Copula Function Research Articles

A Long-Term Power Supply Risk Evaluation Method for China Regional Power System Based on Probabilistic Production Simulation

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.

Optimization mechanism of metakaolin on micro-mechanical properties and pore characteristics of steel fiber-cement matrix interface

The study investigated the influence of aeolian sand and metakaolin (MK) on the micro-mechanical properties and pore structure characteristics of the interfacial transition zone (ITZ) between steel fibers and matrix in steel fiber-reinforced cement-based materials by nanoindentation, scanning electron microscopy (SEM) imaging, and energy-dispersive spectroscopy (EDS) analysis. The results demonstrated that including MK and aeolian sand led to a denser and uniform micro-structure in the ITZ, along with reducing the porosity and refining the pore size within there. Furthermore, MK could enhance the micromechanical properties of the C-(A)-S-H) gel by increasing the Al/Si ratio. Finally, the Copula function further confirmed the correlation between porosity (pore size >0.2 μm) and micro-mechanical properties in the ITZ. This study provides a theoretical basis for the engineering application of aeolian sand and metakaolin, and proves the feasibility of porosity to characterize the microscopic mechanical properties of the interfacial transition zone of cement-based materials. Thus, it has specific engineering value and academic significance.

Assessment of Hydrological and Meteorological Composite Drought Characteristics Based on Baseflow and Precipitation

Traditional univariate drought indices may not be sufficient to reflect comprehensive information on drought. Therefore, this paper proposes a new composite drought index that can comprehensively characterize meteorological and hydrological drought. In this study, the new drought index was established by combining the standardized precipitation index (SPI) and the standardized baseflow index (SBI) for the Jiaojiang River Basin (JRB) using the copula function. The prediction model was established by training random forests on past data, and the driving force behind the combined drought index was explored through the LIME algorithm. The results show that the established composite drought index combines the advantages of SPI and SBI in drought forecasting. The monthly and annual droughts in the JRB showed an increasing trend from 1991 to 2020, but the temporal characteristics of the changes in each subregion were different. The accuracies of the trained random forest model for heavy drought in Baizhiao (BZA) and Shaduan (SD) stations were 83% and 88%, respectively. Furthermore, the Local Interpretable Model-Agnostic Explanations (LIME) interpretation identified the essential precipitation, baseflow, and evapotranspiration features that affect drought. This study provides reliable and valid multivariate indicators for drought monitoring and can be applied to drought prediction in other regions.

System reliability analysis of Cable-stayed bridge using pair Copula and ICLHS with unequal weights under seismic loading

Considering the correlation between different components, quantifying the failure probability of a high-level hyper-static structure under non-stationary seismic loading is of great significance in guiding earthquake-resistant design and fortification of bridges. In this study, a high-pier and large-span railway cable-stayed bridge in the mountainous region were employed to evaluate the seismic performance. First, the basic theoretical methods of the non-stationary random seismic motion simulation, equivalent extreme value theory based on extreme value distribution (EVD), improved Latin hypercube sampling (ICLHS) with unequal weighted fractional moment estimation and Pair Copula technique are presented. Second, the finite element analysis model of the cable-stayed bridge was established based on the OpenSees platform, the seismic response samples of the bridge were calculated under three levels of seismic ground motion to determine the first passage failure probability of the main components of the bridge under the design failure modes. Finally, the Pair Copula function was introduced to estimate the seismic system reliability of the cable-stayed bridge. The optimal Copula type that can describe the correlation between the bridge components was selected through goodness-of-fit tests such as AIC, BIC, and the minimum distance method. The failure probability of the entire system was obtained by calculating the selected Copula function type under seismic loading.

Cyber Risk in Insurance: A Quantum Modeling

In this research, we consider cyber risk in insurance using a quantum approach, with a focus on the differences between reported cyber claims and the number of cyber attacks that caused them. Unlike the traditional probabilistic approach, quantum modeling makes it possible to deal with non-commutative event paths. We investigate the classification of cyber claims according to different cyber risk behaviors to enable more precise analysis and management of cyber risks. Additionally, we examine how historical cyber claims can be utilized through the application of copula functions for dependent insurance claims. We also discuss classification, likelihood estimation, and risk-loss calculation within the context of dependent insurance claim data.

