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In engineering practice, some weighted-k-out-of-n: G systems often show multiple performance levels, resulting in multi-failure-mode mechanism. For these complex degradation systems, a novel reliability model with heterogeneous components is proposed on account of the number of working components, the total weight, and the sojourn time in some a stage, where different weights and thresholds are taken into account in different stages, and so do failure modes. After presenting the indispensable definitions of first passage time and the system lifetime, the system reliability indexes are derived by the Copula method to depict the dependence between components. Furthermore, maximum likelihood estimation of the Copula parameters is adopted and a Copula-based expression for component importance in each type is obtained. Finally, the reliability of the overall supercharging subsystem of an advanced fighter air energy system, as an example of weighted-k-out-of-n: G system with multi-failure modes is analyzed by the suggested model, which offers the comparison of independent and dependent cases. Meanwhile, sensitivity analysis is available to point out the effects on the system. [ABSTRACT FROM AUTHOR]

Thanks to their ability to capture complex dependence structures, copulas are frequently used to glue random variables into a joint model with arbitrary marginal distributions. More recently, they have been applied to solve statistical learning problems such as regression or classification. Framing such approaches as solutions of estimating equations, we generalize them in a unified framework. We can then obtain simultaneous, coherent inferences across multiple regression-like problems. We derive consistency, asymptotic normality, and validity of the bootstrap for corresponding estimators. The conditions allow for both continuous and discrete data as well as parametric, nonparametric, and semiparametric estimators of the copula and marginal distributions. The versatility of this methodology is illustrated by several theoretical examples, a simulation study, and an application to financial portfolio allocation. for this article are available online. [ABSTRACT FROM AUTHOR]

Investigating dependence structures across various fields holds paramount importance. Consequently, the creation of new copula families plays a crucial role in developing more flexible stochastic models that address the limitations of traditional and sometimes impractical assumptions. The present article derives some reasonable conditions for validating a copula of the ratio-type form u v / (1 − θ f (u) g (v)) . It includes numerous examples and discusses the admissible range of parameter θ , showcasing the diversity of copulas generated through this framework, such as Archimedean, non-Archimedean, positive dependent, and negative dependent copulas. The exploration extends to the upper bound of a general family of copulas, u v / (1 − θ ϕ (u , v)) , and important properties of the copula are discussed, including singularity, measures of association, tail dependence, and monotonicity. Furthermore, an extensive simulation study is presented, comparing the performance of three different estimators based on maximum likelihood, ρ -inversion, and the moment copula method. [ABSTRACT FROM AUTHOR]

This paper proposes an efficient method for quantifying the stratigraphic uncertainties and modeling the geological formations based on boreholes. Two Markov chains are used to describe the soil transitions along different directions, and the transition probability matrices (TPMs) of the Markov chains are analytically expressed by copulas. This copula expression is efficient since it can represent a large TPM by a few unknown parameters. Due to the analytical expression of the TPMs, the likelihood function of the Markov chain model is given in an explicit form. The estimation of the TPMs is then re-casted as a multi-objective constrained optimization problem that aims to maximize the likelihoods of two independent Markov chains subject to a set of parameter constraints. Unlike the method which determines the TPMs by counting the number of transitions between soil types, the proposed method is more statistically sound. Moreover, a random path sampling method is presented to avoid the directional effect problem in simulations. The soil type at a location is inferred from its nearest known neighbors along the cardinal directions. A general form of the conditional probability, based on Pickard's theorem and Bayes rule, is presented for the soil type generation. The proposed stratigraphic characterization and simulation method is applied to real borehole data collected from a construction site in Wuhan, China. It is illustrated that the proposed method is accurate in prediction and does not show an inclination during simulation. [ABSTRACT FROM AUTHOR]

Redundancy allocations strategies are commonly used in reliability engineering to enhance the performance of reliability systems. This paper makes use the hierarchical copula to characterize the dependence structures among the components and hot/cold standbys in series systems. Optimal allocation policies of hot and cold standbys are presented for series systems with dependent components and redundancies under certain conditions imposed on the hierarchical copula as well as the components and spares reliabilities. It is shown that, under certain mild conditions imposed on the copula function and lifetimes of components and redundancies, the optimal allocations should be following the way that the resultant lifetimes for all nodes should be balanced as much as possible to enhance the system's performance. The results extend some related ones well‐known in the literature from the independent case to the dependent setting. Some numerical examples are presented as illustrations. We also present a real application in optimal allocations of redundant wires for improving the reliability of high‐voltage electricity transmission network systems. [ABSTRACT FROM AUTHOR]

To qualify the risk of extreme weather events for power supply security during the long-term power system transformation process, this paper proposes a risk probability evaluation method based on probabilistic production simulation. Firstly, the internal relationship of extreme weather intensity and duration is depicted using the copula function, and the influences of extreme weather on power security are described using the guaranteed power output ability coefficient, which can provide the extreme scenario basis for probabilistic production simulation. Then, a probabilistic production simulation method is proposed, which includes a typical-year scenario and extreme weather events. Meanwhile, an index system is proposed to qualify the power security level, which applies the loss of load expectation (LOLE) and time of loss of load expectation (TOLE) under different scenarios and other indices to reveal the long-term power security trend. Finally, the long-term power supply risks for the Yunnan provincial power system are analyzed using the proposed method, validating that the proposed method is capable of characterizing the influences of extreme weather on power security. The security level of different long-term power transformation schemes is evaluated. [ABSTRACT FROM AUTHOR]

Consider a fixed number of clustered areas identified by their geographical coordinates that are monitored for the occurrences of an event such as a pandemic, epidemic, or migration. Data collected on units at all areas include covariates and environmental factors. We apply a probit transformation to the time to event and embed an isotropic spatial correlation function into our models for better modeling as compared to existing methodologies that use frailty or copula. Composite likelihood technique is employed for the construction of a multivariate Gaussian random field that preserves the spatial correlation function. The data are analyzed using counting process and geostatistical formulation that led to a class of weighted pairwise semiparametric estimating functions. The estimators of model parameters are shown to be consistent and asymptotically normally distributed under infill-type asymptotic spatial statistics. Detailed small sample numerical studies that are in agreement with theoretical results are provided. The foregoing procedures are applied to the leukemia survival data in Northeast England. A comparison to existing methodologies provides improvement. [ABSTRACT FROM AUTHOR]

