Trivariate Copulas Research Articles

Modelling the joint accuracy of two diagnostic tests using evidence synthesis methods incorporating linked electronic health records

Background and ObjectivesTo compare the accuracy of different diagnostic pathways comprising of multiple diagnostic tests, statistical methods that evaluate combined or sequential test accuracy are required. This can only be achieved through joint synthesis of diagnostic accuracy data on all tests of interest, using models that account for dependencies between multiple tests. The aim of this research is to develop joint models to assess the diagnostic accuracy of two tests, using real-world evidence to inform within-study dependencies between tests. ApproachUsing a case study in Alzheimer’s disease dementia (ADD), we will develop Bayesian meta-analysis models to evaluate the combined diagnostic accuracy of two tests. A novel application of trivariate copulas will be used to capture within-study dependencies between two tests assessed in the same patients. Individual participant data from Dementias Platform UK (DPUK) Data Portal – a repository of linked electronic health record data optimised for dementia research – will be used to estimate dependencies between diagnostic tests for ADD. ResultsThe DPUK Data Portal includes records of over 3 million people from 51 long-term cohort studies, including individual-level test results for ADD verified against confirmed diagnoses. Analysis will focus on modelling the diagnostic accuracy of different testing strategies for ADD, combining cognitive function, imaging, and biomarker tests. Conclusions and ImplicationsResults of this study have the potential to inform healthcare decision-making regarding optimal diagnostic pathways by enabling the comparison of different testing strategies. The incorporation of real-world data from the DPUK Data Portal makes better use of the available evidence.

Projected changes in compound hot-dry events depend on the dry indicator considered

The intensification of compound hot-dry events due to climate change is a pressing concern, underscoring the need for precise analysis. However, the impact of different dry indicators on projections of these events has not been quantitatively evaluated, nor has its importance been compared with other sources of uncertainty. Here we examine the sensitivity of projected changes in compound hot-dry events to different dry indicators. We use data from 22 Coupled Model Intercomparison Project Phase 6 (CMIP6) models to characterize global dry conditions based on precipitation, runoff, soil moisture, and a multivariate index combining these variables through trivariate copulas. Our findings reveal large differences in projected changes in the likelihood of compound hot-dry events across different dry indicators. While model uncertainty remains the primary source of uncertainty for compound hot-dry event projections, the uncertainty associated with dry indicators is also substantial, surpassing scenario uncertainty in specific regions.

Enhancing flood risk assessment in the Johor River Basin through trivariate copula

ABSTRACT Copulas are a vital statistical tool, particularly in hydrology, for understanding complex relationships among flood characteristics. This study focuses on three key flood features: peak discharge, flood volume, and flood duration, using trivariate copulas to capture their interdependencies. This is crucial because bivariate and univariate analyses fall short in considering all three factors simultaneously. To handle extreme flood values, L-moment is proposed over maximum likelihood estimation and inference function margin due to its enhanced reliability and susceptibility to outliers and extreme values. Akaike information criterion was employed to identify the best-fit marginal distribution and copula. The Lognormal distribution effectively models peak discharge, while Weibull and generalized extreme value distributions fit flood volume and duration best, respectively. Various copula families, including elliptical and Archimedean, are assessed, where Clayton copula emerge as the most suitable. This analysis demonstrates that when more flood features are considered together, the return period increases, indicating the reduced likelihood of occurrence. The trivariate case of the AND-joint return period surpasses the trivariate case of the OR-joint return period where the TP, V, DAND=5, 405.93 years, while TP, V, DOR=500.46 years. This comprehensive approach enhances hydrological modeling and decision-making for water resource management and flood mitigation projects.

