Bivariate Copulas Research Articles

Nonparametric Approach to Copula Estimation in Compounding The Joint Impact of Storm Surge and Rainfall Events in Coastal Flood Analysis

The joint probability modelling of storm surges and rainfall events is the main task in assessing compound flood risk in low-lying coastal areas. These extreme or non-extreme events may not be dangerous if considered individually but can intensify flooding impact if they occur simultaneously or successively. Recently, the copula approach has been widely accepted in compound flooding but is often limited to parametric or semiparametric distribution settings in a limited number of cases. However, both parametric and semiparametric approaches assume the prior distribution type for univariate marginals and copula joint density. In that case, there is a high risk of misspecification if the underlying assumption is violated. In addition, both approaches suffer from a lack of flexibility. This study uses bivariate copula density in the nonparametric distribution setting. The joint copula structure is approximated nonparametrically by employing the Bernstein copula estimator and Beta kernel copula density, and their performances are also compared. The proposed model is tested with 46 years of rainfall and storm surge observations collected on Canada's west coast. The marginal distribution of the selected flood variables is modelled using nonparametric kernel density estimation (KDE). Based on the different model compatibility tests, the Bernstein copula with normal KDE margins defined the joint dependence structure well. The selected nonparametric copula model is further employed to estimate joint and conditional return periods. It is found that flood hazard characteristics occurrence simultaneously is less frequent in AND-joint cases than in OR-joint cases. Also, the derived model is further used to estimate failure probability (FP) statistics to assess the variation of bivariate hydrologic risk during the project lifetime. It is found that FP statistics could be underestimated when neglecting the compound effect of storm surge and rainfall in the coastal flood risk.

Assessment of hydrological drought return periods with bivariate copulas in the Tigris river basin, Turkey

Assessment of hydrological drought return periods with bivariate copulas in the Tigris river basin, Turkey

Bivariate hydrologic risk analysis for the Xiangxi River in Three Gorges Reservoir Area, China

BackgroundHydrological extremes such as floods generally have multidimensional attributes with complex dependence structures. This leads to the urgent demand of hydrological risk analysis within a multivariate context. In this study, the bivariate hydrologic risk framework is proposed based on the bivariate copula method. In the proposed risk analysis framework, bivariate flood frequency would be analyzed for different flood variable pairs (i.e., flood peak-volume, flood peak-duration, flood volume-duration), and the bivariate hydrologic risk is then derived based on the joint return period of a flood variable pair. The distribution of one flood variable conditional on another flood variable can also be obtained through the copula method.ResultsThe proposed method is applied to the risk analysis for the Xiangxi River in the Three Gorges Reservoir area, China, based on 50 years streamflow measurements. The results indicate that the bivariate risk for flood peak flow-duration would keep constant for some time and then decrease as the increase of the flood duration. The bivariate risk for flood peak-volume holds a similar trend with the bivariate risk of flood peak-duration. The probability density functions (PDFs) of the flood volume and duration conditional on flood peak can also be generated through the best fitted copula function.ConclusionThe results indicate that the distributions of flood volume would be highly influenced by the flood peak flows, in which the flood volume would be expected to increase as the increase of flood return period. Conversely, the distribution of the flood duration would not change significantly with the variation in the flood peak return period. The obtained conclusions from the bivariate hydrologic analysis can provide decision support for flood control and mitigation.

The Devil Is in the Tail Dependence: An Assessment of Multivariate Copula‐Based Frameworks and Dependence Concepts for Coastal Compound Flood Dynamics

AbstractA hurricane event can often produce both intense rainfall and a storm tide that can cause a major compound flooding threat to coastlines. This paper examined applications of multivariate copula‐based time series models using data observed during Hurricane Irma (2017) along the coastlines of Florida, Georgia, and South Carolina, United States. Multivariate time series models were developed using bivariate copulas wherein storm tide and rainfall data were modeled using LOWESS‐based autoregressive moving average (ARMA). n samples of observed data were then synthesized using a Monte Carlo approach in which the empirical copula and the parametric estimate of the copula were obtained to approximate two‐sided p‐values using the Rosenblatt probability integral transform method. Analysis suggested that proper selection of the underlying LOWESS‐based ARMA model was the crucial aspect for modeling compound flooding wherein Archimedean, Elliptical, and Extreme Value copulas all offered consistent flexibility in terms of dependence modeling. As a backdrop to compound flood probabilities, this research also outlined both theoretical and applied frameworks for the calculation of non‐exceedance probabilities in a multidimensional environment using classical isofrequency probability assumptions for the “AND” (a bivariate joint probability) and Survival Kendall definitions. Random realizations from storm copulas combined with multivariate non‐exceedance probability definitions ultimately showed there were periods of temporal yet cyclical high intensities that lasted 1–2 hr. Lastly, a discussion is presented on the broader application of the proposed methodology within the field of engineering design and risk management which may serve as a catalyst for the continued research in compound flooding.

