Joint Probability Distribution Model Research Articles

Maximum entropy-based modeling of community-level hazard responses for civil infrastructures

Perturbed by natural hazards, community-level infrastructure networks operate like many-body systems, with behaviors emerging from coupling individual component dynamics with group correlations and interactions. It follows that we can borrow methods from statistical physics to study the response of infrastructure systems to natural disasters. This study aims to construct a joint probability distribution model to describe the post-hazard state of infrastructure networks and propose an efficient surrogate model of the joint distribution for large-scale systems. Specifically, we present maximum entropy modeling of the regional impact of natural hazards on civil infrastructures. Provided with the current state of knowledge, the principle of maximum entropy yields the “most unbiased“ joint distribution model for the performances of infrastructures. In the general form, the model can handle multivariate performance states and higher-order correlations. In a particular yet typical scenario of binary performance state variables with knowledge of their mean and pairwise correlation, the joint distribution reduces to the Ising model in statistical physics. In this context, we propose using a dichotomized Gaussian model as an efficient surrogate for the maximum entropy model, facilitating the application to large systems. Using the proposed method, we investigate the seismic collective behavior of a large-scale road network (with 8,694 nodes and 26,964 links) in San Francisco, showcasing the non-trivial collective behaviors of infrastructure systems.

Parameter estimation of grouting pressure and surface subsidence on the reliability of shield tunnel excavation under incomplete probability information

Most of the results obtained from traditional tunnel reliability models are based on conclusions of deterministic parameters, and ignoring the correlation between parameters tends to generate unreliable conclusions for the analysis. This study investigated the influence of grouting pressure and surface subsidence on the reliability of shield tunnel excavation considering the dependency structures between parameters. A probabilistic analysis approach based on copula method is applied to evaluate the bivariate dependency structure. Firstly, 88 sets of measured data are selected from various tunnel projects at home and abroad, and their statistical parameters and marginal distribution functions are determined. Secondly, a bivariate joint probability distribution model of shield grouting excavation is established, and Monte Carlo Simulation is applied to explore the influence of the selection of the copula function on the failure probability of the shield grouting excavation. The results indicate that under incomplete probabilistic information, disregarding the negative correlation between the parameters will underestimate the reliability of shield grouting construction and yield an overly conservative reliability design. Subsequently, varying the magnitude of the parameter limit values to explore the effect on the failure probability will be advantageous for reliability analysis in dynamic construction. Finally, by varying the sample size of the simulation and original measured data, it is found that excessively small simulated and original sample sizes can lead to inaccurate reliability assessment.

Fragility Analysis of a Transmission Tower-Line System Subjected to Wind and Ice Loads Considering Fatigue Damage

Disasters such as ice and wind can pose a serious threat to the normal operation of power transmission lines. Wind-induced fatigue damage can further reduce the load-carrying capacity of transmission towers, increasing their fragility to ice and wind disasters. To assess the comprehensive capability of transmission towers to resist extreme ice and wind disasters, this paper proposes a failure probability evaluation framework for transmission towers considering wind-induced fatigue damage under the coupled effect of ice and wind. Taking a certain transmission line in Hunan as an example, the failure probability caused by ice and wind disasters is calculated for different years of service. First, based on the historical meteorological data in three cities in Hunan, a joint probability model of wind speed and wind direction considering their correlation is established using the copula function. Then, based on this probability model, the wind-induced fatigue damage of transmission towers is calculated using the Miner linear damage theory and the S-N curve. Subsequently, the fragility of the tower under the coupled load of ice and wind is calculated for different years of service. Finally, by combining the structural fragility function with the joint probability distribution model of ice thickness and wind speed, the collapse probability of the transmission tower under the action of ice and wind disasters is calculated. The results indicate that the influence of wind-induced fatigue damage cannot be ignored when transmission towers encounter ice and wind disasters. With increasing service time, the ability of transmission towers to resist ice and wind disasters gradually decreases, and the failure probability also increases under extreme ice and wind conditions.

