Gumbel-Hougaard Copula Research Articles

Comprehensive evaluation of the response relationship between meteorological drought and hydrological drought in the Yalong River Basin, China

Extreme climate events are becoming increasingly severe owing to global warming. In recent years, the Yalong River has experienced frequent droughts, which has seriously hindered the development of hydropower. Based on standardized precipitation evapotranspiration index (SPEI) and standardized runoff index (SRI) values, this study employed transfer entropy, multivariate Copula functions, and a cloud model to study the response relationship and risk situation between meteorological and hydrological droughts. The results showed that: (1) The smaller the scale of SPEI and SRI, the more sensitive the identification of drought and the faster the detection of drought trend changes. (2) Combined with small-scale drought index analyses, the trend and severity of meteorological and hydrological droughts propagating from the upper and middle reaches to the lower reaches is increasing from year to year. (3) The Gumbel-Hougaard Copula function reveals the relationship between meteorological and hydrological drought, with the longest joint return period of 13–15 months. (4) In cloud model risk evaluation, the degree, uncertainty and stochasticity of drought risk in the basin are increasing, and the risk degree of interannual meteorological and seasonal hydrological droughts in the basin was high. This study provides theoretical support for optimizing water resource allocation and drought risk situation analysis in the basin.

Analyzing the Reliability of a Pumping System through the Application of Copula Analysis

This paper derives and evaluates the reliability measures for a model pumping system composed of six subsystems linked in series: subsystems 1, 2, and 6, each comprising one unit, while subsystems 3 and 5 each contain three units operating in parallel. Subsystem 4 contains three units effectively operating in series. Unit failure rates are assumed to be constant, while repair rates are modeled by either a general distribution or, in the event of system failure, a Gumbel–Hougaard copula. The Laplace transform and additional supplementary variable approaches are employed to solve the system. The conventional measures of reliability are computed for a range of parameter values and as functions of time. In addition, tables are presented containing parameters for different cases, accompanied by a discussion of how they were chosen and their impact on the results.

Bayes analysis of one-shot device testing data with correlated failure modes using copula models

Copula models are capable of modeling the dependence structure among the random variables, a phenomenon that is often required in the statistical analysis. Such models are the flexible substitutes of multivariate distributions because they model both the marginal distributions and the joint dependence structure distinctly. Because of such important features, the models are recognized as popular tools in a variety of situations including reliability engineering and survival analysis. The present paper studies a Bayesian approach using three Archimedean copulas, namely, the Gumbel Hougaard copula, the Frank copula and the Joe copula for analyzing one-shot device testing data with two correlated failure modes collected from a constant stress accelerated life test. One-shot devices are units that can be used only once and destroyed immediately after the use. Obviously, one obtains either left or right censored data on the failure times instead of actual failure times of the devices. Finally, all the considered copula models are compared using the Bayesian model selection tools. A real dataset is analyzed as an illustration of the proposed Bayesian methodology.

Copula-Based Joint Flood Frequency Analysis: The Case of Guder River, Upper Blue Nile Basin, Ethiopia

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.

Convexity of linear joint chance constrained optimization with elliptically distributed dependent rows

In this paper, we study the convexity of the linear joint chance constraints. We assume that the constraint row vectors are elliptically distributed. Further, the dependence of the rows is modeled by a family of Archimedean copulas, namely, the Gumbel–Hougaard copulas. Under mild assumptions, we prove the eventual convexity of the feasibility set.

A neurodynamic algorithm for dependent joint chance constrained geometric programs

This research studies the use of copula theory to model dependencies in joint probabilistic constrained geometric programs with dependent rows. The row vectors are assumed to follow an elliptical distribution, and their dependencies are modeled through a Gumbel–Hougaard copula. We use a log transformation to convert the chance-constrained geometric program into a deterministic optimization problem. Then we solve the resulting deterministic program using a dynamical neural network. The stability and convergence of the proposed neural network approach are demonstrated. The primary characteristic of our framework is its ability to solve the dependent joint chance constrained geometric programs without resorting to any convex approximation methods. This feature sets our approach apart from the current state-of-the-art solving techniques. The neurodynamic algorithm is finally applied to solve three geometric optimization problems.