Impacts of climate change on global meteorological multi-year droughts using the last millennium simulation as a baseline

Multi-year droughts have been widely studied all over the world for their extraordinary long duration and irreparable impacts. However, they are difficult to be estimated due to the short period of observed time series, especially in the context of climate change and on a global scale. The advanced development of climate model simulations with a long period makes it possible to investigate the multi-year drought change on a global scale. Therefore, this study quantifies the impacts of climate change on global meteorological multi-year droughts for the historical period (1850–2005) and the future period (2006–2100) using the pre-industrial last millennium period (850–1849) as a baseline. Meteorological droughts longer than 5-year are identified using a three-dimensional (latitude-longitude-time) method based on a 12-month scale Standard Precipitation Index (SPI12). Return periods are estimated using tri-variate copulas based on drought duration, area, and severity. The results show that multi-year droughts occur more in North America, South America, Asia, and Africa, accounting for more than 85% during 850–2100. The return period of 61% and 70% multi-year droughts for the historical and future periods becomes shorter than that for the last millennium, indicating more severe drought conditions for these two periods. The main change for the future period is the higher incidence of multi-year droughts with a longer duration and greater severity, especially for North America, Asia, and Africa. Besides, the shift of multi-year drought centroids in Central America, East Asia, Southern Africa, and the Mediterranean implies more severe drought conditions for the future period. Overall, this study indicates that climate change has the potential effects to exacerbate global meteorological multi-year droughts for the future period compared to the last millennium.

Coincidence probability analysis of hydrologic low-flow under the changing environment in the Wei River Basin

Hydrologic characteristics including extreme-flow events in many rivers around the world have been altered due to the climate change and human activities. Most available research concerns about the extreme event trend, frequency and duration in a single river rather than the synchronicity or coincidence probability among different tributaries. Accurately knowing the coincidence probability of hydrologic extreme-flow events is vital for better water resources allocation and project design. Joint distribution constructed by the copulas function is a widely used method to conduct this issue. However, studies conducted by the copulas function are mostly carried out in a bivariate environment and ignore the nonstationary. These may not comprehensively reflect the hydrologic characteristics under the changing environment. This paper analyzes the related hydrologic low-flow changes considering the nonstationary under the changing environment in the Wei River Basin, China, where the climate tends to be drier. These analyses are obtained by using trivariate copulas function and Kendall’s return period method to derive the joint distribution of hydrologic low-flow in a seasonal time scale. The results mainly show that the coincidence probabilities of low-flow among three tributaries of the Wei River in four seasons after the change point are basically higher than those before the change point. It may indicate that the low-flow negative impacts in the whole Wei River Basin in four seasons may be intensified (which needs to be paid more attention) under the changing environment.

If worst comes to worst: Co-movement of global stock markets in the US-China trade war

This paper investigates the co-movement characteristics of global stock markets in the context of the US-China trade war. By applying a set of different trivariate Copulas, our results suggest that markets co-move symmetrically in the pre-trade war period, but exhibit negative downside movements and heavy tails during the trade war. Furthermore, we find evidence for left-tail dependency structures during that period. Most importantly, this study finds that the trade war poses a systematic risk on global markets, which potentially can trigger simultaneous market downside trends. Our results are robust across different European equity market indices.

If Worst Comes to Worst: Co-Movement of Global Stock Markets in the US-China Trade War

This paper investigates the co-movement characteristics of global stock markets in the context of the US-China trade war. By applying a set of different trivariate Copulas, our results suggest that markets co-move symmetrically in the pre-trade war period, but exhibit negative downside movements and heavy tails during the trade war. Furthermore, we find evidence for left-tail dependency structures during that period. Most importantly, this study finds that the trade war poses a systematic risk on global markets, which potentially can trigger simultaneous market downside trends. Our results are robust across different European equity market indices.

Trivariate distribution modelling of flood characteristics using copula function—A case study for Kelantan River basin in Malaysia