A Joint Probability Distribution for Multivariate Wind-Wave Conditions and Discussions on Uncertainties

Abstract This article presents a joint statistical model, which is needed in probabilistic design and structural risk assessment, that has been fitted to data of wind and wave conditions for an offshore location off South Brittany. The data are from a numerical model and contain hourly values for several wind and wave variables over a period of 32 years. The joint distribution presented in this article considers the variables wind direction, mean wind speed, significant wave height, wave direction, and peak period. A conditional model for turbulence given wind speed is introduced to yield an additional variable for the joint model. The joint model is constructed as a product of marginal and conditional models for the various variables. Additionally, the fitted models will be used to construct environmental contours for some of the variables. For significant wave height, various models are used to obtain different extreme value estimates, illustrating the uncertainties involved in extrapolating statistical models beyond the support of the data, and a discussion on the use of nonparametric copulas for the joint distribution is presented. Moreover, bootstrap has been performed to estimate the uncertainty in estimated model parameters from sampling variability. The effect of changing which variable to model as the marginal in a conditional model is illustrated by switching from wind speed to significant wave height. Such joint distribution models are important inputs for design of offshore structures, and in particular for offshore wind turbines, and the influence of the joint model in design is illustrated by a simple case study. This article is an extension of the conference paper by Vanem et al. (2023, “A Joint Probability Distribution Model for Multivariate Wind and Wave Conditions,” 42nd International Conference on Ocean, Offshore and Arctic Engineering).

Probabilistic modeling of wind characteristics for long-span cable-stayed bridges based on field measurements considering deck disturbance effects

To assess the reliability of wind resistance behavior for long-span cable-stayed bridges, this study establishes a joint probability distribution model for 10-minute mean wind speed and turbulence intensity based on one year of field monitoring data. Firstly, the distribution characteristics of wind parameters at the deck height are investigated. Notably, substantial abnormalities are observed in the anemometer data obtained from the upstream and downstream sides of the mid-span deck. Thus, a novel component-decomposition method is proposed to eliminate these inconsistencies. The decrease in measured mean wind speed and the increase in turbulence intensity resulting from the bridge deck disturbance is considered to restore the actual wind parameters. Finally, the joint probability distribution model is developed using the Gaussian Copula function. The results demonstrate the efficacy of the proposed method in mitigating the bridge deck disturbance on field measurements, thereby establishing accurate and reliable probability distribution models.

A Practical Approach of Probabilistic Seismic Hazard Analysis for Vector IMs Regarding Mainshock with Potentially Largest Aftershock

ABSTRACT Seismic hazard and risk may be underestimated if aftershocks are ignored within the traditional probabilistic seismic hazard analysis (PSHA). In this study, a practical approach is proposed to calculate the mean annual rate of the joint exceedance of different ground motion vector intensity measures (IMs) for mainshocks and their potentially largest aftershock. The proposed approach is built based on an indirect vector-valued PSHA framework, which was originally used to compute the joint hazard of multiple IMs for mainshocks. Within the proposed approach, the overall hazard caused by mainshock-aftershock scenarios is calculated by including two key components, namely: 1) the occurrence rate of a given hazard level of mainshock as calculated by the standard PSHA; and 2) the occurrence probability of a given hazard level of aftershock conditioned on the mainshock, which is calculated by establishing the joint distribution of the IMs for both mainshocks and aftershocks using the Copula theory. Based on 662 groups of mainshock-aftershock ground motions from 13 earthquake events in the NGA-West2 database, the empirical joint probability distribution model between the peak ground accelerations of mainshocks and aftershocks (i.e. PGA MS and PGA AS) is developed. The Clayton Copula function is calibrated as the optimal choice and the marginal distributions of PGA MS or PGA AS could be properly modelled using the generalized extreme value distributions. The proposed approach is implemented to construct the joint hazard surface regarding both PGA MS and PGA AS for a hypothetical site. The results indicate that using different types of copula functions and marginal distributions of PGA MS and PGA AS has a noticeable impact on the final joint hazard level. Moreover, a coarse assumption that PGA MS and PGA AS follow a joint Gaussian distribution in the logarithm scale would lead to considerable bias in the mainshock-aftershock joint hazard especially at low mean annual rates of exceedance.

