Copula joint function and its application in probability seismic hazard analysis

A mature mathematical technique called copula joint function is introduced in this paper, which is commonly used in the financial risk analysis to estimate uncertainty. The joint function is generalized to the n-dimensional Frank’s copula. In addition, we adopt two attenuation models proposed by YU and Boore et al, respectively, and construct a two-dimensional copula joint probabilistic function as an example to illustrate the uncertainty treatment at low probability. The results show that copula joint function gives us a better prediction of peak ground motion than that resultant from the simple linear weight technique which is commonly used in the traditional logic-tree treatment of model uncertainties. In light of widespread application in the risk analysis from financial investment to insurance assessment, we believe that the copula-based technique will have a potential application in the seismic hazard analysis.

Asymptotic Confidence Bands for Copulas Based on the Local Linear Kernel Estimator

In this paper, we establish asymptotically optimal simultaneous confidence bands for the copula function based on the local linear kernel estimator proposed by Chen and Huang [1]. For this, we prove under smoothness conditions on the derivatives of the copula a uniform in bandwidth law of the iterated logarithm for the maximal deviation of this estimator from its expectation. We also show that the bias term converges uniformly to zero with a precise rate. The performance of these bands is illustrated by a simulation study. An application based on pseudo-panel data is also provided for modeling the dependence structure of Senegalese households’ expense data in 2001 and 2006.

Responses of vegetation yield to precipitation and reference evapotranspiration in a desert steppe in Inner Mongolia,China

Drought,which restricts the sustainable development of agriculture,ecological health,and social economy,is affected by a variety of factors.It is widely accepted that a single variable cannot fully reflect the characteristics of drought events.Studying precipitation,reference evapotranspiration(ET_(0)),and vegetation yield can derive information to help conserve water resources in grassland ecosystems in arid and semi-arid regions.In this study,the interactions of precipitation,ET_(0),and vegetation yield in Darhan Muminggan Joint Banner(DMJB),a desert steppe in Inner Mongolia Autonomous Region,China were explored using two-dimensional(2D)and three-dimensional(3D)joint distribution models.Three types of Copula functions were applied to quantitatively analyze the joint distribution probability of different combinations of precipitation,ET_(0),and vegetation yield.For the precipitation–ET_(0)dry–wet type,the 2D joint distribution probability with precipitation≤245.69 mm/a or ET_(0)≥959.20 mm/a in DMJB was approximately 0.60,while the joint distribution probability with precipitation≤245.69 mm/a and ET_(0)≥959.20 mm/a was approximately 0.20.Correspondingly,the joint return period that at least one of the two events(precipitation was dry or ET_(0)was wet)occurred was 2 a,and the co-occurrence return period that both events(precipitation was dry and ET_(0)was wet)occurred was 5 a.Under this condition,the interval between dry and wet events would be short,the water supply and demand were unbalanced,and the water demand of vegetation would not be met.In addition,when precipitation remained stable and ET_(0)increased,the 3D joint distribution probability that vegetation yield would decrease due to water shortage in the precipitation–ET_(0)dry–wet years could reach up to 0.60–0.70.In future work,irrigation activities and water allocation criteria need to be implemented to increase vegetation yield and the safety of water resources in the desert steppe of Inner Mongolia.

Propagation characteristics from meteorological drought to agricultural drought over the Heihe River Basin, Northwest China

In the context of global warming,drought events occur frequently.In order to better understanding the process and mechanism of drought occurrence and evolution,scholars have dedicated much attention on drought propagation,mainly focusing on drought propagation time and propagation probability.However,there are relatively few studies on the sensitivities of drought propagation to seasons and drought levels.Therefore,we took the Heihe River Basin(HRB)of Northwest China as the case study area to quantify the propagation time and propagation probability from meteorological drought to agricultural drought during the period of 1981–2020,and subsequently explore their sensitivities to seasons(irrigation and non-irrigation seasons)and drought levels.The correlation coefficient method and Copula-based interval conditional probability model were employed to determine the drought propagation time and propagation probability.The results determined the average drought propagation time as 8 months in the whole basin,which was reduced by 2 months(i.e.,6 months)on average during the irrigation season and prolonged by 2 months(i.e.,10 months)during the non-irrigation season.Propagation probability was sensitive to both seasons and drought levels,and the sensitivities had noticeable spatial differences in the whole basin.The propagation probability of agricultural drought at different levels generally increased with the meteorological drought levels for the upstream,midstream,and southern downstream regions of the HRB.Lesser agricultural droughts were more likely to be triggered during the irrigation season,while severer agricultural droughts were occurred mostly during the non-irrigation season.The research results are helpful to understand the characteristics of drought propagation and provide a scientific basis for the prevention and control of droughts.This study is of great significance for the rational planning of local water resources and maintaining good ecological environment in the HRB.