Vector Autoregressive Model Using Non-Normal Copula Errors

This article focuses on a bivariate vector autoregressive model of lag 1 with non-normal errors. We propose a bivariate error distribution using two identical marginals through a copula function. The copula used for the study is the Farlie-Gumbel-Morgenstern copula, which is considered to be one of the efficient class of copula in describing the dependence between two random variables. The model parameters are estimated using the method of inference functions for margins, and the finite sample properties of the model is illustrated through simulation studies. A real life example is considered to illustrate the applications of the proposed model. The adequacy of the copula model is examined using the two sample version of kolmogorov smirnov test statistic.

Modeling and Reliability Analysis of MEMS Gyroscope Rotor Parameters under Vibrational Stress.

Vibrational environments can cause drift or changes in Micro-Electro-Mechanical System (MEMS) gyroscope rotor parameters, potentially impacting their performance. To improve the effective use of MEMS gyroscopes, this study introduced a method for evaluating the reliability of parameter degradation under vibration. We analyzed the working principle of MEMS gyroscope rotors and investigated how vibration affects their parameters. Focusing on zero bias and scale factor as key performance indicators, we developed an accelerated degradation model using the distributional assumption method. We then collected degradation data for these parameters under various vibration conditions. Using the Copula function, we established a reliability assessment approach to evaluate the degradation of the MEMS gyroscope rotor's zero bias and scale factor under vibration, enabling the determination of reliability for these parameters. Experimental findings confirmed that increasing stress levels lead to reduced failure times and increased failure rates for MEMS gyroscope rotors, with significant changes observed in the zero bias parameter. Our evaluation method effectively characterizes changes in the reliability of the MEMS gyroscope rotor's scale factor and zero bias over time, providing valuable information for practical applications of MEMS gyroscopes.

Phase II control chart for monitoring Gumbel’s bivariate exponential distribution

Schemes for monitoring two or more time between events (TBE) data are becoming increasingly significant in statistical process monitoring. In practice, parameters must be estimated prior to the start of process monitoring. In this paper, we first propose a multivariate time between events (MTBE) control chart based on the one-sample log-likelihood ratio test for monitoring the Gumbel’s bivariate exponential (GBE) distribution, and then we analyze the effect of parameter estimation on this chart. To avoid the effect of parameter estimation, we then propose a method based on the two-sample log-likelihood ratio test for monitoring the shifts occurring in either the scale parameter or the dependence parameter or both for the GBE distribution process. The proposed method is designed with two-sample log-likelihood ratio test for univariate marginal distribution function and Gumbel copula function. A Max-type function-based approach is implemented for joint monitoring of margin distribution and copula-related structure. In contrast to the traditional schemes used to monitor bivariate processes, the strength of our proposed technique lies in the ability to identify exactly which component of the scale parameter or dependence parameter is responsible for the source of the signal. The performance of the proposed chart is analyzed via Monte Carlo simulations. Finally, two examples are mentioned to illustrate the implementation process of the proposed scheme.

Reliability analysis of rolling bearings considering failure mode correlations

AbstractAs an essential mechanical component, a rolling bearing can exhibit multiple failure modes that may occur independently or in correlation with one another. A reliability analysis method that meticulously accounts for the interdependencies among various bearing failure modes is presented in this paper. The examination of wear and fatigue failure mechanisms in rolling bearings is carried out using the Physics of Failure (PoF) approach. By considering the influence of uncertain variables, the limit state functions for individual failure modes are formulated through the application of stress‐strength interference theory. In the context of wear failure, the limit state function is derived using working clearance as the characteristic quantity. On the other hand, the limit state function for fatigue failure is constructed with a focus on fatigue damage accumulation. The Copula function is used to characterize the relationship between wear failure and fatigue failure, and a reliability calculation model for rolling bearings is developed, considering the correlation between these failure modes. Ultimately, the proposed method is utilized to assess the reliability of bearings under two different sets of test conditions. The feasibility of this method is confirmed through test data, demonstrating its effectiveness in predicting bearing reliability. Through the application of this method, engineers can optimize bearing size parameters, select appropriate initial clearances, and enhance the reliability design of bearing.