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.

Maximal asymmetry of bivariate copulas and consequences to measures of dependence

Abstract In this article, we focus on copulas underlying maximal non-exchangeable pairs ( X , Y ) \left(X,Y) of continuous random variables X , Y X,Y either in the sense of the uniform metric d ∞ {d}_{\infty } or the conditioning-based metrics D p {D}_{p} , and analyze their possible extent of dependence quantified by the recently introduced dependence measures ζ 1 {\zeta }_{1} and ξ \xi . Considering maximal d ∞ {d}_{\infty } -asymmetry we obtain ζ 1 ∈ 5 6 , 1 {\zeta }_{1}\in \left[\frac{5}{6},1\right] and ξ ∈ 2 3 , 1 \xi \in \left[\frac{2}{3},1\right] , and in the case of maximal D 1 {D}_{1} -asymmetry we obtain ζ 1 ∈ 3 4 , 1 {\zeta }_{1}\in \left[\frac{3}{4},1\right] and ξ ∈ 1 2 , 1 \xi \in \left(\frac{1}{2},1\right] , implying that maximal asymmetry implies a very high degree of dependence in both cases. Furthermore, we study various topological properties of the family of copulas with maximal D 1 {D}_{1} -asymmetry and derive some surprising properties for maximal D p {D}_{p} -asymmetric copulas.

Joint Modelling of Flood Hydrograph Peak, Volume and Duration Using Copulas—Case Study of Sava and Drava River in Croatia, Europe

Morphodynamic changes in the riverbed may be accelerated by the climate change-induced effects, mostly through the increase of the frequency of extreme climatic events such as floods. This can lead to scouring of the riverbed around the bridge substructure and consequently reduces its overall stability. In order to better understand hydromorphological processes at the local scale, the influence of floods on bridge scour requires a detailed analysis of several interacting flood hydrograph characteristics. This paper presents a multivariate analysis of the annual maximum (AM) flood discharge data at four gauging stations on the Drava and Sava Rivers in Croatia (Europe). As part of the hydrograph analysis, multiple baseflow separation methods were tested. Flood volumes and durations were derived after extracting the baseflow from measured discharge data. Suitable marginal distribution functions were fitted to the peak discharge (Q), flood volume (V) and duration (D) data. Bivariate copula analyses were conducted for the next pairs: peak discharge and volume (Q–V), hydrograph volume and duration (V–D) and peak discharge and hydrograph duration (Q–D). The results of the bivariate copula analyses were used to derive joint return periods for different flood variable combinations, which may serve as a preliminary analysis for the pilot bridges of the R3PEAT project where the aim is to investigate the influences on the riverbed erosion around bridges with installed scour countermeasures. Hence, a design hydrograph was derived that could be used as input data in the hydraulic model for the investigation of the bridge scour dynamics within the project and a preliminary methodology is proposed to be applied. The results indicate that bivariate frequency analysis can be very sensitive to the selected baseflow separation methodology. Therefore, future studies should test multiple baseflow separation methods and visually inspect the performance.

Use of monitoring extreme strain data for failure probability analysis of concrete bridge girder: An optimal vine Gaussian-copula model

In this article, a new analysis method for the failure probability of bridge girder is presented. Firstly, considering the correlation among multiple monitored variables, the optimal regular vine model is established based on the monitored extreme strain data; secondly, the optimal regular vine model about the multiple performance functions corresponding to the failure modes of the multiple monitored points, is given, further, through combining the proposed optimal regular vine model with bivariate Gaussian copula model and first order second moment method, the bridge girder failure probability considering the correlation of the performance functions, is analyzed; finally, the monitored extreme strain data of an existing bridge is provided to illustrate the feasibility and application of the proposed model and method.

Use of copula to model within-study association in bivariate meta-analysis of binomial data at the aggregate level: ABayesian approach and application to surrogate endpoint evaluation.