Joint probability distribution of winds and waves from wave simulation of 20 years (1989-2008) in Bohai Bay

The joint probability distribution of wind speed and significant wave height in the Bohai Bay was investigated by comparing the Gurnbel logistic model, the Gumbel-Hougaard （GH） copula function, and the Clayton copula function. Twenty years of wind data from 1989 to 2008 were collected from the European Centre for Medium-Range Weather Forecasts （ECMWF） database and the blended wind data of the Quick Scatterometer （QSCAT） satellite data set and re-analysis data from the United States National Centers for Environmental Prediction （NCEP）. Several typhoons were taken into account and merged with the background wind fields from the ECMWF or QSCAT/NCEP database. The 20-year data of significant wave height were calculated with the unstructured-grid version of the third-generation wind wave model Simulating WAves Nearshore （SWAN） under extreme wind process conditions. The Gumbel distribution was used for univariate and marginal distributions. The distribution parameters were estimated with the method of L-moments. Based on the marginal distributions, the joint probability distributions, the associated return periods, and the conditional probability distributions were obtained. The GH copula function was found to be optimal according to the ordinary least squares （OLS） test. The results show that wind waves are the prevailing type of wave in the Bohai Bay.

A new early warning method for dam displacement behavior based on non-normal distribution function

Traditional methods for early warning of dam displacements usually assume that residual displacements follow a normal distribution.This assumption deviates from the reality,thereby affecting the reliability of early warning results and leading to misjudgments of dam displacement behavior.To solve this problem,this study proposed an early warning method using a non-normal distribution function.A new early warning index was developed using cumulative distribution function(CDF)values.The method of kernel density estimation was used to calculate the CDF values of residual displacements at a single point.The copula function was used to compute the CDF values of residual displacements at multiple points.Numerical results showed that,with residual displacements in a non-normal distribution,the early warning method proposed in this study accurately reflected the dam displacement behavior and effectively reduced the frequency of false alarms.This method is expected to aid in the safe operation of dams.

Dynamic Fatigue Reliability Analysis of Transmission Gear Considering Failure Dependence

Multiple failuremodes and strength degradation are usually inherent in the gear transmission system,which brings new challenges for conducting fatigue reliability analysis and design.This paper proposes a novel dynamic fatigue reliability analysis method for failure dependence and strength degradation based on the combination of theCopula function and Gamma process.Firstly,the dynamic simulation model of the gear transmission system is established to obtain the dynamic stress-time history.The Gamma process is then used to describe the strength degradation to establish the dynamic stress-strength interference model.The marginal distribution functions of tooth contact fatigue and dedendumbending fatigue are calculated respectively based on the dynamic interferencemodel.Finally,the joint distribution of the two failure modes can be obtained by the t-Copula function to characterize the failure dependence,and so the dynamic fatigue reliability considering failure dependence can be estimated.The effectiveness of the proposed method is illustrated with examples.The results reveal the temporal law of reliability and the effects of failure dependence on dynamic fatigue reliability.

Probability Distribution Characteristics of Extreme Rainfall in Beijiang River Basin

7 kinds of probability distribution functions were used to fit extreme rainfall indexes R1d（ the maximum 1-d rainfall in the year） and R5d（ the maximum continuous 5-d rainfall in the year） at 18 observation stations of Beijiang River basin,and linear moment method was used to estimate parameters. According to fitting goodness test,the best probability distribution function was determined. On this basis,spatial analysis of design values of R1d and R5d with 50-a and 100-a reappearance periods was conducted. Via further selection,GH Copula was taken as connection function,and R1d-R5d joint probability distribution in the basin was studied. The results showed that R1d or R5d probability with 50-a and 100-a reappearance periods was larger in Wengyuan and Qingyuan,while R1d and R5d co-occurrence probability with 50-a and 100-a reappearance periods was larger in central north region.

Wind farm polymerization influences on security and economic operation in power system based on Copula function

Wind speed dependences on different areas in a wind farm have influences on security and economic operation in power system.In order to simulate the correlation of wind speed series between different positions,this paper applies Copula function and rank correlation matrix methods to measure the coherence of wind speed in a wind farm.The correlated wind sample space is established.According to active power output characteristics of wind turbines,the polymerization model in a wind farm can be achieved.Monte Carlo optimal power flow is applied to IEEE-30 and IEEE-300 bus systems based on the principle of energy saving dispatching.The study shows that the accuracy of outputs is improved,thus reducing the fluctuation ranges in unit generating costs and power flow in branches while considering wind speed polymerization.This approach provides a new method to improve the effectiveness of energy saving dispatching and system operation arrangement.Results have been tested to be effective.

