Copula Selection Research Articles

Constructing multivariate distribution of rainfall characteristics: A Bayesian vine algorithm

Constructing a multivariate distribution model for rainfall characteristics is an important task for risk-based hydrologic design. However, most stochastic rainfall models are limited to bivariate frequentist distribution and cannot be easily generalized to multivariate (≥3) Bayesian settings. Towards this problem, this study presents a Bayesian vine algorithm. The presented algorithm translates the high-dimensional Bayesian update task into multiple lower-dimensional update tasks using the vine approach. The algorithm also computes the evidence, allowing Bayesian model selection of marginals and copulas in the joint distribution. Two numerical examples, a synthetic rainfall data from Canada, and a real rainfall data from China are utilized to demonstrate the algorithm’s capability. The results suggest that the algorithm can satisfactorily identify and characterize the underlying marginal and dependence for the rainfall characteristics. Since the high-dimensional Bayesian update task is translated into lower-dimensional Bayesian updates, the algorithm is scalable, stable, and computationally efficient in higher dimensions. A 20-year continuous hourly rainfall record can be characterized for its depth, duration, and intensity in around 60 s without any convergence or tuning issues. Overall, this study provides a powerful Bayesian tool for constructing stochastic rainfall models, which in turn can be used in multiple civil engineering disciplines concerned with rainfall interaction. This study does not consider temporal/serial dependence but only the cross-dependence among the rainfall characteristics.

Spatial modeling sensitivity analysis: Copula selection for wind speed dependence

As the adoption of wind energy as a key renewable energy source accelerates, precise power flow analysis becomes crucial for accurate power delivery forecasting. This paper addresses the inherent uncertainties in wind speed data at different wind farm locations by conducting a sensitivity analysis to assess wind farm pairs. The analysis accommodates various data sizes, namely, short, medium, and large, and diverse spatial relationships between wind farms. By leveraging National Renewable Energy Laboratory wind speed data from nine distinct wind farms, the dependence structure between wind farm pairs is modeled using copulas. This modeling takes both the wind speed knowledge level and the various spatial interplays among the wind farm pairs into consideration. The findings indicate an inverse proportionality between the strength of dependence and the distance separating the wind farm pairs.

Quantifying compound flood event uncertainties in a wave and tidally dominated coastal region: The impacts of copula selection, sampling, record length, and precipitation gauge selection

AbstractCoastal flooding is a growing hazard. Compound event characterization and uncertainty quantification are critical to accurate flood risk assessment. This study presents univariate, conditional, and joint probabilities for observed water levels, precipitation, and waves. Design events for 10‐ and 100‐year marine water level and precipitation events are developed. A total water level formulation explicitly accounting for wave impacts is presented. Uncertainties associated with sampling method, copula selection, data record length, and utilized rainfall gauge are determined. Eight copulas are used to quantify multivariate uncertainty. Generally, copulas present similar results, except the BB5. Sampling method uncertainty was quantified using four sampling types; annual maximum, annual coinciding, wet season monthly maximum, and wet season monthly coinciding sampling. Annual coinciding sampling typically produced the lowest event magnitude estimates. Uncertainty associated with record length was explored by partitioning a 100‐year record into various subsets. Withholding 30 years of observations (i.e., records of less than 70 years) resulted in substantial variability of both the 10‐ and 100‐year return period estimates. Approximately equidistant rainfall gauges led to large event estimate differences, suggesting microclimatology and gauge selection play a key role in characterizing compound events. Generally, event estimate uncertainty was dominated by sampling method and rainfall gauge selection.

Assessing the Impact of Copula Selection on Reliability Measures of Type P(X < Y) with Generalized Extreme Value Marginals

In reliability studies, we are interested in the behaviour of a system when it interacts with its surrounding environment. To assess the system’s behaviour in a reliability sense, we can take the system’s intrinsic quality as strength and the outcome of interactions as stress. Failure is observed whenever stress exceeds strength. Taking Y as a random variable representing the stress the system experiences and random variable X as its strength, the probability of not failing can be taken as a proxy for the reliability of the component and given as P(Y<X)=1−P(X<Y). This way, in the present paper, it is considered that X and Y follow generalized extreme value distributions, which represent a family of continuous probability distributions that have been extensively applied in engineering and economic contexts. Our contribution deals with a more general scenario where stress and strength are not independent and copulas are used to model the dependence between the involved random variables. In such modelling framework, we explored the proper selection of copula models characterizing the dependence structure. The Gumbel–Hougaard, Frank, and Clayton copulas were used for modelling bivariate data sets. In each case, information criteria were considered to compare the modelling capabilities of each copula. Two economic applications, as well as an engineering one, on real data sets are discussed. Overall, an easy-to-use methodological framework is described, allowing practitioners to apply it to their own research projects.