Baseflow is a critical hydrological variable linked to a variety of environmental factors. The construction of a dam may modify baseflow generation and adjust its environmental dependence via the regulation of discharge. However, its effects on the relationships among air temperature, water temperature, and baseflow remain unclear. Here, we use copula functions to explore the relationships among these variables in the Yangtze River and show how they have changed in response to the Three Gorges Dam (TGD). Because of the buffering effects of the baseflow, baseflow is negatively correlated to water temperature, and the relationship between baseflow and water temperature lessens as air temperature rises. Construction of the TGD attenuates the relationship between baseflow and water temperature when air temperature is low, and this attenuation is especially prominent in winter and spring. Our findings will enhance environmental protection in the Yangtze River and improve the ecological operations of the TGD. [ABSTRACT FROM AUTHOR]

In this study, we propose a new formula for spread option pricing with the dependence of two assets described by a copula function. The proposed method's advantage lies in its requirement of solely computing one-dimensional integrals. Any univariate stock price process, admitting an affine characteristic function, can be used in our formula to get an efficient numerical pricing procedure for a spread option. In the numerical analysis we present a comparison with the Monte Carlo simulation method to assess the performance of our approach, assuming that the univariate stock price follows three widely applied models: variance gamma, Heston's stochastic volatility and affine Heston–Nandi GARCH(1,1) models. [ABSTRACT FROM AUTHOR]

The traditional modeling of dependence among branches in non-life insurance has historically rested on the assumption of independence between risks. However, recent studies have brought to light that overlooking the interdependence between risks is not prudent. An illustrative example is found in automobile insurance, where a discernible connection exists between the number of claims in the Automobile Damage branch and those in the Automobile Civil Liability branch. This paper aims to delve into the utilization of copula functions for modeling the interrelationships among claims charges across diverse branches within a non-life insurance portfolio. Moreover, it seeks to assess the consequential impacts of these dependencies on the capital requirements of AXA Insurance in the Moroccan market. Our research concentrates on four pivotal branches within AXA Insurance, namely "work accidents", "automobile civil liability", "disability", and "fire, marine, aviation, and transport". Through a thorough analysis encompassing dependency measurements, independence tests, and various graphical representations, we have substantiated the existence of interdependencies among the claim costs associated with these four risk branches. Importantly, our findings underscore that assuming independence leads to a slight underestimation of the capital requirement needed for effective risk accumulation management by the insurer. [ABSTRACT FROM AUTHOR]

In recent years, deep learning (DL) approaches have been proven effective in addressing high nonlinear relationships within complex systems. Although various scientific studies have primarily focused on optimizing model architecture and enhancing its computational efficiency through hybridization, they have neglected the crucial aspect of selecting appropriate predictor variables (PV) and their influence on the model's predictions. In this paper, a new multi-phase stochastic DL model is designed with a feature selection framework for monthly streamflow prediction. The multi-phase hybrid MCMC–BC–BiLSTM–BiGRU model is based on bidirectional long short-term memory (BiLSTM) and bidirectional gated recurrent unit (BiGRU) integrated with Markov Chain Monte Carlo (MCMC) based bivariate copulas (BC). First, Ant Colony Optimization (ACO) algorithm was employed for screening optimal PV, and then significant antecedent streamflow was coupled with it to generate optimized input combinations. In addition, 25 different MCMC–BC models were adopted for each input combination to determine the dependency of previous month's streamflow on current and future streamflow. Finally, best-fitted MCMC–BC model was integrated with hybrid BILSTM–BiGRU model to predict streamflow at nine different catchments in the Victoria region of the Upper Murray Basin (UMB), Australia. The experimental results show that the multi-phase hybrid MCMC–BC–BiLSTM–BiGRU model showed remarkable performance in comparison to its benchmark counterparts with respect to different robust statistical error metrics. Therefore, this study demonstrates that stochastic DL techniques can be effectively employed as a promising alternative predictive tool for upstream discharge predictions with high consistency and accuracy, specifically when a statistically significant relationship with the antecedent streamflow exists. [ABSTRACT FROM AUTHOR]

Groundwater serves as the primary source of freshwater in arid and semi-arid regions, rendering them resilient against precipitation scarcity. Various aspects of the drought phenomenon necessitate the application of multivariate analysis and composite drought indices. These composite indices afford a more comprehensive understanding of various drought types. However, few studies have considered the integration of meteorological and groundwater drought information. This research delves into determining a threshold for triggering the transition from meteorological drought to groundwater drought in an arid region (Minab plain). It introduces the Joint Deficit Meteo-Groundwater Drought Index, a result of combining two droughts based on copula function. Furthermore, trivariate drought frequency analysis (severity-duration-peak) is conducted based on three scenarios in three clusters of the Minab Plain. Drought risk is then assessed using conditional probability. The results indicate that the northern and central parts of the Minab Plain face a high risk of meteo-groundwater drought. With increasing drought duration and severity, the peak also rises, placing the entire region at risk of widespread drought. A comprehensive understanding of meteo-groundwater drought and an assessment of its characteristics provide insights into the signals of propagation from meteorological drought to groundwater, facilitating the management of groundwater and surface water resources to mitigate the effects of drought. [ABSTRACT FROM AUTHOR]