Water resources operational planning, managements or either flood defence infrastructure designs often demand the estimations of flow exceedance probability for visualizing the risk of flood episodes. Numerous literature often incorporated copulas for the development of bivariate joint dependence structure among the flood characteristics, flood peak flow, volume and duration, but it could be more realistic and comprehensive if we focus all the three mutually correlated flood characteristics simultaneously. Actually, the inclusion of more flood characteristics could provide better and much justifiable information in correlation and dependency modelling. In this study, trivariate copulas are incorporated and applied to a case study to analyse flood episodes in the Kelantan River basin at Gulliemard bridge gauge station in Malaysia. Firstly, for describing best-fitted bivariate copulas for establishing the joint dependence structure of each flood attribute pairs, the Gaussian copula is recognized most justifiable model for peak-volume pair and the Frank copula for peak-duration and volume-duration pairs. After that, the trivariate joint distribution is modelled using one Archimedean copula, the Frank copula and one Elliptical copula, the Gaussian copula. Based on Cramer-von-Mises-type statistics, Sn and p-value, the Gaussian copula best representing the trivariate dependence structure of flood and which further employed in the deriving of trivariate joint and conditional return periods.

A Copula-based Index to Measure Directional Reflection Asymmetry for Trivariate Copulas

A Copula-based Index to Measure Directional Reflection Asymmetry for Trivariate Copulas

Copula-based trivariate drought frequency analysis approach in seven climatic sub-regions of mainland China over 1961–2013

In drought analysis, many types of research have applied bivariate analysis. However, a more effective appraisal is achievable by employing three mutually correlated drought variables (i.e., duration, severity, peak) concurrently. This study adopted trivariate copulas for multivariate drought frequency estimation in seven climatic sub-regions and mainland China between 1961 and 2013. The joint distribution is modeled using nine trivariate copula functions, namely Clayton, Ali–Mikhail–Haq (AMH), Gumbel–Hougaard, Frank, M3, M4, M5, M6, and M12. The performance of the different copulas was evaluated using the root mean square error (RMSE), Akaike information criterion (AIC), bias, and graphical test. The results showed that trivariate Clayton copula performed better in modeling joint dependence structure of drought variables in contrast to other class of copulas in sub-regions I–VII and mainland China. Therefore, the Clayton copula–based drought joint distributions were used for the multivariate drought frequency analysis, regarding probabilities and return periods. The joint probabilities in the “OR” cases were greater than that of the “AND” cases in all the seven sub-regions and mainland China. There is every possibility that drought event occurrence will be given for “OR” cases. Accordingly, counter measures of drought hazards need to be set for proactive responses especially in sub-regions III, V, and VI. The trivariate return periods were compared with the univariate and bivariate return periods and discussed the need of multivariate drought frequency analysis. In conclusion, the Clayton trivariate copula is an excellent choice for efficient drought risk evaluation, and results are very handy in the design of water resource hydrologic systems under severe and extreme drought conditions.

Joint occurrence probability analysis of typhoon-induced storm surges and rainstorms using trivariate Archimedean copulas

In this paper, the trivariate frequency analysis method for joint probability assessment of typhoon-induced storm surges and rainstorms is presented. The proposed method includes three major processes. The first process is to the find the best fitted marginal cumulative distributions for wind speeds (W), storm surges (S) and rainstorms (R), respectively. It is found that the Lognormal distribution is more suitable for storm surges and rainstorms, and Weibull distribution is more suitable for wind speeds. The second process is to estimate the joint probability distributions of W, S and R using four trivariate copulas, namely, Gumbel, Clayton, Frank and AMH. It is found that the joint probability can be best modeled by Gumbel copula. The last process is to calculate the conditional joint distributions of typhoon-induced storm surges and rainstorms which is useful for government agencies for improving disaster system management during typhoon events. An effective Particle Swarm Optimization (PSO) is proposed for to estimate the parameters of marginal cumulative distributions and copula functions. Analysis results based on the maximum likelihood method (MLM) and log-likelihood function (LLF) demonstrate the effectiveness of the PSO.

Comment on 'A vine copula mixed effect model for trivariate meta-analysis of diagnostic test accuracy studies accounting for disease prevalence'.

A bivariate copula mixed model has been recently proposed to synthesize diagnostic test accuracy studies and it has been shown that it is superior to the standard generalized linear mixed model in this context. Here, we call trivariate vine copulas to extend the bivariate meta-analysis of diagnostic test accuracy studies by accounting for disease prevalence. Our vine copula mixed model includes the trivariate generalized linear mixed model as a special case and can also operate on the original scale of sensitivity, specificity, and disease prevalence. Our general methodology is illustrated by re-analyzing the data of two published meta-analyses. Our study suggests that there can be an improvement on trivariate generalized linear mixed model in fit to data and makes the argument for moving to vine copula random effects models especially because of their richness, including reflection asymmetric tail dependence, and computational feasibility despite their three dimensionality.