A copula-based approach to instream ecological flow determination considering inter- and intra-annual runoff variability

ABSTRACT Instream ecological flow (IEF) provides a flow reference value for maintaining the stability of river ecosystems and enriching biodiversity. Existing methods for determining the IEF often consider only one of the intra-annual runoff distribution characteristics or the inter-annual runoff distribution characteristics. In this study, a copula-based ecological runoff determination method is proposed to consider both annual and monthly runoff magnitudes. The marginal distribution functions of annual and monthly runoff are first determined using the maximum likelihood method. Then the copula function is used to construct the joint distribution function of annual and monthly runoff series. Using the flow duration curve method, the IEF under different annual and monthly runoff exceedance probabilities is calculated. The probability values are combined with the initial IEF values to determine the final IEF process. A case study is conducted using real-world data from the Tongtian River in China. The results show that the total annual basic ecological water demands of the Tongtian River in abundant, flat, and dry water years are 138, 102 and 72.2 billion m3. The proposed method effectively avoids the disturbance of extreme runoff.

A neural network copula function approach for solving joint basic probability assignment in structural reliability analysis

AbstractApplying evidence theory to structural reliability analysis under epistemic uncertainty, it is necessary to consider the correlation of evidence variables. Among them, solving the joint basic probability assignment (BPA) of the evidence variables is a crucial link. In this study, a solution method of joint BPA based on neural network copula function is proposed. This method is to automatically construct copula function through neural network, which avoids the process of selecting the optimal copula function. Firstly, the neural network copula function is constructed based on the sample set of evidence variables. Then, the expression for solving the joint BPA using the neural network copula function is derived through vectors. Furthermore, the expression is used to map the marginal BPA of evidence variables to joint BPA, thus realizing the solution of joint BPA. Finally, the effectiveness of this method is verified by three examples. The results show that the neural network copula function describes the data distribution better than the optimal copula function selected by the traditional method. In addition, there is actually an error in solving the reliability intervals using the traditional optimal copula function method, whereas the results of this paper's neural network copula function method are more accurate and better for decision making.

Reliability analysis method of energy storage stations considering the correlation of battery clusters

In the context of dual carbon, in order to address the volatility of new energy output, energy storage power stations (ESPSs) are thriving. However, it is challenging to assess the dependability of ESPS operation because of the steady functioning of individual battery clusters (BCs), which will have an impact on how safely and steadily the power system is evaluated. Therefore, a reliability model for energy storage BC units based on the structural analysis of battery structure modules is constructed in this manuscript. Then, a method for constructing a correlation model of energy storage units based on the Copula function was proposed. Finally, a simulation analysis based on the ESPSs of pertinent demonstrative application projects in Fujian Province is used as an example to confirm the viability and efficacy of the proposed reliability analysis approach.

AI‐Based Ensemble Flood Forecasts and Its Implementation in Multi‐Objective Robust Optimization Operation for Reservoir Flood Control

AbstractInforming reservoirs with forecasts is highly important for real‐time flood control. This study proposed a forecast‐informed methodology framework for reservoir flood control operation under uncertainty. A new combination of two post‐processing methods, that is, the Cloud model and error‐based copula functions, were developed to merge individual AI‐based forecasts to ensemble flood forecasts, so called stochastic errors‐based Cloud (SE‐Cloud). A multi‐objective robust optimization model (MRO) integrating the risk, resilience, and vulnerability was then proposed to tackle flood control problems under ensemble forecasts; for comparison, a two‐objective stochastic optimization model (TSO) was developed to minimize the expected highest reservoir level and peak release. The proposed methodology was applied to the Lishimen reservoir in the Shifeng River subbasin, China, aiming to comprehensively verify the relationships among deterministic forecasts, ensemble forecasts, and flood control performance. Results showed that the Cloud model could effectively integrate different models and improve forecast accuracy. But a higher deterministic forecast quality did not consistently result in improved flood control performance. SE‐Cloud could capture the peak flow and effectively characterize forecast uncertainties and increased hypervolume values by 13.14%–39.65% compared to the Cloud model, indicating the superiority of ensemble forecasts in generating robust solutions over individual deterministic forecasts. MRO released more inflow than TSO, decreasing the expected highest water level by 0.05 m and incrementing the expected peak release by 4.29%. However, with downstream resilience value remaining at zero, it is demonstrated that MRO improving upstream vulnerability did not necessarily diminish resilience. The enhanced robustness highlights the potential of AI‐based ensemble forecasts in flood control.