The quality and quantity of groundwater resources are often assessed by sampling a limited number of wells drilled sparsely across a plain. So far, various methods have been developed for assessing and modeling groundwater quality, each of which has its capabilities and limitations. In the present study, the copula functions were applied to multivariate analysis of groundwater quality variables (including SAR, K, Mg, Na, Ca, Cl, EC, pH, TDS, SO4, TH and HCO3) in Shahrekord plain, Iran. For this purpose, the quality data from 24 wells distributed across the Shahrekord plain during the period of 1990–2020 were used. For bivariate analyses of groundwater quality, first the fitness of some common distributions in hydrology were examined to the quality variables and the appropriate marginal distribution was determined by Kolmogorov–Smirnov test (K–S). The results revealed that the Generalized Extreme Value distribution has an acceptable fitness on groundwater quality variables of Shahrekord plain. The results of correlation analysis indicated that the highest correlation based on Spearman Rho, Kendall Tau and Pearson correlation coefficient is related to the paired variables of SAR-Na and EC-TDS with correlation coefficient greater than 0.9. Therefore, these variables were selected for further analysis. In the next step, bivariate distribution of two different quality variables in a well (point analysis) and also bivariate distribution of a quality variable in two different wells (inter-well analysis) were created using copula functions. To do this, the fitness of different copulas (including Plackett, Joe, Clayton, Frank, Farlie–Gumbel–Morgenstern, Ali–Mikhail–Haq, Gumbel, Gumbel–Hougaard, Gumbel–Barnett, Philip–Gumbel and Galambos) was tested to construct a bivariate distribution of quality variables. For choosing the best fitted copula on studied paired variables, the theoretical copula values was compared with the corresponding values of empirical copula based on the goodness of fit criteria. Based on the results of point analysis, the Joe copula function was chosen as the best fitted copula function for multivariate analysis of two quality variables in a well. For inter-well analysis (IWA), the Clayton copula function was selected for creating bivariate distribution of a quality variable in two different wells. Comparing the performance of the proposed IWA method with geostatistical methods showed that in addition to the IWA method having acceptable accuracy, it has a higher efficiency than geostatistical methods, especially in areas where the number of sampling wells is less. [ABSTRACT FROM AUTHOR]

Demand-side flexible load resources, such as Electric Vehicles (EVs) and Air Conditioners (ACs), offer significant potential for enhancing flexibility in the power system, thereby promoting the full integration of renewable energy. To this end, this paper proposes an optimal allocation method for demand-side flexible resources to enhance renewable energy consumption. Firstly, the adjustable flexibility of these resources is modeled based on the generalized energy storage model. Secondly, we generate random scenarios for wind, solar, and load, considering variable correlations based on non-parametric probability predictions of random variables combined with Copula function sampling. Next, we establish the optimal allocation model for demand-side flexible resources, considering the simulated operation of these random scenarios. Finally, we optimize the demand-side resource transformation plan year by year based on the growth trend forecast results of renewable energy installed capacity in Jiangsu Province from 2025 to 2031. [ABSTRACT FROM AUTHOR]

This study examines the consistency metrics used to evaluate the durability of a spirulina production plant, which consists of seven subsystems: cultivation pond, paddlewheel, filter unit, washing unit, spray dryer, ribbon blender, and packaging. By studying the spirulina firm, we can repair it by discovering future failures. We can increase spirulina production so that untimely failure can be prevented and production can be increased. There are two types of system failures: partial and total. While a full failure renders the system incapable of operating, a partial failure is thought to degrade the system. In contrast, repair rates follow two different types of distributions: an ordinary and an exponential distribution. The system in a partially failed or degraded condition is thought to be repaired using general time distribution. In contrast, fully failed systems are thought to be fixed using the Gumbel-Hougaard family copula distribution. Using the supplementary variable approach, the system is examined. A Chapmen-Kolmogorov differential equation is created and solved by applying the Gumbel-Haugaard family Copula approach, employing the schematic representation of the system's state. supplementary variable approaches are applied to develop and resolve the differential equations related to transition diagrams, which are significant to this research. Reliability, availability, profitability, and MTTF are the critical performance metrics for the spirulina production plant. Moreover, sensitivity analysis is carried out for MTTF. [ABSTRACT FROM AUTHOR]

Quantitatively analyze the uncertainties in turbulence models is of great meaning for research and engineering. In this paper, we develop a Bayesian uncertainty analysis framework based on copulas. The proposed framework is applied to the parametric uncertainty quantification and calibration of the shear-stress transport turbulence model for the flow over a Gaussian bump. A copula function is used to establish the correlation between variables and construct the weight function, which enables greater consideration of Bayesian inference. Prior analysis is conducted based on Sobol indices with the nonintrusive polynomial chaos method, demonstrating the important effects of κ , a 1 , and β ∗ on the flow, both for the wall force coefficients and the Reynolds shear stress. Next, the wall force coefficients and Reynolds shear stress are used as training sets to verify the effectiveness of the method. The results show that modeling the relationship based on copulas is necessary and effective, and that the weight function adaptively balances the influence of physical quantities at different inference locations. The proposed method effectively corrects the computational deviations of the corresponding physical quantities. In addition, the confidence interval for the posterior sample is increased, enhancing the likelihood of obtaining results close to the actual values. • A copula-based Bayesian uncertainty quantification framework with adaptive weight function is proposed. • Gaussian bump is a recently proposed meaningful geometry shape, attracting wide concerns, but has not been studied enough by now. • Detailed and innovative prior and posterior analysis is conducted. • The physical mechanisms of the parameters for separated flow are analyzed. [ABSTRACT FROM AUTHOR]

Random fields are useful mathematical tools for representing natural phenomena with complex dependence structures in space and/or time. In particular, the Gaussian random field is commonly used due to its attractive properties and mathematical tractability. However, this assumption seems to be restrictive when dealing with counting data. To deal with this situation, we propose a random field with a Poisson marginal distribution considering a sequence of independent copies of a random field with an exponential marginal distribution as "inter-arrival times" in the counting renewal processes framework. Our proposal can be viewed as a spatial generalization of the Poisson counting process. Unlike the classical hierarchical Poisson Log-Gaussian model, our proposal generates a (non)-stationary random field that is mean square continuous and with Poisson marginal distributions. For the proposed Poisson spatial random field, analytic expressions for the covariance function and the bivariate distribution are provided. In an extensive simulation study, we investigate the weighted pairwise likelihood as a method for estimating the Poisson random field parameters. Finally, the effectiveness of our methodology is illustrated by an analysis of reindeer pellet-group survey data, where a zero-inflated version of the proposed model is compared with zero-inflated Poisson Log-Gaussian and Poisson Gaussian copula models. for this article, including technical proofs and R code for reproducing the work, are available as an online supplement. [ABSTRACT FROM AUTHOR]

Conditions are obtained for a Gaussian vector autoregressive time series of order k, VAR(k), to have univariate margins that are autoregressive of order k or lower‐dimensional margins that are also VAR(k). This can lead to d‐dimensional VAR(k) models that are closed with respect to a given partition {S1,...,Sn} of {1,...,d} by specifying marginal serial dependence and some cross‐sectional dependence parameters. The special closure property allows one to fit the subprocesses of multi‐variate time series before assembling them by fitting the dependence structure between the subprocesses. We revisit the use of the Gaussian copula of the stationary joint distribution of observations in the VAR(k) process with non‐Gaussian univariate margins but under the constraint of closure under margins. This construction allows more flexibility in handling higher‐dimensional time series and a multi‐stage estimation procedure can be used. The proposed class of models is applied to a macro‐economic data set and compared with the relevant benchmark models. [ABSTRACT FROM AUTHOR]