A New Agricultural Drought Disaster Risk Assessment Framework: Coupled a Copula Function to Select Return Periods and the Jensen Model to Calculate Yield Loss

China is one of the regions with the most frequent drought disasters and serious social and economic losses. Agricultural drought is the most serious natural disaster. Due to climate change, the regional agricultural drought risk assessment has always been the focus of the academic circle. This study takes Zunyi City as an example, which is the most typical city of karst landform development. The monthly precipitation data set of ground meteorological observation stations in Zunyi City from 1956 to 2020 was selected, and the drought characteristic variables were extracted by the coupled use of the precipitation anomaly percentage (Pa) index and the theory of runs. A copula function was applied to establish the joint distribution model of characteristic variables, obtaining the drought frequency and drought return periods. Combined with the Jensen model, the agricultural drought loss rate under different drought return periods in the target year (2020) was calculated and evaluated. The results showed that the Gumbel-Hougaard copula function was suitable for the joint distribution of drought joint variables in Zunyi City. From 1956 to 2020, fewer droughts occurred in Zhengan and Wuchuan, and the most droughts took place in Fenggang, Meitan, and Yuqing. The average drought duration in each county was about 1.5 months, and the average drought severity was about 0.35 in spatial distribution. Crop loss rate caused by drought increased and the affected area expanded with the increase of drought return periods (5, 10, 20, 50, and 100 years) in temporal distribution. Meanwhile, the drought disaster was most drastic in the eastern region, followed by the south, north, west, and central area. The results were highly consistent with the historical drought in Zunyi City, which verified the validity of the model. This study could provide scientific knowledge for drought resistance and reasonable mitigation programing for the security of the regional agricultural production and the sustainability of social and economic development.

Availability modeling of systems incorporating uncertainty & multiple failure modes using critical repair time & copula

This paper investigates the availability of a system incorporating uncertainty in the inspection period, including multiple failure modes (FMs). We decrease or increase the inspection period depending on the occurrence or nonoccurrence of system failure during any particular inspection, i.e. if, during an inspection, a defect is detected, we reduce the inspection period for the next inspection, and if no defect is found, we increase the inspection period. Also, the proposed model is examined under a pre-defined critical repair time incorporating the Gumbel-Hougaard copula. This paper considers a repairable system having α number of FMs. Random failure times are associated with every single FM, and whenever the system collapses from the ath FM, a subsequent corrective repair is conducted, which requires a random amount of time (a = 1, 2,…, α). Then the point and long-run availability of the proposed model are derived, and a sensitivity analysis is done to examine how inspection time affects the system’s availability. Additionally, the system’s average long-run cost rate is analyzed, and the model is employed to determine the optimal inspection period that optimizes the system’s ALRCR. Lastly, the results are verified via a numerical illustration of a power transformer.

Copula-based probability for occurrence of the event due to one of dependent NHPPs before the other, past a truncated time

The bivariate Gumbel-Hougaard copula is considered for two dependent non-homogenous Poisson processes, The purpose of this article is to provide a computational tool for estimation of the probability that the next event arrival after a truncated time τ, is due to a specific process. There are many situations in which non-homogenous Poisson processes could be dependent, such as failure times of a two-component repairable system in which the failure of the system at a given time is due to only one of the components. Power Law is used as the intensity function for both non-homogenous Poisson Processes. Maximum likelihood and Bayes estimators for the probability based on gamma priors are derived. It is verified through simulations that Bayes estimator outperforms maximum likelihood estimator with regard to their accuracy. Two examples for illustration of computation via Mathematica code (in the Appendix) are provided.

Solving linear programs with joint probabilistic constraints with dependent rows using a dynamical neural network

In this paper, we study joint probabilistic constrained linear programs with dependent rows. We take the special case where the dependence between the rows is driven by Gumbel–Hougaard copula. We first transform the problem into a deterministic biconvex optimization problem. Then, we solve the obtained problem using a dynamical neural network based on the partial KKT system. We show the stability and the convergence of the proposed neural network. The main feature of our approach is to solve the joint probabilistic constraints problem without the use of any convex approximations. We finally use a problem of profit maximization to evaluate the performances of our approach.