A Copula Based Unsupervised Domain Adaptation for Image Classification

In this paper, we present an unsupervised domain adaptation algorithm for image classification using principal component analysis (PCA) and Gaussian copula function alignment. The motivation of the proposed algorithm stems from the idea of CORAL algorithm which extracts domain invariant features by aligning the correlation structure between a source and a target domain. However, it suffers from the fact that highly skewed marginal distributions happen to distort the correlation structure so that it may cause a negative transfer. Therefore we utilize a copula function that enables us to analyze separately the dependency structure and the marginals by Sklar’s theorem. In particular, we propose to align the Gaussian copula correlations in the copula feature spaces instead of aligning the correlation matrices in the original space. Considering the extremely skewed distribution of SURF image features in Office-Caltech10 data set we apply PCA first in order to extract some skewness-mitigated principal features and then derive copula features to align for domain adaptation by using CORAL idea with Gaussian copula correlation matrices. The proposed method showed a good classification accuracy when applied to image classification problem in an unsupervised domain adaptation setting.

Vine Copula-Based Multivariate Distribution of Rainfall Intensity, Wind Speed, and Wind Direction for Optimizing Qatari Meteorological Stations

This study employs copula functions to establish the dependency structure of the joint distribution among rainfall intensity, wind speed, and wind direction in Qatar. Based on a Vine Copula, the trivariate distribution between rainfall intensity, wind speed, and wind direction is found to exhibit a root-mean-square error (RMSE) of 0.0072 on the observed vs. modeled cumulative probabilities using ranked normalized observations. It is also found that the winter Shamal winds are most pronounced during rainfall. However, a secondary component of easterly winds known as the Kaus winds is also found to exert an important influence. This wind pattern is observable during rainfall at all the selected stations, albeit with minor variations. It is also found that rainfall stations where the rainfall is obstructed in any way from northwest to north and from east to southeast significantly influence the rainfall measurements. Specific rain gauges in Qatar are found to be situated in disrupted surroundings, such as meteorological stations close to passing traffic, where road spray could infiltrate the rain gauge funnel, impacting the accuracy of rainfall measurements. The study results necessitated the relocation of approximately half of these roadside gauges to mitigate wind-induced biases from road spray. An evaluation of operations is recommended for approximately 80 meteorological stations responsible for measuring rainfall in Qatar. The methodology devised in this study holds potential for application to other Middle Eastern countries and regions with similar climates.

Nonparametric estimation of the copula function with bivariate twice censored data

In this work, we are interested in the nonparametric estimation of the copula function in the presence of bivariate twice censored data. Assuming that the copula functions of the right and the left censoring variables are known, we propose an estimator of the joint distribution function of the variables of interest, then we derive an estimator of their copula function. Using a representation of the proposed estimator of the joint distribution function as a sum of independent and identically distributed variables, we establish the weak convergence of the empirical copula that we introduce.