This article discusses the nonparametric estimation of a bivariate density function using copula information under right censoring. We propose an adaptive estimator based on wavelet methods and the formulae for the asymptotic mean integrated squared error(MISE) is used to get the near optimal rate on a large functional class of regular densities. In particular, the asymptotic formulae for MISE in the context of kernel density estimators is derived in the case of censoring. Finally, the consistency of the proposed estimators is established and its effectiveness is validated through a numerical simulations. [ABSTRACT FROM AUTHOR]

After a transmission line is covered by ice in winter, ice-shedding and vibration occurs under special meteorological and external dynamic conditions, which leads to intense transmission line shaking. Transmission line ice-shedding and vibration often cause line flashover trips and outages. In January 2018, three 500 kV transmission lines, namely, the 500 kV Guanli line, the 500 kV Dushan line, and the 500 kV Guanqiao line, tripped and cut off due to ice-shedding and vibration in Anhui province, seriously threatening the safe operation of a large power grid. Current studies mainly focus on analyzing the influence factors and characteristics of line ice-shedding and investigating suppression measures, but they only analyze the correlation between each influencing factor and icing or shedding, and do not consider the coupling effects between multiple factors. In this paper, the key influencing factors and the probability distribution of transmission line ice-shedding were analyzed, and a multiple-factor coupling fault probability calculation model of line ice-shedding based on Copula function was proposed. The fault probability was calculated directly by considering multiple influence factors at the same time, which effectively overcame the error caused by multi-factor transformation in fuzzy membership degree and other methods. It provided an important decision-making basis for preventing and controlling transmission line ice-shedding faults. [ABSTRACT FROM AUTHOR]

In a high proportion of wind–photovoltaic–hydro hybrid power systems, fluctuation and dispersion make it difficult to accurately quantify the output characteristics. Therefore, in this study, a cloud model and copula correlation coefficient matrix were constructed for a hybrid power generation system based on the output data. Multiple backward cloud transformation based on the sampling-with-replacement method was proposed to calculate the improved entropy and hyperentropy to analyze the fluctuation range and dispersion degree quantitatively. A similarity index was proposed to evaluate the similarity between wind power, PV power, and hydropower. A suitable copula function was selected, and the Kendall and Spearman coefficients show the correlation relationships of the hybrid systems. The temporal and spatial characteristics of the hybrid systems were analyzed based on the two models. A typical example in Qinghai proved the effectiveness and applicability of the method. The results show that the correlation between photovoltaic power and hydropower is better and that, in summer, hydropower can be used to adjust the output of renewable energy. [ABSTRACT FROM AUTHOR]

The novel circumstance-driven bivariate integer-valued autoregressive (CuBINAR) model for non-stationary count time series is proposed. The non-stationarity of the bivariate count process is defined by a joint categorical sequence, which expresses the current state of the process. Additional cross-dependence can be generated via cross-dependent innovations. The model can also be equipped with a marginal bivariate Poisson distribution to make it suitable for low-count time series. Important stochastic properties of the new model are derived. The Yule–Walker and conditional maximum likelihood method are adopted to estimate the unknown parameters. The consistency of these estimators is established, and their finite-sample performance is investigated by a simulation study. The scope and application of the model are illustrated by a real-world data example on sales counts, where a soap product in different stores with a common circumstance factor is investigated. [ABSTRACT FROM AUTHOR]

Objectives: The traditional method uses a single monitoring model to conduct the early warning of dam displacement behavior. However, the single monitoring model may not reflect the real behavior of the dam due to its low accuracy, thus causing false alarms. Methods: In order to improve the reliability of early warning results, a new multi-model early warning method is proposed. The hydraulic-season-time (HST) model and autoregressive moving average (ARMA) model, which are the most common models in engineering, are taken as the basic model. First, the advantages and disadvantages of the two models in dealing with dam displacement monitoring data are analyzed. Then the one-dimensional residual distribution of the two models is discussed by kernel density estimation. On this basis, the frequency analysis of the joint residuals of HST-ARMA is carried out and it shows the joint residuals is weakly correlated in non-tail, but strongly correlated in tail. Finally, the Copula function is used to fit the joint residuals of HSTARMA, and the joint distribution function is obtained, which realizes the joint early warning of dam displacement behavior with multiple models. Results: It is verified by several measuring points on the wire alignment system of concrete dam. The case study shows that if a single HST model or ARMA model is used for early warning, the false alarm rate can reach 23.17%-27.94% due to the influence of modeling sequence features and model structure features. And if the HST-ARMA method is used, the false alarm rate can be reduced to 0.00%-0.63% due to combine the advantages of different models. Conclusions: The new method can combine the advantages of different models and avoid the disadvantages of the single model, thus effectively reducing the frequency of false alarms. The warning results of HST-ARMA method are more reliable and can more truly reflect the dam displacement behavior. It can provide reference for improving the dam safety management level. [ABSTRACT FROM AUTHOR]

In the reliability analysis of competing risks model, it is usually assumed that the components are independent and failure causes are observable. However, this assumption may be unrealistic in some applications. In this article, based on the tampered failure rate (TFR) model, we introduce a constant-stress partially accelerated life test (PALT) model of dependent competing risks with masked data. The dependent structure is described by Gumbel copula function and the life distribution of components is described by the Weibull distribution. The maximum likelihood method is used to estimate the dependent parameter and unknown parameters of the model, so as to derive the estimation of the reliability function (RF). Confidence intervals (CIs) of the model parameters are also obtained via asymptotic normality theory and bias-corrected percentile bootstrap (Boot-BCP) technique. Simulation results show that the dependent model with masked data improves the estimation accuracy of system reliability compared with the independent model. Finally, a real data analysis indicates that the statistical inference model and methods have good performance. [ABSTRACT FROM AUTHOR]