Reliability Copula Model for Wind Turbine Gearbox Based on Failure Correlation

On the basis of each gear's failure correlation, the reliability Copula model of a wind turbine gearbox is established and a 1.5 MW wind turbine gearbox is taken as the research object. Firstly, based on the dynamic reliability model of mechanical parts, each gear's life distribution function of a wind turbine gearbox is obtained.The life distribution function can be used as the marginal distributions of the system's joint distribution. Secondly,Copula function is introduced to describe the failure correlation between parts, and the appropriate Copula function is selected according to the shape characters of Copula probability density function. Finally, the wind turbine gearbox system is divided into three parts according to the failure correlation of each gear. The Sklar theorem and the thought of step by step analysis are used to obtain the reliability Copula model for a wind turbine gearbox based on failure correlation.

Analysing taxi customer-search behaviour using Copula-based joint model

The drivers of vacant taxis tend to cruise the road network searching new passengers,which leads to additional traffic congestion,air pollution and other problems.This study introduces a Copula-based joint model to analyse destination selection and route choice behaviour in the customer-search process.A multinomial logit model is used to analyse the destination selection behaviour,and a path size logit model is used to explore the routes choice behaviour.Accordingly,the joint model applied Copula function is then established to analyse the correlation between these two behaviours.The destination customer generation rate,destination distance,route customer generation rate,path travel time,cumulative intersection delay,path size and route length are selected as explanatory variables.The taxis trajectory data were collected by global positioning system in Xidan District of Beijing City from September 2014 to February 2015.Using the log-likelihood,Bayesian information criterion as evaluation indexes to measure the fitting result,the joint model applied Copula function has the highest goodness-of-fit.The effect of explanatory variables on customer search behaviour is discussed based on the parameter estimation results.The results of this study are helpful to understand the customer-search behaviour of taxi drivers to reduce operating costs and improve the efficiency of the taxi operation system.

A Fully Pipelined Probability Density Function Engine for Gaussian Copula Model

The Gaussian Copula Probability Density Function （PDF） plays an important role in the fields of finance, hydrological modeling, biomedical study, and texture retrieval. However, the existing schemes for evaluating the Gaussian Copula PDF are all computationally-demanding and generally the most time-consuming part in the corresponding applications. In this paper, we propose an FPGA-based design to accelerate the computation of the Gaussian Copula PDF. Specifically, the evaluation of the Gaussian Copula PDF is mapped into a fully-pipelined FPGA dataflow engine by using three optimization steps： transforming the calculation pattern, eliminating constant computations from hardware logic, and extending calculations to multiple pipelines. In the experiments on 10 typical large-scale data sets, our FPGA-based solution shows a maximum of 1870 times speedup over a well-tuned single- core CPU-based solution, and 610 times speedup over a well-optimized parallel quad-core CPU-based solution when processing two-dimensional data.

Dependence structure between LIBOR rates by copula method

This paper discusses the correlation structure between London Interbank Offered Rates （LIBOR） by using the copula function. We start from one simplified model of A. Brace, D. Gatarek, and M. Musiela （1997） and find out that the copula function between two LIBOR rates can be expressed as a sum of an infinite series, where the main term is a distribution function with Gaussian copula. Partial differential equation method is used for deriving the copula expansion. Numerical results show that the copula of the LIBOR rates and Gaussian copula are very close in the central region and differ in the tail, and the Gaussian copula approximation to the copula function between the LIBOR rates provides satisfying results in the normal situation.

Economic Policy Uncertainty:Cross-Country Linkages and Spillover Effects on Economic Development in Some Belt and Road Countries

With the increase of economic environment uncertainty,it is of great importance to study the linkage and spillover effects of economic policy uncertainty among countries.Especially,this article selects eight countries along the Belt and Road as the core countries(China,Korea,Croatia,India,Russia,Greece,Pakistan,and Singapore)and four countries(Germany,France,Japan,and UK)as the peripheral countries,and then copula technique and mixed-frequency global vector autoregressive model are employed to analyze the correlation and the spillover effect of the economic policy uncertainty(EPU)for the twelve selected countries,respectively.The proposed empirical findings show clearly that the EPU correlation among the eight core Belt and Road countries is stronger and the spillover effect of the core countries to the peripheral countries is statistically significant.As a result,for harmonious and win-win development,the Belt and Road countries should pay a close attention to the EPU,because the stability of the EPU promotes greatly the economy development.