Operational Risk Measurement: An Optimal Timescale Selection Method Based on Information Entropy

Operational risk capital is typically allocated on an annual timescale in extant research, while ignoring that the dependence between operational risk events may differ on different timescales, such as semi-annual and quarterly. This study examines whether operational risk dependence structures differ on multiple timescales and proposes an optimal timescale selection method based on information entropy theory to obtain more reasonable operational risk measurement results. Different dependence modeling methods, including linear correlation and copula functions, are employed to analyze the dependence structures on multiple timescales. The information changes between different timescales are calculated to select the optimal timescale. The empirical analysis is based on 2,024 operational risk events from 1994 to 2017 in the Chinese Operational Loss Database (COLD), which is one of the largest external operational risk datasets for the Chinese banking industry. The results reveal significant differences in operational risk dependence at different timescales which affect the operational risk measurement results. Crucially, the quarterly timescale should be considered for more reasonable measurements. The findings provide important insights for considering a more reasonable allocation of operational risk capital under multiple timescale dependence.

Failure probability analysis of high fill levee considering multiple uncertainties and correlated failure modes

Such complex causative factors in current failure probability models are represented by simply random uncertainty and completely independent or correlation of failure modes, which can often limit the model utility. In this study, we developed a methodology to construct failure probability models for high fill levees, incorporating the identification of uncertainties and an analysis of failure modes. Based on quantification of stochastic-grey-fuzzy uncertainties, probability analysis involved with overtopping, instability and seepage failure modes was implemented combined with probability and non-probability methods. Given that the interaction among failure modes typically exhibits nonlinear behavior, rather than linear correlation or complete independence, a simple methodology for the binary Copula function was established and implemented in MATLAB. This methodology was applied to the high fill segments of a long-distance water transfer project characterized by high population density. It shows that the failure probability of a single failure mode is overestimated when uncertainties are not considered, because of the randomness and fuzziness of some parameters and the greyness of information. Meanwhile, it is found that the magnitude of failure probability related to levee breach is overestimated without respect to failure modes correlation, especially when the probabilities of seepage and instability are both significant and closely aligned.

Extraction of intensity-duration for short-term extreme rainfalls from daily and yearly extreme rainfalls using copula functions

Extraction of intensity-duration for short-term extreme rainfalls from daily and yearly extreme rainfalls using copula functions

Using Copula functions to predict climatic change impacts on floods in river source regions

Flood frequency in river source regions is significantly affected by rainfall and snowmelt as part of climatic changes. A traditional univariate flood frequency analysis cannot reflect the complexity of floods, and when used in isolation, it can only underestimate flood risk. For effective flood prevention and mitigation, it is essential to consider the combined effects of precipitation and snowmelt. Copula functions can effectively quantify the joint distribution relationship between floods and their associated variables without restrictions on their distribution characteristics. This study uses copula functions to consider a multivariate probability distribution model of flood peak flow (Q) with cumulative snowmelt (CSm) and cumulative precipitation (CPr) for the Hutubi River basin located in northern Xinjiang, China. The joint frequencies of rainfall and snowmelt floods are predicted using copula models based on the Coupled Model Intercomparison Project Phase 6 data. The results show that Q has a significant positive correlation with 24-d CSm (r = 0.559, p = 0.002) and 23-d CPr (r = 0.965, p < 0.05). Flood frequency will increase in the future, and mid- (2050–2074) and long-term (2075–2099) floods will be more severe than those in the near-term (2025–2049). The probability of flood occurrence is higher under the SSP2-4.5 and SSP1-2.6 scenarios than under SSP5-8.5. Precipitation during the historical period (1990–2014) led to extreme floods, and increasing future precipitation trends are found to be insignificant. Snowmelt increases with rising temperatures and occurs earlier than estimated, leading to an earlier flood period in the basin and more frequent snowmelt floods. The Q under the joint return period is larger than that during the same univariate return period. This difference indicates that neglecting the interaction between precipitation and snowmelt for floods leads to an underestimation of the flood risk (with underestimations ranging from 0.3% to 22%). The underestimations decrease with an increase in the return period. The joint risks of rainfall or snowmelt according to various flood periods should be considered for rivers with multi-source runoff recharge in flood control design. This study reveals the joint impact of precipitation and snowmelt on extreme floods under climate change in river source regions. This study also provides a scientific basis for regional flood prevention and mitigation strategies, as well as for the rational allocation of water resources.