In the present study, a new methodology has been introduced for predicting the simultaneous removal optimal conditions of Acid red 33 (AR33), Reactive orange 7 (RO7), Acid yellow 3 (AY3), and Malachite green (MG) dyes. Since industrial wastewater contains a variety of dyes, simultaneous prediction of their removal efficiency is a practical issue. The goal of this research was to address challenges associated with modeling this type of removal process. In the first part of this methodology, the c-vine copula function was used to generate synthetic data since the use of most modeling methods requires a large number of data, and studying different experimental conditions is time- and cost-consuming. Besides, Artificial neural network, Adaptive neuro-fuzzy inference system, Least-square support vector machine, and Long-short term memory algorithms were applied for modeling the dyes' removal efficiency using experimental and synthetic data. Utilizing synthetic data significantly improved the accuracy of modeling. For the purpose of finding the optimal amount of dyes' removal efficiency simultaneously, the removal efficiency of each dye is defined as an objective function. This leads to a problem requiring multi-objective optimization. Hence, the Multi-Objective Adaptive Guided Differential Evolution was employed for multi-objective optimization of dyes' removal efficiency. The optimization process was carried out based on the four objective functions including AR33, RO7, AY3, and MG dyes' removal efficiency. The results indicated a direct relationship between dyes' removal efficiency on the Pareto front as well as a highly oscillating and nonlinear relationship with decision variables (pH, NaCl concentration, current, and Time) under optimal conditions. Finally, the average results of Technique for Order of Preference by Similarity to Ideal Solution and VIekriterijumsko KOmpromisno Rangiranje were utilized to select the best optimal conditions from the Pareto front. With pH, NaCl, Time, and current inputs at 6.60, 0.67 (g/L), 96.93 (min), 0.01 (A) (optimal conditions), the best removal efficiencies of 98.88, 37.02, 91.19, and 98.33 (%) were obtained for AR33, RO7, AY3, and MG dyes, respectively. The introduced method demonstrated excellent promise for predicting optimal conditions for dyes' removal efficiency as well as for making wastewater treatment systems easier to design. [Display omitted] • A new methodology for optimization of AR33, RO7, AY3, and MG dyes removal efficiency. • Utilization of copula function in the generation of a large number of synthetic data. • The ANN, ANFIS, LSSVM-GS, and LSTM modeled the removal efficiency of dyes. • MOAGDE has been used for multi-objective optimization. • The use of TOPSIS and VIKOR to select the best solution of optimal Pareto front. [ABSTRACT FROM AUTHOR]

Local association measures provide useful insights in time-varying changes in association, especially between time-to-event variables. Such local dependence between two correlated random variables can be measured using the cross ratio function. The cross ratio function is defined as the ratio of conditional hazard functions which have been estimated using Bernstein polynomials before. Alternatively, the cross ratio function can be expressed in terms of (derivatives of) the joint survival function of the two random variables. In this paper, we discuss an alternative Bernstein-based plug-in estimator of the cross ratio function in which each of the ingredients is estimated separately. Next to asymptotic normality of the nonparametric estimator, a simulation study is used to assess its finite-sample performance. Finally, the novel estimator is applied to a real-life data application. [ABSTRACT FROM AUTHOR]

In this paper, we study the estimation of a scale parameter from a sample of lifetimes of coherent systems with a fixed structure. We assume that the components are independent and identically distributed having a common distribution which belongs to a scale parameter family. Some results are obtained as well for dependent (exchangeable) components. To this end, we will use the representations for the distribution function of a coherent system based on signatures. We prove that the efficiency of the estimators depends on the structure of the system and on the scale parameter family. In the dependence case, it also depends on the baseline copula function. [ABSTRACT FROM AUTHOR]

This paper investigated reliability modeling for systems subject to dependent competing risks considering that variation of the failure threshold is not considered in most studies on competing failure reliability. Firstly, the variation of degradation quantity under shocks was analyzed, and the variation of the threshold was considered on this basis. Secondly, the cumulative degradation under the influence of the random shock process was analyzed. The attractive property of Copula functions is symmetry. Then, a linear Wiener process model was applied to model performance degradation failure, and a multi-performance degradation correlated-competition model based on a Copula function was constructed, which considered the correlated competition between multi-performance degradation failures. Lastly, a micromotor system was used to analyze the applicability of the proposed model for bivariate instances, demonstrating the rationality and effectiveness of the proposed model. [ABSTRACT FROM AUTHOR]

In this paper, sufficient conditions are presented for comparing the residual lifetimes and inactivity times of dependent random variables at different fixed times, with respect to hazard rate, mean residual life and likelihood ratio orders. The stochastic comparisons of residual lifetimes at different random times are discussed next. Several examples are presented to illustrate the established results. [ABSTRACT FROM AUTHOR]

The complex topology of river networks and the numerous factors influencing streamflow make it challenging to forecast streamflow in large river basins. Improving the accuracy of streamflow forecasting in these basins is of great importance for water resource planning and management. In this paper, considering the spatial variability of impacts of human activities on streamflow, a streamflow forecasting method with a hybrid physical process-mathematical statistic is proposed to obtain better results by realizing the complementary advantages of conventional methods and the streamflow into Hongze Lake in China is forecasted by the proposed method. Firstly, the physical process-based Soil and Water Assessment Tool (SWAT) is used to forecast the streamflow of tributaries with few water storage projects such as reservoirs in the catchment area. Secondly, considering the correlation between mainstream streamflow and tributaries streamflow as well as the correlation between mainstream streamflow and its previous streamflow, streamflow forecasting models based on the vine copula function are constructed to forecast the mainstream streamflow and its confidence intervals, where there are many water storage projects in the catchment area. Then, the total streamflow into Hongze Lake and its confidence intervals are obtained by coupling the outputs of the above two parts. Finally, statistical indicators are chosen to evaluate the forecasting effects in terms of the credibility of deterministic forecasting as well as the reliability and acuity of probabilistic forecasting. The results demonstrate that the proposed method outperforms existing SWAT and long and short-term memory (LSTM) neural network methods in terms of forecasting performance. Consequently, it presents an effective alternative for addressing complex hydrological forecasting tasks in large river basins. [ABSTRACT FROM AUTHOR]