A Generalized Framework for Risk-Based Extreme Load Analysis in Offshore System Design

Abstract The primary aim of this research is to consider the correlation among environmental factors in calculating 100 and 1000 years of extreme load design criteria. This is done by considering load as energy transferred from external environment to the offshore system. Also, incorporating spatial and temporal dependence of environmental variables in the context of offshore design. A bivariate extreme value distribution and a conditional joint return level function are developed using the Gumbel–Hougaard copula. The offshore design risk criteria are developed for the finer grid locations (0.1 deg × 0.1 deg latitude/longitude grid) considering joint extreme wind and wave energy. The developed approach is tested using data for the Flemish Pass basin off the east coast of Canada. Along with the primary aim, the impact of climate change is investigated (time and space variability) by implementing the proposed methodology in two cases: the periods from 1959 to 1988 and 1989 to 2018. This study observed that climate change has caused 30% less correlation between wind speed and wave height in recent years (1989–2018) compared to the period of 1959–1988. The proposed extreme design wind speed is 39.7 m/s, and significant wave height is 16.4 m; their joint exceeding probability is 5.80 × 10−5 over an annual basis for a scenario of 100-year.

Power and Sample Size Calculations for the Restricted Mean Time Analysis of Prioritized Composite Endpoints

As a new way of reporting treatment effect, the restricted mean time in favor (RMT-IF) of treatment measures the net average time the treated have had a less serious outcome than the untreated over a specified time window. With multiple outcomes of differing severity, this offers a more interpretable and data-efficient alternative to the prototypical restricted mean (event-free) survival time. To facilitate its adoption in actual trials, we develop simple approaches to power and sample size calculations and implement them in user-friendly R programs. In doing so we model the bivariate outcomes of death and a nonfatal event using a Gumbel–Hougaard copula with component-wise proportional hazards structures, under which the RMT-IF estimand is derived in closed form. In a standard set-up for censoring, the variance of the nonparametric effect-size estimator is simplified and computed via a hybrid of numerical and Monte Carlo integrations, allowing us to compute the power and sample size as functions of component-wise hazard ratios. Simulation studies show that these formulas provide accurate approximations in realistic settings. To illustrate our methods, we consider designing a new trial to evaluate treatment effect on the composite outcomes of death and cancer relapse in lymph node-positive breast cancer patients, with baseline parameters calculated from a previous study.

Effect of Dependency on the Estimation of P[Y<X] in Exponential Stress-strength Models

We consider an expression for the probability R=P(Y<X) where the random variables X and Y denote strength and stress, respectively. Our aim is to study the effect of the dependency between X and Y on R. We assume that X and Y follow exponential distributions and their dependency is modeled by a copula with the dependency parameter theta. We obtain a closed-form expression for R for Farlie-Gumbel-Morgenstern (FGM), Ali-Mikhail-Haq (AMH), Gumbel's bivariate exponential copulas and compute R for Gumbel-Hougaard (GH) copula using a Monte-Carlo integration technique. We plot a graph of R versus theta to study the effect of dependency on R. We estimate R by plugging in the estimates of the marginal parameters and theta in its expression. The estimates of the marginal parameters are based on the marginal likelihood. The estimates of theta are obtained from two different methods; one is based on the conditional likelihood and the other on the method of moments using Blomqvist's beta. Asymptotic distribution of both the estimators of R is obtained. For illustration purpose, we apply our results to a real data set.

Multi-model ensemble prediction of pan evaporation based on the Copula Bayesian Model Averaging approach