Capacity Assessment of a Transmission Tower Considering the Wind-Ice Probability Distributions

Extreme weather, such as wind and ice, threatens the safety and stability of transmission networks. In this paper, the joint probability distribution model of wind speed and ice thickness are established, based on which the bearing capacity of a transmission tower under combined wind and ice loads are evaluated. The results show that the developed probability distribution models can well characterize the dependence structure of wind speed and ice thickness, and the safety of the transmission tower under combined wind-ice loads can be guaranteed within its design life.

Multi-hazard joint probability distribution model for wind speed, wind direction and rain intensity

Multiple disasters such as strong wind and torrential rain pose great threats to civil infrastructures. However, most existing studies ignored the dependence structure between them, as well as the effect of wind direction. From the dimension of the engineering sector, this paper introduces the vine copula to model the joint probability distribution (JPD) of wind speed, wind direction and rain intensity based on the field data in Yangjiang, China during 1971–2020. First, the profiles of wind and rain in the studied area are statistically analyzed, and the original rainfall amounts are converted into short-term rain intensity. Then, the marginal distributions of individual variables and their pairwise dependence structures are built, followed by the development of the trivariate joint distribution model. The results show that the constructed vine copula-based model can well characterize the dependence structure between wind speed, wind direction and rain intensity. Meanwhile, the JPD characteristics of wind speed and rain intensity show significant variations depending on wind direction, thus the effect of wind direction cannot be neglected. The proposed JPD model will be conducive for reasonable and precise performance assessment of structures subjected to multiple hazards of wind and rain actions.

Typhoon-Resistant Performance Assessment of Coastal Rural Residential Keel Brick Walls Reinforced with High Ductility Concrete

The keel brick wall belongs to the rural residential exterior wall unique to the southeast coastal region of China. The wooden keel inside the wall and the external tie members can improve the stability of the whole wall. However, due to the age of construction, the wooden keel inside the existing residential keel brick walls has basically rotted and lost the tie with the external members, which seriously reduces the stability of the whole wall. Moreover, the collapse of keel brick walls caused by typhoons has become more and more frequent in recent years, seriously endangering the lives and properties of residents. Therefore, it is necessary to carry out reinforcement and repair for the existing residential buildings with keel brick walls. In this paper, the keel brick walls of rural houses in Ningbo area were taken as the research object, a framework for evaluating the performance of keel brick walls against typhoons was established, tests on the lateral force resistance of keel brick walls reinforced with high ductility concrete and mortar were carried out. With the help of typhoon full-track simulation and wind field simulation technology, the 600-year near-surface typhoon wind speeds affecting Ningbo area were obtained. A joint probability distribution model of multi-directional extreme wind speed was constructed using t-Copula function to determine the mixed wind climate of Ningbo area. The extreme wind loads that keel brick walls can withstand under four types of reinforcement were finally given. The results show that the high ductility concrete reinforcement has great advantages over mortar reinforcement and avoids brittle damage of the masonry structure. The keel brick walls reinforced with single-sided mortar can withstand wind loads of a 3-year return period, the keel brick walls reinforced with single-sided high ductile concrete can withstand extreme wind loads of a 110-year return period, while the keel brick walls reinforced with double-sided mortar as well as double-sided high ductile concrete can withstand extreme wind loads of well over a 110-year return period.

A statistical model of routine process monitoring variates and conformance assessment for radiation processing

Modeling a routine radiation process monitoring variate consists of identifying a statistical model that accurately forecasts process variate solution space and provides an inference structure by which processed product conformance to dose range specifications can be assessed with a high degree of surety. This paper shall identify and describe the structural characteristics of the process variates associated with the various forms of routine radiation process monitoring and demonstrate the Bivariate Normal Joint Probability distribution model applicability to routine radiation processing end to end. The routine radiation process end to end consists of:a.Performance qualification dose mapping (PQ)b.Capability/reliability assessmentc.Process target optimization (PTO)d.Statistical process control (SPC) of the routine processe.Product conformance assessment of processed products to dose range specifications.The significance of this model is it is a probabilistic risk/confidence assessment of the process output variate. This treats the sampled variate output as expressing the aggregate variance of all expressing predictor variables (common cause), accurately treats output variates as random variates (solution space) and avoids the errors of the current routine radiation process monitoring industry model.