The rank‐tracking probability (RTP) is a useful statistical index for measuring the 'tracking ability' of longitudinal disease risk factors in biomedical studies. Two existing unstructured smoothing methods for estimating the RTP in literature require sufficient observations at any two design time points, which may be violated in practice. We consider the dynamic estimation methods based on semiparametric copula modeling and smoothing method and compare the corresponding estimators under three smoothing ways: parametric‐smoothing (C‐PAS), probability‐smoothing (C‐PRS) and RTP‐smoothing (C‐RS). The proposed estimators are consistent under some mild assumptions when the true copula model is known. Among them, the C‐PAS, though performs best under the true copula model, is not robust for the copula model selection. The C‐PRS and C‐RS estimators are preferred under the selected copula model, which have much smaller mean squared errors than unstructured smoothing methods. Moreover, the C‐RS estimator performs slightly better than C‐PRS estimator. Some numerical simulations and an application of a longitudinal epidemiological study are carried out to illustrate the performances of the proposed methods. [ABSTRACT FROM AUTHOR]

According to the shortage of independent hypothesis allocation models between influence factors and subsystems, a reliability allocation method considering influence factor correlation and subsystem failure correlation is proposed. The CNC (Computer Numerical Control) machine tool is taken as the research object. Using Copula function to analyze failure correlation analysis between subsystems, establish subsystem complete reliability model and integrated importance measure model under failure correlation. The four factors of influencing reliability allocation are subsystem complexity, integrated importance measure of subsystem based on failure correlation, failure frequency, and specific gravity ratio of failure downtime. The correlation analysis among the influencing factors was carried out by using gray correlation theory, and the subjective weight of the subsystem was determined. The IEW‐VIKOR (Information Entropy Weight‐Vlsekriterijumska Optimizacija I Kompromisno Resenje) method is used to determine the objective weights of subsystems. The comprehensive weight of the subsystem is also obtained through combinatorial empowerment. The reliability allocation of the CNC machine tool subsystem was carried out according to the failure correlation reliability model, and the results of failure independence and failure correlation hypothesis allocation were compared and analyzed. When the reliability index is the same, the reliability allocation method proposed in this paper can assign lower reliability to each subsystem, which is of great significance to improve the reliability of CNC machine tools and reduce the difficulty of reliability design [ABSTRACT FROM AUTHOR]

For the fatigue reliability analysis of aeroengine blade-disc systems, the traditional direct integral modelling methods or separate independent modelling methods will lead to low computational efficiency or accuracy. In this work, a physics-informed ensemble learning (PIEL) method is proposed, i.e. firstly, based on the physical characteristics of blade-disc systems, the complex multi-component reliability analysis is split into a series of single-component reliability analyses; moreover, the PIEL model is established by introducing the mapping of multiple constitutive responses and the multi-material physical characteristics into the ensemble learning; finally, the PIEL-based system reliability framework is established by quantifying the failure correlation with the Copula function. The reliability analysis of a typical aeroengine high-pressure turbine blade-disc system is regarded as an example to verify the effectiveness of the proposed method. Compared with the direct Monte Carlo, support vector regression, neural network, ensemble learning and physics-informed neural network, the proposed method exhibits the highest computing accuracy and efficiency, and is validated to be an efficient method for the reliability analysis of blade-disc systems. The current work can provide a novel insight for physics-informed modelling and fatigue reliability analyses. This article is part of the theme issue 'Physics-informed machine learning and its structural integrity applications (Part 1)'. [ABSTRACT FROM AUTHOR]

Multivariate frequency analysis is essential for hydraulic design and water resource management. Copulas have been widely utilized for frequency analysis. However, it is difficult to estimate the parameters of copulas when a real-valued or discrete marginal probability density function (PDF) is unavailable. Employing the matrix-based Renyi's α -order entropy functional, a new estimation method (mutual information estimation, MIE) for the parameters of a bivariate mixed copula is derived by investigating the relationship between Shannon's entropy of the mixed copula and mutual information. MIE requires only the value of mutual information that can be directly estimated from two precipitation sequences without marginal and joint PDF estimation. Based on the annual precipitation sequences at four pairs of neighbouring counties in Qingyang City, four experiments are performed to test the effectiveness of MIE and compared with maximum likelihood estimation (MLE), which requires accurate marginal PDF estimation. All the experiments show that the performance of MIE is even better than that of MLE in three pairs of counties. MIE performs nearly as well as MLE in Huachi and Heshui Counties. Moreover, the mixed copula by MIE is applied to predict the encounter probability of rich–poor precipitation for four pairs of counties. The results show that the values of synchronous probability are much greater than those of asynchronous probability for all four pairs of counties and that the synchronous probability varies from 0.60 to 0.68. Renyi's α -order entropy functional-based derivation contributes to a new method for multivariate frequency analysis of hydrology. [ABSTRACT FROM AUTHOR]

The lack of effective failure correlation analysis is one main reason for the gap between the reliability models and the actual complex systems with mixed static and dynamic characteristics. Takagi and Sugeno (T-S) dynamic fault tree is one powerful tool to analyze the static and dynamic failure logic relationship but it assumes the failure probability of the event is independent. Therefore, this paper proposes a multi-dimensional T-S dynamic fault tree analysis method involving failure correlation. The method integrates the failure probability distribution function of basic events with multi-factors and the multi-dimensional copula function, and the important measure of this method is also deduced. The reliability model expression for systems with failure correlations, both in series and in parallel, is discussed and verified. Compare the proposed method with the assumption that the probability of a failure event is independent. This method solves the problem of a large error when ignoring the failure correlation between parts and the degree of the correlation between variables can be characterized. The reliability analysis can be conducted on complex systems affected both by multi-factors and failure correlations. The proposed method is applied to the reliability analysis of a hydraulic height adjustment system and the correctness and superiority of the method are verified. [ABSTRACT FROM AUTHOR]

This study introduces methodologies for constructing joint probability distribution functions utilizing the Copula function and neural networks, and evaluates their efficacy in marine and civil engineering projects. Through an analytical comparison of both models using a numerical example, it is revealed that the neural network model exhibits superior adaptability to large sample sizes. This adaptability is attributed to the neural network's ability to learn complex relationships within the data, which is especially beneficial when dealing with large datasets. The neural network model also demonstrates higher accuracy in constructing joint probability distribution functions compared to the Copula function model. In marine and civil engineering, the adaptability and accuracy of neural networks are of paramount importance due to the variable and complex nature of weather patterns. A practical engineering application is presented, wherein a joint probabilistic distribution neural network model of wind velocity and rain intensity is established for the Lanzhou–Xinjiang high-speed railroad in China. This model illustrates the promising application of neural networks in engineering projects where weather factors play a critical role. Subsequent to the construction of the joint probability distribution functions, a feature importance analysis is incorporated to quantify the contribution of different weather parameters such as wind velocity and rain intensity to the joint distribution function. This analysis provides an objective assessment of the relative importance of various weather factors and offers data-driven insights that are essential for engineering applications where weather conditions are a significant consideration. The study concludes by highlighting the potential benefits of neural network models in marine and civil engineering, suggesting areas for future exploration. [ABSTRACT FROM AUTHOR]