Effect of copula dependence structure on the failure modes of slopes in spatially variable soils

The copula dependence structure underlying the correlated shear strength parameters of soils can significantly affect the results of slope reliability analysis and risk assessment of slope failure. Currently, how do copula dependence structures affect the failure modes of slopes in spatially variable soils still remains an open question. This study systematically investigates the effect of copula dependence structures on the failure modes of a cohesive-frictional slope in spatially variable soils via an adapted random finite difference method (RFDM) that incorporates the copula-based random fields. The proposed RFDM proceeds with the simulation of cross-correlated random fields considering the copula dependence between the shear strength parameters of the slope. The strength reduction analyses are then performed within the framework of Monte Carlo simulation to obtain the stability analysis results, including the critical slip surface (CSS) and associated factor of safety (FS) and sliding volume. Finally, various statistical analyses on the number and distribution of CSS, FS and sliding volume as well as the failure modes versus different statistics of soil properties are analyzed considering different dependence structures. The results show that the overall distribution range of CSS obtained by the commonly used Gaussian copula is more concentrated, whereas the result by the No. 16 copula is the widest. The differences in the overall distribution range of CSS among different copulas decrease with the increase of the scale of fluctuation, coefficient of variation, and cross-correlation coefficient of the strength parameters. The proportions of different failure modes are comparable for the Gaussian, Plackett and Fran copulas, whereas the No.16 copula has the smallest proportion for the shallow failure and largest proportions for the deep and multi-slip failures. The influence of copula dependence structure on the slope failure mode is more significant than the statistical parameters of soil properties. The results provide useful references for copula selection when the underlying dependence structure is difficult to be determined.

¿(Está/Es) Difícil?: Variable Use of Ser and Estar by Heritage Learners of Spanish

The current study examines variation in copula selection in Spanish by looking at the written productions of three groups of language learners in the United States, including heritage learners, those with English as an L1, and international students with English as an L2. Research on copula variation in Spanish has pinpointed several key linguistic and social factors that influence selection; this study aims to apply these findings to heritage learners in order to determine how their acquisition differs from that of non-native language learners. This analysis used the COWS-L2H corpus of Spanish from the University of California, Davis. Examining over 8000 tokens of [adjective + copula] constructions in variable contexts where both ser and estar were used, the study tracks how linguistic and extralinguistic factors condition copula selection within the three learner groups and how these results compare to previous findings. Seven factors were predictive of copula selection: resultant state, frame of reference, adjective class, experience with study abroad, essay prompt, student age, and course level. Heritage learner copula use was found to be governed by a different set of predictors than that of learners, hinting at the variable motivations and backgrounds that influence use and reflect the identity goals of these speakers.

Selection and application of Copula functions with dependence on its tail right to the joint frequency analysis (Q, V) of annual floods

través del análisis de frecuencias bivariado del gasto máximo (Q) y el volumen escurrido (V) anuales de las crecientes registradas. Las funciones Cópula (FC) son modelos probabilísticos basados en la dependencia entre Q y V, que establecen fácilmente su distribución bivariada con base en las funciones marginales previamente adoptadas o distribuciones de cualquier tipo, iguales o diferentes. La aplicación de las FC en las estimaciones hidrológicas mostró que un aspecto decisivo en su selección idónea está relacionado con la dependencia en el extremo derecho de los datos ("λ" _U^CFG) y la que tienen ("λ" _U) ciertas FC. Por lo anterior, en este estudio se exponen las FC: t de Student, Gumbel-Hougaard, Clayton Asociada y Joe, que muestran valores de "λ" _U que van en aumento. Se contrastan los valores de "λ" _U contra los "λ" _U^CFG obtenidos en 16 registros reales aleatorios de Q y V, para establecer la aplicabilidad de cada FC citada. Además, se procesa como aplicación numérica el registro de 26 datos de Q y V anuales de las crecientes de entrada a la presa Adolfo López Mateos (Humaya) del estado de Sinaloa, México. Por último, se exponen las conclusiones, las cuales destacan las ventajas de las FC en los análisis de frecuencias bivariados de crecientes.