Diagonal plane sections of trivariate copulas

We introduce the notion of diagonal plane section of a trivariate copula as an additional tool to describe its tail dependence behavior. This notion extends the concept of diagonal section of a bivariate copula. We provide existence results for trivariate copulas with a given diagonal plane function. Related results are discussed, especially about extensions to a higher dimensional setting.

Copula based drought frequency analysis considering the spatio-temporal variability in Southwest China

Summary Drought frequency analysis is a prerequisite for drought resistance planning and drought risk management. Drought is a spatio-temporal dynamic process, usually characterized by its duration, spatial extent, and severity. Copula based multivariate frequency analysis has been widely used to calculate drought frequency. However, the spatial extent is scarcely considered in previous studies, due to the fact that drought event is usually identified either for a fixed spatial scale or for a fixed temporal scale. This study develops a regional drought frequency analysis model based on trivariate copulas by considering the spatio-temporal variations of drought events. Drought duration, drought affect area, and drought severity are identified first, and their trivariate joint distribution is constructed later. The model is applied for drought frequency analysis in Southwest China during 1961–2012. A variety of probability distribution functions and copula functions (including elliptical, symmetric and asymmetric Archimedean) are used as candidate choices, and the most appropriate ones are selected based on goodness of fit using different methods. The robustness of drought frequency analysis is then evaluated and discussed. The results show that drought frequency analysis needs to fully consider the three characteristic parameters (duration, affect area, and severity) reflecting drought spatio-temporal variability. And the drought return period estimated by the copula-based trivariate frequency analysis appropriately integrates the effects of drought duration, affect area and severity, which is a reliable drought statistical measurement. The 2009–2010 drought, which has a return period of about 94 years, is the most severe one in Southwest China during the period of 1961–2012. The Joe and Gumbel copulas are found to be more suitable to estimate the joint distribution of drought duration, affect area and severity, and the Asymmetric (nested) function forms perform better than the symmetric functions.

Meta-analysis of diagnostic tests accounting for disease prevalence: a new model using trivariate copulas.

In real life and somewhat contrary to biostatistical textbook knowledge, sensitivity and specificity (and not only predictive values) of diagnostic tests can vary with the underlying prevalence of disease. In meta-analysis of diagnostic studies, accounting for this fact naturally leads to a trivariate expansion of the traditional bivariate logistic regression model with random study effects. In this paper, a new model is proposed using trivariate copulas and beta-binomial marginal distributions for sensitivity, specificity, and prevalence as an expansion of the bivariate model. Two different copulas are used, the trivariate Gaussian copula and a trivariate vine copula based on the bivariate Plackett copula. This model has a closed-form likelihood, so standard software (e.g., SAS PROC NLMIXED) can be used. The results of a simulation study have shown that the copula models perform at least as good but frequently better than the standard model. The methods are illustrated by two examples.

Analysis of ENSO-based climate variability in modulating drought risks over western Rajasthan in India

This paper investigates the role of El Nino-Southern Oscillation (ENSO)-based climate variability in modulating multivariate drought risks in the drought-prone region of Western Rajasthan in India. Droughts are multivariate phenomenon, often characterized by severity, duration and peak. By using multivariate ENSO index, annual drought events are partitioned into three climatic states – El Nino, La Nina and neutral phases. For multivariate probabilistic representation of drought characteristics, trivariate copulas are employed, which have the ability to preserve the dependence structure of drought variables under uncertain environment. The first copula model is developed without accounting the climate state information to obtain joint and conditional return periods of drought characteristics. Then, copula-based models are developed for each climate state to estimate the joint and conditional probabilities of drought characteristics under each ENSO state. Results of the study suggest that the inclusion of ENSO-based climate variability is helpful in knowing the associated drought risks, and useful for management of water resources in the region.