Mission Success Probability is a top‐level indicator that integrates reliability, maintainability, and spare parts security during the system's mission. Numerical simulation is used for mission success evaluation of complex system because it can consider any form of system structure and component failure distribution model. In actual engineering, system components usually deteriorate or fail with the extension of service time and need to be repaired or replaced, so there are dynamic changes in component status during the life cycle. To evaluate the mission success in such dynamic scenarios, it is necessary to repeat the simulation based on the whole system once the local parameters are adjusted. Despite the continuous improvement in computational performance, traditional numerical simulations are not well matched for rapid evaluation and timely decision making in dynamic scenarios. Therefore, an evaluation method that mixes proxy model and simulation sampling is proposed in this paper. The method uses the simulation to complete the data acquisition, and then analyzes the data by inverse function sampling and Copula function to obtain the model training samples, and finally realizes the approximate description of the local unit simulation process by proxy model. The experimental results show that the method can reduce the repetitive simulation scale from the whole system to local unit while ensuring high accuracy, avoiding many repeated calculations in variable scenarios. [ABSTRACT FROM AUTHOR]

Hydrological droughts are closely related to the meteorological ones in the arid and semi-arid areas. It is of great significance to reveal the response relationship between the meteorological and hydrological droughts in drought warning and water resource management. In this study, the changing patterns of drought were determined in the Xiangjiang River Basin of western China using Copula functions. Daily precipitation data was collected from 40 meteorological stations, while the monthly runoff data was from three hydrological stations (Laobutou, Hengyang, and Xiangtan stations). The standardized precipitation index (SPI) and standardized runoff index (SRI) were adopted to describe meteorological and hydrological droughts. The Pearson correlation coefficient was used to determine the response time to drought. The run theory was used to identify, merge, and remove the drought events. The Mann-Kendall trend test and wavelet analysis were employed to reveal the trend and periodicity of drought in different seasons in the upper, middle, and lower reaches. Furthermore, seven widely used distribution functions were selected to fit the duration and severity sequences of meteorological and hydrological drought that were obtained from SPI and SRI. The optimal parameter estimation was obtained in each marginal distribution using the maximum likelihood. There were strong links between meteorological and hydrological droughts using three Archimedean Copula functions (Frank, Gumbel, and Clayton Copula) by coupling the response probability curve in a Bayesian network model. Finally, the model performance was evaluated using the Akaike information criterion (AIC) and ordinary least squares (OLS). The main results demonstrate: 1) The run theory [0.5, 0, −0.5] was achieved to better identify the drought events, with a matching rate of over 90% between meteorological and hydrological droughts. There was a two-month propagation time from the meteorological to hydrological drought in the whole watershed. The SPI and SRI effectively detected the drought events and better matched them with the historical drought statistics from the different periods (in 1963, 1986, and 2003). 2) The frequency of meteorological drought was higher than that of hydrological drought. The duration and severity of hydrological drought were greater than those of meteorological one, where the downstream area experienced the most severe hydrological drought, but the midstream is the most severely affected by meteorological drought. The difference was attributed to the higher evapotranspiration and high level of socio-economic development in the downstream. 3) There was a tendency toward wetness of the historical data in the basin, which was consistent with the remote sensing monitoring of terrestrial water storage anomaly. However, there was a distinct seasonal variation in the inter-annual wetness of the basin. The upper and middle reaches showed a drier trend in the autumn, while the middle reaches were in a trend toward dryness in the summer. Additionally, there were several outstanding oscillation periods of dry and wet, namely 3-5, 6-10, and 18-21 a. 4) The optimal joint distribution of meteorological-hydrological drought severity in the upstream and downstream area was the Gumbel Copula function, while the Frank Copula function was the optimal joint distribution for the rest. 5) The response probability of hydrological drought duration and severity increased with the increase of characteristic variables in the meteorological drought. There was a stable response probability of hydrological drought duration when the duration of meteorological drought exceeded a specific threshold. This study is of great significance in understanding the mechanism of drought in the Xiangjiang River Basin as well as guiding the drought prevention and fighting work in the region. [ABSTRACT FROM AUTHOR]

The Copula concept has long been used in many applications, especially in the financial field. This concept was first used in 1959 by Sklar in his mathematical work and greatly assisted in the applications of financial and insurance areas. The copula functions have been widely used in dependence modeling. In this study, we look at how the copula began to develop from a basic form to a more advanced form through studies that previous researchers have made. Throughout this study, we find various types of the copula, and each exhibits its own characteristics lying under two main families, Elliptical and Archimedean copulas. Our findings suggest that copula is vital in solving problems in statistical dependence measures and joint marginal distribution functions. This comprehensive study served as a review paper on the development of copulas from their initial existence to their latest evolution. [ABSTRACT FROM AUTHOR]

Due to the large climatic and geographical differences, the occurrence time of maximum 1-day precipitation (RM1day) in different zones of the Huaihe River Bain (HRB) is different. In this study, in order to analyze the characteristics of precipitation encounters, we used the mixture of von Mises distributions and Copula functions to analyze the occurrence time distribution, interval time distribution, and joint encounter probability of RM1day in the HRB. We divide the upstream area of HRB into Zone1, the midstream area into Zone2-1 and Zone2-2, and the downstream area into Zone3 and Zone4. The results show that the RM1day of Zone1, Zone2-1, Zone2-2, Zone3, and Zone4 are mainly concentrated on July 13, July 9, July 6, July 19, and July 14, respectively. The probability appearing RM1day of Zone1 first to Zone2-1, Zone1 first to Zone2-2, Zone2-2 first to Zone2-1, and Zone4 first to Zone3 is 50.84%, 40.0%, 49.1%, and 42.4%, respectively. It shows that most of the RM1day occurred in the southern sites of the HRB earlier than the northern sites. Zone1 and Zone2-2, Zone1 and Zone2-1, Zone2-2 and Zone2-1, and Zone4 and Zone3 have the highest encounter probability of RM1day in July, with 0.34%, 0.56%, 0.37%, and 0.42%, respectively. Therefore, we need to pay more focus to the risk of extreme precipitation encounter in July. This study provides an important reference for flood control and anti-logging in the HRB. [ABSTRACT FROM AUTHOR]