Estimating a joint probability distribution model of fluctuating wind speeds of monsoons from field-measured wind speed data

Modelling the complete probabilistic information of wind speed processes, which can describe the joint probability distributions of wind speeds in the time domain, is significant and challenging. In this study, a joint probability distribution model of fluctuating wind speeds of monsoons is directly estimated from field-measured wind speed samples. The fluctuating wind speeds are modelled by a harmonisable process in the frequency domain. The marginal probability distribution of the wind speed frequency components is represented by a derived mixture distribution model consisting of the Gaussian distribution and generalised Pareto distribution. The probabilistic dependence among the frequency components is modelled by the D-vine copula distribution. The results show that the measured samples are nonstationary in the time domain and non-Gaussian in both time and frequency domains. The probabilistic dependence among the frequency components is moderate but cannot be ignored. The estimated joint probability distribution model can characterise the mean value, standard deviation, kurtosis coefficient, probability density functions (PDFs) of the measured samples and the PDF of the measured maximum wind speed in the time domain. The wind skewness coefficient and the PDF of the minimum wind speed from the obtained joint probability distribution model have certain differences from the measured results.

Typhoon-Induced Fragility Analysis of Transmission Tower in Ningbo Area Considering the Effect of Long-Term Corrosion

The purpose of this paper was to investigate the influence of long-term corrosion on the deterioration of wind resistance of a steel transmission tower during its service life. An analytical model for predicting the long-term corrosion depth of carbon steel was established, and the corrosion depth of carbon steel in the Ningbo area was predicted based on the local atmospheric environment data. With the help of typhoon full-track simulation and wind field simulation technology, a joint probability distribution model of multidirectional extreme wind speeds was constructed using the t-Copula function to determine the typhoon climate of the transmission tower site. Finite element models of the ZM4 cathead transmission tower under 30/60/90 corrosion years were then established, respectively, according to the predicted corrosion depth of carbon steel in Ningbo. Three damage modes, i.e., minor damage, moderate damage and severe damage, corresponding to the transmission tower under wind loads, were defined, and pushover analyses were used to determine the limit values of each damage mode so as to obtain the typhoon-induced fragility curves of the transmission tower within 30/60/90 corrosion years. The results show that the increase in corrosion age leads to a deterioration in the nominal mechanical properties of the transmission tower components, making the damage probability to the transmission tower increase. Under the typhoon wind loads of a 50-year return period in the most unfavorable wind direction in Ningbo, the probability of moderate damage of the tower is within 10% and the probability of minor damage is controlled between 10% and 40%.

Pair-Copula-based trivariate joint probability model of wind speed, wind direction and angle of attack

The temporal and spatial distribution of wind fields in the deep-cut gorge bridge site is complex, and the impact of different incoming flows on the structure is also essentially different. Therefore, it is essential to comprehensively consider the correlation between wind direction, wind speed, and angle of attack. The joint probability models between average wind parameters are first established using the Pair-Copula decomposition based on the field-measured data. Then, the concept of the probability-segmentation-based first order inverse reliability method (PSB-IFORM) is proposed for calculating the environmental surface of the multi-peak joint distribution model. The results show that the mixed von Mises distribution is ideal to fit the wind direction with multi-peak characteristics. The trivariate joint probability model constructed in this paper ultimately represents the correlation of average wind parameters, and the most unfavorable combination of design wind parameters can be found on the environmental surface. Furthermore, the proposed PSB-IFORM can effectively avoid the defects of the inverse first-order reliability method (IFORM) and the conservative highest probability contour (HDC) method in solving the multi-peak joint probability distribution model. This study is of particular interest to researchers and engineers engaged in wind resistance of mountain structures.