Probabilistic assessment of flood risks using trivariate copulas

In this paper, a copula-based methodology is presented for probabilistic assessment of flood risks and investigated the performance of trivariate copulas in modeling dependence structure of flood properties. The flood is a multi-attribute natural hazard and is characterized by mutually correlated flood properties peak flow, volume, and duration of flood hydrograph. For assessing flood risk, many studies have used bivariate analysis, but a more effective assessment can be possible considering all three mutually correlated flood properties simultaneously. This study adopts trivariate copulas for multivariate analysis of flood risks, and applied to a case study of flood flows of Delaware River basin at Port Jervis, NY, USA. On evaluation of various probability distributions for representation of flood variables, it is found that the flood peak flow and volumes can be best represented by Frechet distribution, whereas flood duration by log-normal distribution. The joint distribution is modeled using four trivariate copulas, namely, three fully nested form of Archimedean copulas: Clayton, Gumbel–Hougaard, Frank copulas; and one elliptical copula: Student’s t copula. Based on distance-based performance measures, graphical tests, and tail-dependence measures, it is found that the Student’s t copula best representing the trivariate dependence structure of flood properties as compared to the other copulas. Similar results are found for bivariate copula modeling of flood variables pairs, where Student’s t copula performed better than the other copulas. The obtained copula-based joint distributions are used for multivariate analysis of flood risks, in terms of primary and secondary return periods. The resultant trivariate return periods are compared with univariate and bivariate return periods, and addressed the necessity of multivariate flood risk analysis. The study concludes that the trivariate copula-based methodology is a viable choice for effective risk assessment of floods.

Drought Analysis under Climate Change Using Copula

AbstractThe joint behavior of drought characteristics under climate change is evaluated using the copula method, which has recently attained popularity in the analysis of complex hydrologic systems with correlated variables. Trivariate copulas are applied, in this study, to analyze the major drought variables, including duration, severity, and intensity, in Oregon’s Upper Klamath River Basin. Among the variables, results show that duration severity exhibits the strongest correlation, whereas duration intensity exhibits the least correlation. The impact of climate change on future droughts is evaluated using five general circulation models (GCMs) under one emission scenario. Despite more intense extreme events that are expected to occur in most parts of the globe in the future, the results of this study show that the Upper Klamath River Basin in the Pacific Northwest will experience less intense droughts affected by climate change. Compared with historical events, an overall decrease in drought duration an...

Comment on “Trivariate statistical analysis of extreme rainfall events via the Plackett family of copulas” by Shih‐Chieh Kao and Rao S. Govindaraju: How are trivariate copulas put to use?

Comment on “Trivariate statistical analysis of extreme rainfall events via the Plackett family of copulas” by Shih‐Chieh Kao and Rao S. Govindaraju: How are trivariate copulas put to use?

Drought Analysis Using Trivariate Copulas Conditional on Climatic States

Droughts are an inevitable consequence of meteorological variability, and the design of water resource infrastructure and management strategies to mitigate their effects requires assessment of the risk. Crucial characteristics of droughts are related to their peak intensities, durations, and severities. These variables are typically correlated and copulas provide a versatile means to model their dependence structure. In Australia, for example, drought severity is associated with the El-Niño Southern Oscillation. Data from two rainfall districts in New South Wales, one on the east and the other on the west of the Great Dividing Range, are considered. These rainfall data are categorized into three states, El-Niño, Neutral, and La-Niña, according to the prevailing Southern Oscillation Index. Gumbel-Hougaard copulas and t-copulas are fitted to the droughts in the three states. The copula parameters are estimated separately for each state, and the differences are analyzed. The goodness-of-fit of the Gumbel-Hougaard and t-copulas are compared, and the limitations of the two copula models are discussed. The times between drought events are also analyzed according to the El-Niño Southern Oscillation state they occur in. The fitted copulas are used to estimate annual recurrence intervals of at least one of the three variables, and of all three variables, exceeding critical values taking into account of the mixture of states.