Joint modelling of longitudinal and time-to-event data is usually described by a joint model which uses shared or correlated latent effects to capture associations between the two processes. Under this framework, the joint distribution of the two processes can be derived straightforwardly by assuming conditional independence given the random effects. Alternative approaches to induce interdependency into sub-models have also been considered in the literature and one such approach is using copulas to introduce non-linear correlation between the marginal distributions of the longitudinal and time-to-event processes. The multivariate Gaussian copula joint model has been proposed in the literature to fit joint data by applying a Monte Carlo expectation-maximisation algorithm. In this paper, we propose an exact likelihood estimation approach to replace the more computationally expensive Monte Carlo expectation-maximisation algorithm and we consider results based on using both the multivariate Gaussian and t copula functions. We also provide a straightforward way to compute dynamic predictions of survival probabilities, showing that our proposed model is comparable in prediction performance to the shared random effects joint model. [ABSTRACT FROM AUTHOR]

Rainfall and temperature extremes have become more frequent and severe in recent times due to changing climate. Since these catastrophic occurrences directly affect a region's hydrology, it is imperative to develop models that can project and explain the joint behavior of climate variables. Copula functions have been used relatively successfully to capture multivariate processes. With climate being a multifaceted process, there is interdependence between variables, making copula use desirable since traditional bivariate distributions do not account for the dependent structure. In this study, we introduced a bivariate exponentiated Teissier distribution based on a Clayton copula. For parameter estimation, the maximum likelihood and inference functions for margin approaches are used. A simulation study that considered various sets of parameters is also conducted in order to select the most efficient parameter estimation method. Last, the applicability of the proposed model is demonstrated using real-world data from flood and temperature processes. After fitting, the log-likelihood, Akaike information criteria (AIC), and Bayesian information criteria (BIC) values of the proposed model are −145.00, 300.00, and 311.71 for flood data, respectively, and −128.71, 267.42, and 275.98 for temperature data, respectively. Estimated parameters are α ^ 1 = 1.730 038 4 , α ^ 2 = 2.164 941 2 , θ ^ 1 = 0.429 843 6 , θ ^ 2 = 0.598 815 7 , δ ^ = 1.662 378 9 for flood data and α ^ 1 = 4745.48 , θ ^ 1 = 0.0988 , α ^ 2 = 26 056.65 , θ ^ 2 = 0.0583 , δ ^ = 0.454 for temperature data. It is concluded that this model may be effectively used for modeling the hydrological processes for calculating the probabilities of flood and extreme temperature events. [ABSTRACT FROM AUTHOR]

In the form of a scholarly exchange with ChatGPT, we cover fundamentals of modeling stochastic dependence with copulas. The conversation is aimed at a broad audience and provides a light introduction to the topic of copula modeling, a field of potential relevance in all areas where more than one random variable appears in the modeling process. Topics covered include the definition, Sklar's theorem, the invariance principle, pseudo-observations, tail dependence and stochastic representations. The conversation also shows to what degree it can be useful (or not) to learn about such concepts by interacting with the current version of a chatbot. [ABSTRACT FROM AUTHOR]

This paper examines the preservation of several aging classes of lifetime distributions in the formation of coherent and mixed systems with independent and identically distributed (i.i.d.) or identically distributed (i.d.) component lifetimes. The increasing mean inactivity time class and the decreasing mean time to failure class are developed for the lifetime of systems with possibly dependent and i.d. component lifetimes. The increasing likelihood ratio property is also discussed for the lifetime of a coherent system with i.i.d. component lifetimes. We present sufficient conditions satisfied by the signature of a coherent system with i.i.d. components with exponential distribution, under which the decreasing mean remaining lifetime, the increasing mean inactivity time, and the decreasing mean time to failure are all satisfied by the lifetime of the system. Illustrative examples are presented to support the established results. [ABSTRACT FROM AUTHOR]

In this article, we study an integral transformation of the copula function. We prove that under certain conditions, the studied integral transformation is also the copula function. Some properties of the integral transformation of a copula function are studied. Also we derive bounds for Kendall's tau and Spearman's rho. [ABSTRACT FROM AUTHOR]

The univariate analysis of hydrological extremes is a well-established practice in developing countries such as Ethiopia. However, for the design of hydrological and hydraulic systems, it is essential to have a thorough understanding of flood event characteristics, including volumes, peaks, time of occurrence, and duration. This study utilizes copula functions for bivariate modeling of flood peak and volume characteristics, examining the performance of four Archimedean copulas in the Guder basin located in Ethiopia from 1987 to 2017. Flood peak and volume were extracted using the theory of runs for analysis of their joint characteristics with the truncation level chosen as equal to the lowest annual maximum event. Univariate distributions with the best fitness on both variables were determined, and results showed that gamma and GEV-fitted flood peaks and lognormal-fitted flood volumes are the most suitable. Four Archimedean copulas were evaluated, and the Gumbel-Hougaard copula was found to be the best fit for the data based on graphical and measurable tests. Bivariate probability and return period were computed in "AND" and "OR" states. The joint return period for flood peak (97.49 m3/s) and volume (77.35 m3/s) was found to be 15 years in the "AND" state and approximately 4 years in the "OR" state. The study also evaluates univariate and conditional return periods, comparing them with the primary one. The copula method was an effective method for distributing marginal variables, highlighting its potential as a valuable tool in flood management. [ABSTRACT FROM AUTHOR]

Stochastic characterizations of functions subject to constraints result in treating them as functions with non-independent variables. By using the distribution function or copula of the input variables that comply with such constraints, we derive two types of partial derivatives of functions with non-independent variables (i.e., actual and dependent derivatives) and argue in favor of the latter. Dependent partial derivatives of functions with non-independent variables rely on the dependent Jacobian matrix of non-independent variables, which is also used to define a tensor metric. The differential geometric framework allows us to derive the gradient, Hessian, and Taylor-type expansions of functions with non-independent variables. [ABSTRACT FROM AUTHOR]