Copula Function Research Articles

Comprehensive evaluation of extreme hydrometeorological events coincidence and their interrelationships in the Hanjiang River Basin, China

Extreme hydrometeorological events are often highly destructive and can cause huge damage to the natural environment and social economy. As climate change and human activities intensify, the frequency and intensity of such extreme events continue to increase. This phenomenon is also shown in the Hanjiang River Basin (HRB) in recent decades. In this study, we focus on the coincidence occurrence patterns of extreme hydrometeorological events. First, we used the standardized index method to extract the annual maximum and minimum values of runoff, precipitation, maximum and minimum temperatures from 1961 to 2019 in the control area of the Baihe (BH) hydrological station, with time scales from 1 month to 12 months. Then through the Copula and nested structure Copula functions, the joint occurrence probability of bivariate and multivariate were analyzed. The results show that too high a maximum temperature will inhibit runoff, while an increase in minimum temperature will promote an increase in runoff to a certain extent. Both the maximum and minimum temperatures in the basin shown a significant increasing trend, and the intensity of hydrological drought also shows an increasing trend. An increase in maximum temperature will aggravate hydrological drought, as the combined joint occurrence probability of them is the greatest. Analysis through the nested structure Copula functions show that when extreme hydrometeorological events both occurred at the median of their intervals, their coincidence occurrence probability is the maximum, and when the time scale is eight months, their coincidence occurrence probability is the highest. This study can provide a reference for joint probability analysis of extreme hydrometeorological events.

Quantifying the Impacts of Dry–Wet Combination Events on Vegetation Vulnerability in the Loess Plateau under a Changing Environment

Extreme drought and flood events, as well as their combined events, pose significant challenges to global sustainable socio-economic development and ecological health. However, the impact of dry–wet combination events (DWCEs) on vegetation vulnerability remains to be investigated. The Loess Plateau (LP) was selected as the study area to explore the response time of vegetation to precipitation index changes by optimal correlation coefficient; then, the impact of different DWCEs on vegetation vulnerability under moderate and severe scenarios was analyzed; finally, a vegetation loss probability model was constructed based on the copula function and Bayesian framework, to quantify the vegetation loss probability under DWCEs stress. The results indicate that: (1) normalized difference vegetation index (NDVI) shows an upward trend in spring, summer, and autumn, with the proportion of areas are 90.5%, 86.2%, and 95.4%, respectively, and show an insignificant trend in winter; (2) the response time of vegetation to precipitation index changes tends to be one or two seasons; (3) moderate scenarios have more influence than severe scenarios, dry-to-wet events (DWEs), wet-to-dry events (WDE) and continuous dry events (CDE) in spring-summer have a significant impact on summer vegetation of Ningxia and Shanxi, and WDE and CDE have a higher impact on autumn vegetation. (4) in terms of the probability of vegetation loss, DWE, and CDE cause higher losses to summer vegetation, while WDE and CDE cause higher losses to autumn vegetation. This study quantifies the impact of adjacent seasonal DWCE stress on future vegetation vulnerability.

Revisiting the estimation of extreme wind speed considering directionality

In the estimation of extreme wind speed, the effect of wind direction is usually ignored. Nevertheless, as a vital parameter of the wind, it should be reasonably taken into consideration. Nowadays, the influence of wind direction can be considered through directional wind speed or directly regarding it as a variable. Based on wind speed data of climate stations, the wind directions are divided into uniform and non-uniform sectors. Copula function is then applied to establish the joint distribution of directional wind speed. The effect of the partition of directional sectors is discussed subsequently. Meanwhile, a joint distribution model of wind speed and wind direction based on copula function is also set up. According to the joint distribution, extreme wind speeds under specified wind directions are estimated based on conditional probability. Besides, extreme wind speeds considering wind directionality based on the aforementioned two methods are compared. Results show that the classification of wind direction will affect the estimation of extreme wind speed. In addition, the estimation of extreme wind speed under specified wind direction based on the joint distribution of wind speed and wind direction does not reflect the actual wind characteristics. Through the comparison of different methods considering directionality, the result based on the joint distribution of directional wind speed seems more reasonable.

Advanced Copula-Based Models for Type II Censored Data: Applications in Industrial and Medical Settings

Copula models are increasingly recognized for their ability to capture complex dependencies among random variables. In this study, we introduce three innovative bivariate models utilizing copula functions: the XLindley (XL) distribution with Frank, Gumbel, and Clayton copulas. The results highlight the fundamental characteristics and effectiveness of these newly introduced bivariate models. Statistical inference for the distribution parameters is conducted using a Type II censored sampling design. This employs maximum likelihood and Bayesian estimation techniques. Asymptotic and credible confidence intervals are calculated, and numerical analysis is performed using the Markov Chain Monte Carlo method. The proposed methodology’s applicability is illustrated by analyzing several real-world datasets. The initial dataset examines burr formation occurrences and consists of two observation sets. Additionally, the second and third datasets contain medical information. The second dataset focuses on diabetic nephropathy, while the third dataset explores infection and recurrence time among kidney patients.