Pan evaporation (Ep) is an efficient and practical tool for planning and managing water resources, understanding the water balance in hydrological processes, and developing irrigation systems, particularly in regions with limited water resources. Accurate prediction of Ep using easy-to-measure meteorological variables is beneficial for any region. The current study aimed to explore the applicability of Copula-based Bayesian Model Averaging (CBMA) for improving probabilistic Ep predictions in different climates of Iran. To this end, the parameters of the Adaptive Neuro-Fuzzy Interface System (ANFIS) were optimized using four meta-heuristic algorithms of Seagull Optimization Algorithm (SOA), Crow search Algorithm (CA), Firefly Algorithm (FFA), and Particle Swarm Optimization (PSO) for finding global solutions. The ANFIS-SOA, ANFIS-CA, ANFIS-FFA, ANFIS-PSO, and ANFIS models were implemented as inputs for employing ensemble CBMA and Bayesian Model Averaging (BMA) methods. Daily meteorological variables of average air temperature (Ta), sunshine hours (SH), relative humidity (RH), wind speed (WS), and Ep from six stations from 2000 to 2003 were applied. Evaluation of five Posterior Distribution Functions (PDFs) and three copula functions showed that Gumbel marginal distribution and Gumbel–Hougaard copula function provide the smallest Kolmogorov–Smirnov statistic indicator at the 5% significance level for all predictive models. The results established that the ensemble CBMA approach exhibited the highest prediction accuracy in all climates, followed by the BMA model, superior to the other individual models. The average root mean square error (RMSE) of the ensemble CBMA model was lower than BMA, ANFIS-SOA, ANFIS-CA, ANFIS-PSO, ANFIS-FFA, ANFIS, by 20.35%, 43.19%, 51.28%, 56.74%, 61.15%, and 64.36%, respectively. Furthermore, the uncertainty analysis indicated the Ep predictions were more confident after applying CBMA. In conclusion, we highly recommend applying the ensemble CBMA model to improve Ep’s prediction performance by standalone and hybrid machine learning models. The combination of Ta-SH was suggested to be used for robust predicting daily Ep in the regions of hot desert, cold desert, and cold semi-arid and Ta-RH in the humid subtropical area regarding the convenience of data acquisition.

Drought identification based on Palmer drought severity index and return period analysis of drought characteristics in Huaibei Plain China

The threshold level method for drought identification is challenging due to the problems of selection of drought index reflecting the drought process associated with water supply and demand as well as the underlying physical meaning of drought thresholds. The frequently used hydrological drought indices (e.g., runoff) are susceptible to being affected by human activities, and drought characteristics are incapable of revealing spatial and temporal comparability. Furthermore, the drought process with the same severity but a longer duration is more likely to be evaluated as a more severe event, which contradicts the actual drought situation. In this study, the Palmer drought severity index (PDSI) method, in which the meteorological factors less influenced by human activities were taken as the input, was adopted to determine the dry/wet states and the PDSI value at each period firstly. The dry/wet states were defined with dry period, wet period, transition period, transition period in dry spell, and transition period in a wet spell. Following that, drought identification criteria were established through the dry/wet states and PDSI value according to the consistency of the identified results and the actual drought situations. Particularly, drought severity and peak intensity were taken as drought characteristics in this paper, and the joint return periods of the characteristics were estimated based on the Gumbel-Hougaard copula function. And eventually, a case study was conducted in Huaibei Plain, China. The results showed that the most severe droughts identified by PDSI had a good consistency with the actual drought situations, drought severity and peak intensity were applicable to reflect the drought impacts. It is worth noting that the implications of the joint return period and the relationships among different types of them. The occurrence probability of a multi-characteristic drought event should be calculated by the integration of joint probability density function over the region corresponding to the event of interest, and the joint frequency of drought characteristics should not be used as the occurrence probability of the drought disaster losses greater (or less) than that caused by the drought with the same characteristics. In addition, drought processes identified by PDSI and standardized precipitation indices (SPI) from monthly and seasonal scales were compared, indicating the drought identified results through PDSI are almost consistent with the actual situations.

How many are enough?: Investigating the effectiveness of multiple conflict indicators for crash frequency-by-severity estimation by automated traffic conflict analysis

Traffic conflict techniques are a viable alternative to crash-based safety assessments and are particularly well suited to evaluating emerging technologies such as connected and automated vehicles for which crash data are sparsely available. Recently, the use of multiple traffic conflict indicators has become common in methodological studies, yet it is often difficult to determine which conflict indicators are appropriate given the application context, and the net benefit, in terms of improved crash prediction accuracy, of considering additional conflict indicators. Addressing these concerns, this study investigates the potential benefits of multiple conflict indicators for conflict-based crash estimation models by using a multivariate extreme value modeling framework (with Gumbel-Hougaard copulas) to estimate crash frequency by severity. The selected conflict indicators include Modified Time-To-Collision (MTTC), Deceleration Rate to Avoid a Collision (DRAC), Proportion of Stopping Distance (PSD) and expected post-collision change in velocity (Delta-V). The proposed framework was applied to estimate the total, severe (Maximum Abbreviated Injury Scale ≥ 3; MAIS3+), and non-severe (MAIS < 3) rear-end crash frequencies at three four-legged signalized intersections in Brisbane, Australia. Rear-end traffic conflicts were extracted from video data using state-of-the-art Computer Vision analytics. Results show that the prediction performance improvements are not necessarily proportional to the number of conflict indicators used in extreme value models. MTTC and DRAC, combined with the severity indicator Delta-V, were the most suitable predictors of rear-end crashes at signalized intersections. Results suggest that instead of adding more and more conflict indicators, careful selection of compatible conflict indicators (considering their functional differences and empirical correlations) is the best way to enhance the predictive performance of conflict-based models.