Bivariate meta-analysis provides a useful framework for combining information across related studies and has been utilized to combine evidence from clinical studies to evaluate treatment efficacy on two outcomes. It has also been used to investigate surrogacy patterns between treatment effects on the surrogate endpoint and the final outcome. Surrogate endpoints play an important role in drug development when they can be used to measure treatment effect early compared to the final outcome and to predict clinical benefit or harm. The standard bivariate meta-analytic approach models the observed treatment effects on the surrogate and the final outcome outcomes jointly, at both the within-study and between-studies levels, using a bivariate normal distribution. For binomial data, a normal approximation on log odds ratio scale can be used. However, this method may lead to biased results when the proportions of events are close to one or zero, affecting the validation of surrogate endpoints. In this article, we explore modeling the two outcomes on the original binomial scale. First, we present a method that uses independent binomial likelihoods to model the within-study variability avoiding to approximate the observed treatment effects. However, the method ignores the within-study association. To overcome this issue, we propose a method using a bivariate copula with binomial marginals, which allows the model to account for the within-study association. We applied the methods to an illustrative example in chronic myeloid leukemia to investigate the surrogate relationship between complete cytogenetic response and event-free-survival.

A Copula-Based Approach for Modelling the Dependence between Inflation and Exchange Rate in Kenya

In this study, we modeled the dependence structure between inflation and exchange rate using the copula approach. To formulate a bivariate copula, we used ARMA+GARCH to model serial dependence for each univariate series of returns. Both for in ation and exchange rate, it was found that the student t distribution was the best marginal distribution. Then, we transformed the standardized residuals from those marginal distributions (student t) into uniform over the range [0; 1]. To estimate the copula, we used a parametric approach. Gumbel copula was found to be the best to capture the dependence. We investigated the time-varying dependence using change-point detection based on copula. We found that there is a significant change in the nature of dependence over the period under consideration. The change in the nature of dependence between the two variables was in line with the prevailing macro-economics conditions during the period under review. We recommend to the future researchers to consider studying time-varying dependence between those two variables and investigate also the change in copula parameters in values with time. We also recommend including other macroeconomic variables while modeling the relationship between in ation and exchange rate.

Modeling Bivariate Dependency in Insurance Data via Copula: A Brief Study

Copulas are a quite flexible and useful tool for modeling the dependence structure between two or more variables or components of bivariate and multivariate vectors, in particular, to predict losses in insurance and finance. In this article, we use the VineCopula package in R to study the dependence structure of some well-known real-life insurance data and identify the best bivariate copula in each case. Associated structural properties of these bivariate copulas are also discussed with a major focus on their tail dependence structure. This study shows that certain types of Archimedean copula with the heavy tail dependence property are a reasonable framework to start in terms modeling insurance claim data both in the bivariate as well as in the case of multivariate domains as appropriate.

Pre- and post-1975 scaling relationships of monsoon and non-monsoon hourly precipitation extremes with coincident temperature across urban India

This study presents a scaling analysis of extremes in hourly precipitation with coincident temperature in 14 urban locations across India for monsoon as well as non-monsoon seasons, and examines the probabilistic association through bivariate copula. Using gridded hourly rainfall and dew point temperature (DPT) datasets from ERA5 Reanalysis and daily maximum temperature from the India Meteorology Department (IMD), it examines how far the scaling relationships of sub-daily precipitation extremes with temperature deviate from the Clausius-Clapeyron (CC) scaling across different climatic regimes in India. Pre- and post-1975 analyses were conducted to evaluate if the scaling relationships have altered in line with the global climate shift of the 1970s. Findings from the study indicate that during the monsoon, the scaling relationship was negative in all the cities except Pune (PNQ) during the post-1975 period. In fact, in most of the locations (e.g., Bhopal (BHO), Jaipur (JAI) and Mumbai (BOM)), the scaling coefficient for monsoon rainfall extremes decreased after the 1975 global climate shift. During the non-monsoon season, the scaling coefficient was slightly positive almost everywhere and approximately equal to the CC scaling rate, both pre- and post-1975; the only exception being in PNQ, where a negative scaling was observed. The joint probabilistic distributions for BOM showed that the non-exceedance probability of extreme rainfall was higher when conditioned on higher temperatures in the monsoon season; however, the opposite was observed in the non-monsoon season pre-1975. A similar distinct seasonal feature was noticed in a few other locations including New Delhi (DEL). Using DPT as the scaling variable, Chennai (MAA), Guwahati (GAU) and Tiruchirappalli (TRZ) showed positive scaling in both pre- and post-1975 in both monsoon and non-monsoon seasons. Unlike with maximum air temperature, super-CC scaling was observed with DPT in the monsoon season for TRZ (pre-and post-1975) and for MAA and GAU (pre-1975) and also in the non-monsoon season for MAA (pre-and post-1975) and for GAU and BOM (post-1975).This analysis is expected to provide useful insights into the regional and seasonal distribution of hydrological extremes for a subsequent assessment of their impacts on the design of urban infrastructure.