Reliability Modelling of Pipeline Failure under the Impact of Submarine Slides-Copula Method

The instability of seabed slope sediments is the main factor influencing the safety of marine resource development. Therefore, to ensure the safe operation of submarine pipelines under complex and uncertain seabed rock and soil conditions, a reliability model was developed to elucidate the trend of impact-related pipeline damage due to submarine slides. Then, a risk assessment of the damage process of submarine slides impacting pipelines was conducted, which is of great significance for the in-depth safety assessment of pipelines impacted by submarine slides. Based on the copula function, a joint probability distribution model considering the correlation among risk variables was established for rational correlation characterization. A probability analysis method of impact-related pipeline damage attributed to submarine slides based on the copula function was proposed. The Monte Carlo simulation (MCS) method was employed to simulate the random uncertainty in limited observation values and accurately determine the reliability of safe pipeline operation under the action of submarine slides. The conclusions were as follows: (1) Based on the copula function, a joint probability distribution model of risk variables with any marginal distribution function and related structure could be developed. (2) The copula function could reasonably characterize relevant nonnormal distribution characteristics of risk variables and could simulate samples conforming to the distribution pattern of the risk variables. (3) The failure probability calculated with the traditional independent normal distribution model was very low, which could result in a notable overestimation of the reliability of submarine pipelines.

SalyPath360: Saliency and scanpath prediction framework for omnidirectional images

This paper introduces a new framework to predict visual attention of omnidirectional images. The key setup of our architecture is the simultaneous prediction of the saliency map and a corresponding scanpath for a given stimulus. The framework implements a fully encoder-decoder convolutional neural network augmented by an attention module to generate representative saliency maps. In addition, an auxiliary network is employed to generate probable viewport center fixation points through the $SoftArgMax$ function. The latter allows to derive fixation points from feature maps. To take advantage of the scanpath prediction, an adaptive joint probability distribution model is then applied to construct the final unbiased saliency map by leveraging the encoder decoder-based saliency map and the scanpath-based saliency heatmap. The proposed framework was evaluated in terms of saliency and scanpath prediction, and the results were compared to state-of-the-art methods on Salient360! dataset. The results showed the relevance of our framework and the benefits of such architecture for further omnidirectional visual attention prediction tasks.

Stochastic Dynamic Response Analysis of the 3D Slopes of Rockfill Dams Based on the Coupling Randomness of Strength Parameters and Seismic Ground Motion

Because rockfill strength and seismic ground motion are dominant factors affecting the slope stability of rockfill dams, it is very important to accurately characterize the distribution of rockfill strength parameters, develop a stochastic ground motion model suitable for rockfill dam engineering, and effectively couple strength parameters and seismic ground motion to precisely evaluate the dynamic reliability of the three-dimensional (3D) slope stability of rockfill dams. In this study, a joint probability distribution model for rockfill strength based on the copula function and a stochastic ground motion model based on the improved Clough-Penzien spectral model were built; the strength parameters and the seismic ground motion were coupled using the GF-discrepancy method, a method for the analysis of dynamic reliability of the 3D slope stability of rockfill dams was proposed based on the generalized probability density evolution method (GPDEM), and the effectiveness of the proposed method was verified. Moreover, the effect of different joint distribution models on the dynamic reliability of the slope stability of rockfill dams was revealed, the effect of the copula function type on the dynamic reliability of the slope stability was analysed, and the differences in the dynamic reliability of the slope stability under parameter randomness, seismic ground motion randomness, and coupling randomness of parameters and seismic ground motion were systematically determined. The results were as follows: the traditional joint distribution models ignored related nonnormal distribution characteristics of rockfill strength parameters, which led to excessively low calculated failure probabilities and overestimations of the reliability of the slope stability; in practice, we found that the optimal copula function should be selected to build the joint probability distribution model, and seismic ground motion randomness must be addressed in addition to parameter randomness.