Perspectives on Compound Flooding in Chinese Estuary Regions

Extreme surges and rainfall represent major driving factors for compound flooding in estuary regions along the Chinese coast.The combined effect of extreme surges and rainfall(that is,compound floods)might lead to greater impacts than if the drivers occurred in isolation.Hence,understanding the frequency and severity of compound flooding is important for improving flood hazard assessment and compound flood resilience in coastal cities.In this study,we examined the dependence between extreme surges and corresponding rainfall events in 26 catchments along the Chinese coastline during typhoon and non-typhoon seasons using copula functions,to identify where the two drivers more often occur together and the implication for flood management in these locations.We found that the interaction between flood drivers is statistically significant in 10catchments located around Hainan Island(south)and Shanghai,where surge peaks occur mainly during the typhoon season and around the Bohai Sea(north),where surge peaks occur mainly during the non-typhoon season.We further applied the copula-based framework to model the dependence between surge peaks and associated rainfall and estimate their joint and conditional probability in two specific locations—Hainan Island and the Bohai Sea,where the correlation between flood drivers is statistically significant.We observed that in Hainan Island where most of the surge peaks occur during the typhoon season,extreme rainfall events during the typhoon season are generally more intense compared to annual maxima rainfall.In contrast,around the Bohai Sea where surge peaks occur mainly outside the typhoon season,rainfall is less intense than annual maxima rainfall.These results show that the interaction between extreme surges and rainfall can provide valuable insight when designing coastal and urban infrastructure,especially in highly populated urban areas prone to both coastal and pluvial flooding,such as many Chinese coastal cities.

Reliability estimation of rotary lip seal in aircraft utility system based on time-varying dependence degradation model and its experimental validation

With several attractive properties, rotary lip seals are widely used in aircraft utility system, and their reliability estimation has drawn more and more attention. This work proposes a reliability estimation approach based on time-varying dependence analysis. The dependence between the two performance indicators of rotary lip seals, namely leakage rate and friction torque, is modeled by time-varying copula function with polynomial to denote the time-varying parameters, and an efficient copula selection approach is utilized to select the optimal copula function. Parameter estimation is carried out based on a Bayesian method and the reliability during the whole lifetime is calculated based on a Monte Carlo method. Degradation test for rotary lip seal is conducted and the proposed model is validated by test data. The optimal copula function and optimal order of polynomial are determined based on test data. Results show that this model is effective in estimating the reliability of rotary lip seals and can achieve a better goodness of fit.

Semi-analytical Formula for Pricing Bilateral Counterparty Risk of CDS with Correlated Credit Risks

Based on the framework of [7], we discuss pricing bilateral counterparty risk of CDS, where each individual default intensity is modeled by a shifted CIR process with jump （3CIR＋＋）, and the correlation between the default times is modeled by a copula function. We present a semi-analytical formula for pricing bilateral counterparty risk of CDS, which is more convenient to compute through calculating multiple numerical integration or using Monte-Carlo simulation without simulating default times. Moreover, we obtain simpler formulae under FGM copulas, Bernstein copulas and CA＇B copulas, which can be applied for speeding up the computation and reducing the pricing error. Numerical results under FGM copulas and CA＇B copulas show that our method performs better both in computation speed and accuracy.

Amplification of Flood Risks by the Compound Effects of Precipitation and Storm Tides Under the Nonstationary Scenario in the Coastal City of Haikou,China

In the context of climate change,coastal cities are at increased risk of extreme precipitation and sea level rise,and their interaction will aggravate coastal floods.Understanding the potential change of compound floods is valuable for flood risk reduction.In this study,an integrated approach coupling the hydrological model and copulabased design of precipitation and storm tides was proposed to assess the compound flood risk in a coastal city—Haikou,China.The copula model,most-likely weight function,and varying parameter distribution were used to obtain the combined design values of precipitation and storm tides under the nonstationary scenario,which were applied to the boundary conditions of the 1D-2D hydrological model.Subsequently,the change of the bivariate return periods,design values,and compound flood risks of precipitation and storm tides were investigated.The results show that the bivariate return period of precipitation and storm tides was reduced by an average of 34%under the nonstationary scenario.The maximum inundation areas and volumes were increased by an average of 31.1%and 45.9%respectively in comparison with the stationary scenario.Furthermore,we identified that the compound effects of precipitation and storm tides would have a greater influence on the flood risk when the bivariate return period is more than 50 years,and the peak time lag had a significant influence on the compound flood risk.The proposed framework is effective in the evaluation and prediction of flood risk in coastal cities,and the results provide some guidance for urban disaster prevention and mitigation.