Conditional probability modeling of intensity measures for offshore mainshock-aftershock sequences

Offshore structures are typically subjected to offshore mainshock (MS)–aftershock (AS) sequences. The realistic statistical modeling of intensity measures (IMs) for offshore MS–AS sequences is crucial for the seismic risk assessment and optimal design of offshore structures. This paper develops a framework to forecast the AS IMs given the offshore MS IMs. In particular, using the IMs of offshore MS–AS seismic motions selected from the Japan's K-NET seismograph network, the standard kernel density estimation (KDE) and transformation KDE (TKDE) (i.e., logarithmic TKDE (log-TKDE) and square root TKDE (sqrt-TKDE)) are employed to estimate the marginal cumulative distributions (MCDs) of IMs. Several bivariate copulas are used to model the joint probability distributions (JPDs). Furthermore, the copula-based conditional distribution is used to calculate the conditional probability for the AS IMs under the conditions of the MS IMs. The results demonstrate the ability of the proposed bivariate model to realistically characterize the statistical characteristic and dependent structures for MS–AS IMs. Finally, given the offshore MS IMs, the copula-based conditional probability can be used to accurately estimate the range and probability of occurrence for offshore AS IMs.

Vine copula statistical disclosure control for mixed-type data

In this paper, we develop a new statistical disclosure control (SDC) method for mixed-type data based on vine copulas. The use of Gaussian and skew-t copulas has been demonstrated to be capable of incorporating information from the marginal distributions of mixed-type variables, whether they are discrete or continuous. In particular, our proposed SDC method using vine copulas generalizes a data perturbation method using an extended skew-t copula. Our vine-SDC method improves the SDC method using the extended skew-t copula by allowing the bivariate copulas in the vine decomposition to take various forms, thus offering a better fit for the joint distribution of the data and more flexibility in data perturbation. An additional advantage of our vine-SDC method is the significant improvement in computational efficiency compared with that using the extended skew-t copula. We discuss some statistical properties of vine copulas and the methodology of vine-SDC. A simulation and a study of real healthcare survey data are provided to explore the performance and strength of vine-SDC and compare it with a common copula-based SDC method.

Generating Continuous Rainfall Time Series with High Temporal Resolution by Using a Stochastic Rainfall Generator with a Copula and Modified Huff Rainfall Curves

In this study, a stochastic rainfall generator was developed to create continuous rainfall time series with a high temporal resolution of 10 min. The rainfall-generation process involved Monte Carlo simulation for stochastically generating rainfall parameters such as rainfall quantity, duration, inter-event time, and type. A bivariate copula was used to preserve the correlation between rainfall quantity and rainfall duration in the generated rainfall series. A modified Huff curve method was used to overcome the drawbacks of rainfall type classification by using the conventional Huff curve method. The number of discarded rainfall events was lower in the modified Huff curve method than in the conventional Huff curve method. Moreover, the modified method includes a new rainfall type that better represents rainfall events with a relatively uniform temporal pattern. The developed rainfall generator was used to reproduce rainfall series for the Yilan River Basin in Taiwan. The statistical indices of the generated rainfall series were close to those of the observed rainfall series. The results obtained for rainfall type classification indicated the necessity and suitability of the proposed new rainfall type. Overall, the developed stochastic rainfall generator can suitably reproduce continuous rainfall time series with a resolution of 10 min.

Copula-Based Modeling of RIS-Assisted Communications: Outage Probability Analysis

Statistical characterization of the signal-to-noise ratio (SNR) of reconfigurable intelligent surface (RIS)-assisted communications in the presence of phase noise is an important open issue. In this letter, we exploit the concept of copula modeling to capture the non-standard dependence features that appear due to the presence of discrete phase noise. In particular, we consider the outage probability of RIS systems in Rayleigh fading channels and provide joint distributions to characterize the dependencies due to the use of finite resolution phase shifters at the RIS. Numerical assessments confirm the validity of closed-form expressions of the outage probability and motivate the use of bivariate copula for further RIS studies.