Risk Assessment of Different Maize (Zea mays L.) Lodging Types in the Northeast and the North China Plain Based on a Joint Probability Distribution Model

Mastering the lodging risk of planting environment is of great significance to the optimal layout of maize varieties and the breeding of lodging resistant varieties. However, the existing lodging risk models are still at the stage of single or multi-factors independent analysis, and lack of assessment for different lodging types. To address this issue, based on the mechanism of different lodging types, the Archimedean copula function was used to describe the joint probability distribution of wind speed and precipitation, and the lodging risk assessment model of maize was established. By comparing the goodness of fit, when the rank correlation coefficient of these two is positive and negative, the corresponding optimal joint probability distribution functions are the Gumbel copula and Frank copula. According to the spatial distribution of lodging risk, the area from Liaodong Bay northward to Tongyu, Jilin province in the Northeast and the North China Plain has a high frequency of lodging, in which the probability of stalk lodging is two to four times that of root lodging. Finally, we discussed how to apply the lodging risk distribution results to optimize the maize variety test sites to improve the efficiency and reliability of the existing test system. The method proposed in this paper comprehensively considers the synergistic effect of multiple factors and can provide technical support for other risk assessment.

Copula-Based Method for Estimating the Minimum Void Ratio Parameters of Tailings Deposits

The pore ratio is an important parameter affecting the stability and safety of tailings reservoirs; however, the relationship between the pore ratio and physical properties of tailings sand has not been researched in-depth. In this paper, using the tailings from a tungsten mine in southern Shaanxi as a case study, the correlation between the minimum void ratio and related parameters is analyzed, based on laboratory test data, and the optimal marginal distribution function of the parameters is determined. The Gumbel-Hougard copula function that best describes the correlation between parameters is identified, and it is used to establish the joint probability distribution model of the three parameters, and the guarantee rate α is introduced to estimate and analyze the minimum void ratio. The results show that the optimal edge distribution of the fine particle content and specific gravity follows a truncated normal distribution, and the optimal edge distribution of the minimum void ratio follows a logarithmic normal distribution. According to AIC criterion, the Gumbel-Hougard copula is the best three-dimensional copula function to fit the minimum void ratio and related parameters. When the guarantee rate α is 0.485, the joint probability distribution model achieves optimal performance in terms of estimating the minimum void ratio. The maximum error of the estimation is 1.99%, which is verified through data, and the estimation meets the requirements for practical engineering. The method proposed in this paper uses the existing measured data to establish a joint probability distribution model and combines the collected fine particle content and specific gravity data with the guarantee rate to estimate the minimum void ratio, providing a novel basis for the study of the physical properties of tailings.

The robust recovery model of distribution network considering correlation between wind speed and load

In order to consider the correlation between distributed power output and load, this paper uses the value-at-risk theory to quantitatively calculate the uncertainty of wind speed and load demand, and establishes a wind speed-load joint probability distribution model using bivariate normal distribution, which will take into account the wind speed uncertainty. The robust decision-making problem of reliability and load participation is characterized as a mixed-integer linear programming problem, which takes the minimization of the equivalent load loss as the optimization goal, establishes the load recovery path as the optimization variable, and takes the wind speed and load as the disturbance variable to account for the wind speed-The load-dependent robust restoration model of the distribution network is solved by the Gurobi high-efficiency solver. Finally, the IEEE69-bus power distribution system with wind power was used to verify the effectiveness of this model and solution method.

A force model of high-speed dry milling CF/PEEK considering fiber distribution characteristics

Carbon fiber reinforced polyetheretherketone (CF/PEEK) is an emerging material that is widely used in the automotive and aviation industry due to its high shock resistance, thermostability, and recyclability. However, its dry-machining requirement with an unclear dynamic material removal process limits the machining efficiency and surface quality. To this end, we firstly established a two-dimensional joint probability distribution model of cutting carbon fibers based on the microstructure of CF/PEEK. Compared with previous brittle models, the plastic model is verified to be more consistent with experimental results. A high-speed dry (HSD) milling force model considering carbon fiber distribution is then developed and is proofed with a high prediction accuracy of about 92.6%. With the proposed model, the milling force can be separated into forces of cutting carbon fibers and PEEK matrix, by which the influences of high-speed dry milling process on carbon fibers and PEEK matrix can be analyzed. The results show that carbon fiber distribution exhibits a significant impact on milling force and is more easily affected by milling parameters compared with the PEEK matrix. It is also verified feasible to HSD milling CF/PEEK, and the work gives the guidance of breaking the low-speed machining limits by improving the machining process.