Reliability and Performance Analysis of a Series-Parallel System Using Gumbel–Hougaard Family Copula

This research looks into the reliability metrics that are used to assess the strength of a solar system’s serial system, which is made up of four subsystems. Each subsystem consists of two parallel active components: two out of two photo-voltaic panels, one of two charge controllers, two of two batteries, and one of two inverters. Both the charge controller and the inverter have two human operators or switches. The Gumbel–Hougaard copula family was used to produce formulations of system dependability metrics such as reliability, mean time to failure (MTTF), availability, and profit function. Numerical examples are presented to show the obtained results and to investigate the impact of various system characteristics. The new study might help homes overcome some of the problems experienced by electric generation systems operating in hostile locations or under adverse weather conditions. A new model was developed, and solar photovoltaic system’s subsystems were analyzed in order to identify the most essential component. It was also indicated how to improve the system.

Gumbel–Hougaard copula-based tetravariate flood frequency analysis for the Hirakud reservoir catchment

ABSTRACT The floods are averted depending upon the effective operations of gate keeping downstream conditions in mind. In this regard inflow hydrographs have a major role to play. The tetravariate flood frequency analysis has been done in this study by taking four variables of a hydrograph i.e. peak (Qp), volume (V), duration (D) and time to peak (Tp). The variable time to peak has given more emphasis in this study, as it characterizes the inflow more effectively and its severity gives less time for operation of dam. In this study, the dependence parameters () of the proposed copula model are determined, finally the values of best fit marginal for all the four variables and dependence parameters () are applied in four-dimensional asymmetric Gumbel–Hougaard copula. The main advantage of using this model is that, it relaxes the restriction of using a similar type of marginal distributions for all the four basic variables. The three hourly inflow data of Hirakud reservoir has been taken to fit the proposed model. Finally, this model is validated using the observed tetravariate probability plotting position model. The result of the copula model is in better agreement with the observed tetravariate probability, ultimately using this copula model, conditional probability is determined.

Residual lifetime prediction for sequential k-out-of-n systems with dependent component lifetimes

This paper deals with residual lifetimes of sequential k-out-of-n systems when component lifetimes are dependent. Given the first j component failure times, an unbiased predictor for the (j + i)-th component failure time is derived. Maximum likelihood prediction of future component failure times is also studied. Estimation of the nuisance parameters is investigated under the conditional proportional hazard rates model with the Exponential baseline distribution and the Gumbel-Hougaard copula. Illustrative examples are also given.

The performance of the copulas in estimating the joint probability of extreme waves and surges along east coasts of the mainland China

In designing coastal and nearshore structures, the joint probability of the wave heights and storm surges is essential in determining the possible highest total water level. The key elements to accurately estimate the joint probability are the appropriate sampling of the extreme values and selection of probability functions for the analysis. This study is to provide a full assessment of the performance of the different methods employed in the joint probability analysis. The bivariate extreme wave height and surge samples are analysed using 2 different probability distributions and the performance of 4 copulas, namely: Gumbel–Hougaard copula, Clayton copula, Frank copula and Galambos copula, is assessed. The possible highest total water levels for 100-year return period along the coastline of the mainland China are estimated by the joint probability method with the Gumbel–Hougaard copula. The results show that the wave heights and surges are highly correlated in the areas of dense typhoon paths. The distributions of the possible highest total water levels show a higher value in the southeast coast and lower value in the north. The results also indicate that at the locations where the sea states are energetic, the joint probability approach can improve the accuracy of design.