The Incorporation of Generalized Linear Models into Bivariate Gaussian Copula and An Application

In non-life insurance mathematics, analyses and premium or reserve calculations are carried out in the presence of dependency between the claim variables in recent years. And, thus over- or underestimation of aggregate loss caused by the assumption of dependency between the claim severity and frequency are prevented. The Gaussian copula function, which is frequently used for dependency modeling, is integrated into the marginal generalized linear models to obtain a mixed copula-based regression model called "copula regression". In this study, a copula regression model is created using a bivariate Gaussian copula, Gamma and Poisson marginal generalized linear models for claim severity and frequency, respectively. An application is performed with a simulated data where there is a dependence between the claim severity and frequency using the R package “CopulaRegression”. The importance of the modeling of dependency between claims is investigated by the comparison of the independent and dependent models and the results of application show that the copula regression model in which dependency is considered has lower relative mean square errors compared the independent marginal generalized linear models.

Dependence Structure between Green Bond Market and Asian Stock Markets: Evidence from India, China and Pakistan

The purpose of the underlying study is to investigate the dependence structure between the selected Asian Stock Market (Pakistan, India, and China) and Green Bonds Market. For this purpose time, a series of daily data of closing prices of all the markets were obtained from 2014 to 2021. Further, the Bivariate copula approach has been used for analyzing the data. The findings of the study reveal that there is a dependence between log-returns of three Asian stock markets (Pakistan, China, and India) and the green bond market as the best fit copula is t-student. The student-t copula work portrays the dependence between arbitrary variables on the two tails. Although this study contributes to the literature by analyzing more countries in the future shall further add to the literature by adding a regional perspective. Lastly, the findings of the study shall aid the investors in portfolio making, risk management, and policymakers as well.

Assessment of inequity in bicyclist crashes using bivariate Bayesian copulas

Introduction: Physical activity associated with active transport modes such as bicycling has major health benefits and can help to reduce health concerns related to sedentary lifestyles, such as cardiovascular disease, Type II diabetes, and obesity, as well as risks of colon and breast cancer, high blood pressure, lipid disorders, osteoporosis, depression, and anxiety. However, as a vulnerable user group, bicyclists experience negative health impacts of transportation policies and infrastructure, such as traffic crashes and exposure to air and noise pollution that is disproportionately distributed within low-income and underserved areas. Method: This study used aggregated (block-group) bicyclist crash data from Harris County, Texas, to analyze how various equity measures are associated with both fatal and injury (FI) and no injury (property damage only) bicyclist crashes that occurred from 2010 to 2017. We used Bayesian bivariate copula-based random effects regression analysis to evaluate these associations. In contrast to more traditional univariate analysis, this novel methodology can consider the effects of factors of interest across different severity levels or crash types to fully understand their effects and how they may differ across categories. Results: The analysis results indicate that the bicyclist exposure, vehicle exposure, population demographics, population density, the percentage of African-Americans, and households below the poverty level are associated with both FI and PDO bicyclist crashes. Conclusions: Although more location and context-specific analyses are required, this study's overall results once again conform with the findings and assumptions in bicycling safety literature that the low-income and racially diverse communities are prone to experience more bicyclist crashes. Practical Applications: The findings of this study may have implications for future transportation and planning policies. These findings can be used to guide the policies and strategies targeting the elimination of inequity in transportation-related health concerns.

Rarity of Joint Probability Between Interest and Inflation Rates in the 1998 Economic Crisis in Indonesia and Their Comparison Over Three Time Periods

After more than twenty years, there has been no economic crisis as severe as 1998 based on inflation and interest rates. It is interesting to compare the conditions before and after the 1998 crisis and the economic conditions in the last decade in Indonesia. Therefore, this study aims to analyze the relationship between inflation and interest rates using a copula-based joint distribution. The joint return period of the 1998 economic crisis is estimated from this joint distribution. The results showed that the Gumbel copula is the most suitable bivariate copula to construct a joint distribution between inflation and interest rates in 1990-2019, with an upper tail dependency of 0.6224. Moreover, the joint return period between inflation and interest rates more severe than 1998 is 389 years with a 95% confidence interval of [47, ∞] years. This result is uncertain because many factors affect inflation and interest rates. The inflation rate decreased after the 1998 crisis. Meanwhile, in the last decade, the inflation and interest rates were much lower than in the two previous periods.