Reliability Analysis for RV Reducer by Combining PCE and Saddlepoint Approximation Considering Multi-Failure Modes

Abstract RV (Rotate Vector) reducer is an essential mechanical transmission device extensively used in industrial machinery, robotics, aerospace and other fields. The dynamic transmission characteristics and strength of the cycloidal pin gear, turning arm bearing of RV reducer significantly affect the motion accuracy and reliability of the whole equipment. Uncertainties from manufacturing and assembly error, working loads add complexity to these effects. Developing effective methods for uncertainty propagation and reliability analysis for the RV reducer is crucial. In this work, the mail failure modes of RV reducer are studied, and an effective reliability analysis method for RV reducer considering the correlation between multi-failure modes by combining polynomial chaos expansions (PCE) and saddlepoint approximation method (SPA) is proposed. This paper develops an uncertainty propagation strategy for RV reducer based on dynamic simulation and PCE method with high accuracy. On this basis, a surrogated cumulant generating function (CGF) and SPA are combined to analyze the stochastic characteristic for the failure behavior. Based on the probability density function (PDF) and cumulative distribution function (CDF) calculated by SPA, copula function is employed to quantify the correlations between the multi-failure modes. Then, the system reliability with multi-failure modes is estimated by SPA and optimal copula function. The proposed method provides an effective reliability assessment technology with high-accuracy for complex system under unknown physical model and distribution characteristics. The validity of the proposed approach is illustrated RV-320E reducer reliability estimation, offering a basis to improve the performance of complex dynamic system. .

Comparative Analysis of Financial Market Volatility and Correlation Risk During the Great Recession and the COVID-19 Pandemic

This paper offers a starting point for reflection on the similarities and differences of the impact on financial markets of the Great Recession of 2008 and of the Covid-19 pandemic of 2020 in terms of volatility and correlation risk among the most significant financial indexes in Europe. More precisely, the dataset employed includes the daily returns of Ftse100, CaC40, Dax30 and FtseMib40, with reference to the two time periods in which the two major crises manifested their effects on the markets. We use two different methodological approaches: the analysis of the daily conditional variance using various families of GARCH models and the study of the weekly realized volatility using HAR models. Furthermore, the estimation of the dependence structure of the GARCH residuals using copula functions is performed.

Dividing a Farm: A Simple Application of Game Theory in Geometry

Two farmers decide to divide a triangle farm between them. The problem is modeled as a simple application of game theory in geometry. The game is defined and is solved, theoretically, in independent case. Simulation results are proposed using the copula function, in the dependent cases. Finally, concluding remarks are proposed.

Ubiquitous Increases in Streamflow and Flooding Magnitude Across the Yellow River Basin

AbstractGlobal human‐induced warming has intensified water circulation in the atmospheric environment and altered the streamflow generation regime. The VIC hydrological model approach for impact assessment of climate change and human activities mainly focuses on variations in streamflow, but ignores other critical flooding characteristics induced by extreme streamflow, especially bivariate flooding characteristics. In this work, the copula functions are employed to structure the flooding risk under the shared socioeconomic pathway (SSP) across the Yellow River basin (YRB). This is based on the multi‐model ensemble (MME) and Delta downscaling outputs (Delta‐MME) of the CMIP6 global climate models (GCMs), as well as the flooding characteristics simulated by VIC hydrological model. Compared to the reference period (1995–2014), Delta‐MME reveals a significant warming and humidifying trend under three SSPs over the YRB. Despite uncertainties originating from climate variables and hydrological model, multiple findings underscore the substantial influence of climate change on the flooding generation regime in YRB. This includes: (a) an increase in the streamflow under all SSPs; (b) a larger flooding peak (Q) and volume (W) under SSP585, with Q and W at the Huayuankou hydrologic station (HYK) increasing by 52.7% and 44.8%, respectively; (c) an advancement in the bivariate flooding risk, particularly in SSP585 where flooding co‐occurrence return period at HYK may be more than 50 times earlier. This study underscores that the urgent need to enhance social resilience to climate change in the YRB.

Comparative Study of Low Flow Frequency Analysis Using Bivariate Copula Model at Soyanggang Dam and Chungju Dam

A univariate analysis that relies solely on precipitation data in low flow frequency analysis is a technique to express meteorological drought, so it is limited to analyzing the characteristics of hydrological drought related to available water resources. In addition, if the data for the model calibration are insufficient, the uncertainty of a single variable limits the construction of a reliable model. To improve this problem, a frequency analysis was performed by constructing a bivariate copula model as a multivariate model with a high correlation between variables targeting reservoir inflows. The methodology utilizes the theory of runs to identify low flow events, establishing a threshold based on the mandatory regional water supply plan, and determining the low flow duration and cumulative water deficit. The Gumbel copula function, effective in capturing correlations between hydrological variables, was applied to derive a joint bivariate probability distribution, facilitating the calculation of combined low flow event return periods. This study compared low flow frequencies at Soyanggang dam (’74–’22) and Chungju dam (’86–‘22), which are in the same Han River basin but have different capacities and water demands, using a bivariate copula model. The top four extreme low flow events for the two adjacent dam basins did not occur in the same year and, in the years of the extreme low flow events at one of the two dam basins, there was an insignificant magnitude at the remaining dam basin. This result is noteworthy because it shows that the possibility of extreme low flow events appearing simultaneously in both watersheds is not as high as expected. The operational efficiency can be improved by setting the coordinated operation rules of the two reservoirs using the copula dependency structure.

