and quality.At present,the construction of ideological and political education in the Probability Theory and Mathematical Statistics course still faces some problems,mainly manifested in the lack of attention paid by teachers to course ideological and political education,insufficient exploitation of ideological and political elements,and the simplification of ideological and political education implementation methods.In order to comprehensively deepen the construction of course ideological and political education in line with the actual needs of Probability Theory and Mathematical Statistics course teaching,we should strengthen the construction of teacher teams,improve teachers’ability to carry out course ideological and political education,integrate educational resources,develop educational resources for ideological and political education,and innovate teaching methods to improve the overall effect of ideological and political education integration.

Research on the Application of PBL+SPOC Blended Teaching Model in Probability and Statistics Course

To cultivate talents with an exploratory spirit and practical skills in the era of information technology,it is imperative to reform teaching methods and approaches.In the teaching process of the Probability and Statistics course,an application-oriented blended teaching model combining problem-based learning and small private online course was explored.By organizing and implementing online and offline teaching activities based on problem-based learning,a multidimensional process-oriented learning assessment system was established.Practice has shown that this model can effectively enhance classroom teaching effectiveness,benefiting the improvement of students’overall skills and mathematical literacy.

Probability Distribution of Ratio of China Aviation Network Edge Vertices Degree and Its Evolutionary Trace Based on Complex Network

In order to reveal the complex network characteristics and evolution principle of China aviation network, the probability distribution and evolution trace of ratio of China aviation network edge vertices degree were studied based on the statistics data of China civil aviation network in 1988, 1994, 2001, 2008 and 2015. According to the theory and method of complex network, the network system was constructed with the city where the airport was located as the network node and the route between cities as the edge of the network. Based on the statistical data, the ratio of edge vertices degree in China aviation network in 1988, 1994, 2001, 2008 and 2015 were calculated. Using the probability statistical analysis method and regression analysis approach, it was found that the ratio of edge vertices degree had linear probability distribution and the two parameters of the probability distribution had linear evolution trace.

Probability Density Analysis of Nonlinear Random Ship Rolling

Ship rolling in random waves is a complicated nonlinear motion that contributes substantially to ship instability and capsizing.The finite element method(FEM)is employed in this paper to solve the Fokker Planck(FP)equations numerically for homoclinic and heteroclinic ship rolling under random waves described as periodic and Gaussian white noise excitations.The transient joint probability density functions(PDFs)and marginal PDFs of the rolling responses are also obtained.The effects of stimulation strength on ship rolling are further investigated from a probabilistic standpoint.The homoclinic ship rolling has two rolling states,the connection between the two peaks of the PDF is observed when the periodic excitation amplitude or the noise intensity is large,and the PDF is remarkably distributed in phase space.These phenomena increase the possibility of a random jump in ship motion states and the uncertainty of ship rolling,and the ship may lose stability due to unforeseeable facts or conditions.Meanwhile,only one rolling state is observed when the ship is in heteroclinic rolling.As the periodic excitation amplitude grows,the PDF concentration increases and drifts away from the beginning location,suggesting that the ship rolling substantially changes in a cycle and its stability is low.The PDF becomes increasingly uniform and covers a large region as the noise intensity increases,reducing the certainty of ship rolling and navigation safety.The current numerical solutions and analyses may be applied to evaluate the stability of a rolling ship in irregular waves and capsize mechanisms.