A statistical analysis method for significant wave height and spectral peak frequency considering the random and time-varying effects based on copula function and Bayesian inference

Random and time-varying effects are important factors for statistical analysis of wave characteristic variables, including the significant wave height and spectral peak frequency. This paper proposes a statistical analysis method for the accurate statistical analysis of the state of the ocean. Several common distributions are applied as candidates for describing a specific variable, denoted as the marginal distribution. The joint distribution for the wave characteristic variables is constructed using copula functions based on the marginal distributions. The probability and unknown parameters of the marginal distributions are then determined by fully Bayesian inference. The best-fitting marginal distribution is selected based on the posterior probabilities of the candidates. Then, unknown parameters of the candidate copula functions are estimated by maximum likelihood estimation. The best-fitting copula function is selected based on Akaike information criterion, root mean squared error and Nash Sutcliffe efficiency. The proposed method is verified using the National Data Buoy Center dataset for 2019. However, this dataset, collected from a network of almost 100 moored buoys and Coastal-Marine Automated Network (CMAN) stations, contains incomplete data. The results reveal that the best-fitting marginal distribution and copula function may vary with the month. The average and maximum values of the improved RMSE using the proposed method are only 0.0064 and 0.0187, respectively. This indicates the high accuracy of the proposed method for the statistical analysis of wave states even though missing some data.

Estimate rainfall return period based on maximum pipe flow: A new framework to address mutual relationship

The unreasonable assumption that the rainfall return period is similar to the runoff return period may lead to unreasonable drainage system design. Hence, this study proposes a computational framework to calculate the rainfall return period according to the runoff return period. Based on the Akaike Information Criterion (AIC) and the optimal probability distribution functions, a novel approach was proposed to combine the rainfall and maximum pipe flow return periods. Copula functions were employed to construct a joint probability distribution for calculating the rainfall return period using the maximum pipe flow return period. Thus, this study reached the following results. (I) The Gamma, Logistic, and Weibull distributions provided better descriptions for the rainfall and maximum pipe flow probability distributions. The Gaussian, Gumbel, and Survival Clayton Copula functions were shown to be the optimal choices. (II) A calculation method for the rainfall return period was constructed based on the Satisfaction Percentage (SP). For instance, using a 2-year as the maximum pipe flow return period, the rainfall return period was determined to be 4.705 years (60 min of rainfall duration, Maluan community) with an SP value of 0.95. (III) The study area showed non-significant impacts (p < 0.05) in selecting the rainfall return periods. Therefore, simulations can be conducted in regions with missing pipe network data using similar underlying surface conditions from other study areas.

Some Results on Bivariate Squared Maximum Sharpe Ratio

The Sharpe ratio is a widely used tool for assessing investment strategy performance. An essential part of investing involves creating an appropriate portfolio by determining the optimal weights for desired assets. Before constructing a portfolio, selecting a set of investment opportunities is crucial. In the absence of a risk-free asset, investment opportunities can be identified based on the Sharpe ratios of risky assets and their correlation. The maximum squared Sharpe ratio serves as a useful metric that summarizes the performance of an investment opportunity in a single value, considering the Sharpe ratios of assets and their correlation coefficients. However, the assumption of a normal distribution in asset returns, as implied by the Sharpe ratio and related metrics, may not always hold in practice. Non-normal returns with a non-linear dependence structure can result in an overestimation or underestimation of these metrics. Copula functions are commonly utilized to address non-normal dependence structures. This study examines the impact of asset dependence on the squared maximum Sharpe ratio using copulas and proposes a copula-based approach to tackle the estimation issue. The performance of the proposed estimator is illustrated through simulation and real-data analysis.

Multiple hypothesis testing in cognitive IoT sensor network

Recent Internet of Things (IoT) research aims to develop generic objects to learn, reason, and perceive their environment. Therefore, a new area has emerged known as cognitive IoT (CIoT). The cognitive Internet of Things integrates IoT with intelligence and behaves as well as humans through intelligent functionality. Several inferential tasks in CIoT require multiple hypothesis testing. The situation becomes cumbersome when the data is massive and heterogeneous. Thus, this research suggests a novel technique for multiple-hypothesis testing that uses a copula function to deal effectively with massive heterogeneous data. In addition, these data may contain missing or corrupted entries. Hence, it introduced probabilistic clustering, which reduces model inefficiency and takes control over the false discovery rate (FDR). Most of the variance from each cluster was extracted using kernel principal component analysis (KPCA) to reduce the processing burden at the fusion centre. Subsequently, it computes the p-value of each cluster's first principal component data and employs the Bonferroni method for multiple hypothesis testing. Finally, this research study evaluates the performance of the proposed algorithm on six-month environmental data, revealing that the proposed technique is efficient in terms of accuracy and computation time compared to other methods in the presence of massive heterogeneous data.

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.