Acquisition of estar + adjective in L2 Spanish by L1 French and Portuguese speakers

Abstract This paper reports the developmental stages of the acquisition of estar in stage-level predicate constructions in the L2 Spanish grammar of native speakers of French and Portuguese. Copular verbs exist in the L1s of both learner groups; however, only in Portuguese is there an aspectual contrast, with copula selection conditions that partially differ from those of ser and estar in Spanish. The study used data extracted from the Corpus de Aprendices de Español (CAES), made up of written texts produced by L2 Spanish learners from CEFR A1 to C1 levels. We attempt to analyse whether, and to what degree, these L2 learners are sensitive to estar copula restrictions in adjective constructions. Our analysis of the written production of 143 L1 French speakers and 361 L1 Portuguese speakers showed differing acquisitional patterns depending on the L1. We consider that the estar overextension, found in the learners’ productions, is related to the process of developing the ability to identify the specific features that distinguish the use of copular verbs in the interlanguage of our learners. Our findings, thus, provide a more fine-grained description of the semantic representation and access of interpretable features in L2 Spanish with special relevance to current hypotheses on Second Language Acquisition such as the Feature Reassembly Hypothesis (Lardiere 2009).

Application of vine copulas to estimate dew point temperature

In this study, the accuracy of the copula-based model in the simulation of the dew point temperature in various climates of Iran was investigated, using simulations based on vine copulas such as C-, D-, and R-vine copulas. By examining the various vine copulas and their tree sequences, the best copula and best tree sequence based on AIC, BIC, and log-likelihood were selected. The results show that based on the complete similarity in our case between C-, D- and R-vine copulas, the selected best C-vine copulas fit well the dependence between the minimum and maximum air temperatures and dew point temperature. The simulation results were analyzed using root mean square error (RMSE), Nash-Sutcliffe efficiency (NSE) coefficient, and violin plots. The results show that the copula-based model has high accuracy at all stations. The min (max) RMSE is related to Kerman (Ahvaz) station with RMSE = 0.396 oC (0.617 oC). Also, the min (max) NSE is related to Ahvaz (Urmia) station with NSE = 0.925 (0.955). Also, according to the violin plot, it is possible to appreciate the acceptable certainty of the copula-based model. Due to the diversity of the tree sequences of vine copulas and the use of the rotated states of the internal vine copulas, as well as the possibility of interfering with the effective parameters in high dimensions, the simulation results are reliable and have no restrictions. This model can be used as the best model to estimate dew point temperature due to the full coverage of the range of changes in data.

Probabilistic Life Prediction of Tunnel Boring Machine under Wearing Conditions with Incomplete Information

This paper developed a data analysis approach to estimate the probabilistic life of an earth pressure balance (EPB) tunnel boring machine (TBM) under wearing conditions with incomplete information. The marginal reliability function of each system component of TBM is derived based on data collected from the site. The structure of the failure framework was determined based on the evaluation of influencing factors, including the wearing of the cutter head panel and screw conveyor. The joint distribution model was built by utilizing the best-fit copula function and the remaining reliable mining distance can be predicted from this model. Real data of the remaining thickness of the wearing resistance structure of the cutter head panel and screw conveyor from an earth pressure balance (EPB) TBM were captured. A realistic metro tunneling project in China was utilized to examine the applicability and effectiveness of the developed approach. The results indicate that: (1) With the selection of normal distribution and Gumbel copula as the best-fit marginal distribution function and copula function, the reliable mining distance was predicted as 4.0834 km when the reliability equaled 0.2. (2) The copula function was necessary to be considered to assess the joint distribution of the reliability function, as the predicted mining distance reduces significantly to 3.9970 km if assumed independent. (3) It enables the user to identify the weak component in the machinery and significantly improve the reliable mining distance to 4.5075 km by increasing the initial thickness of the screw conveyor by 0.5 mm. This approach can be implemented to minimize the risk of unintended TBM breakdown and improve the tunneling efficiency by reducing unnecessary cutter head intervention during the mining process.

Integrated Drought Index based on Vine Copula Modelling

AbstractAs a natural hazard, drought is a complex multivariate phenomenon that does not depend on only one hydrometeorological variable, as well as, considering a particular kind of drought may not be effective for drought management. Considering this, many multivariate drought indices have been developed based on linearity assumptions or conventional copulas assuming symmetric relationships among univariate drought indices. In this study, D‐vine copula was applied to construct a four‐dimensional index, named as Integrated Drought Index (IDI), by combining four univariate drought indices (Standardized Precipitation Index [SPI], Reconnaissance Drought Index [RDI], Standardized Soil moisture Index [SSI] and Standardized stream flow Drought Index [SDI]) to better reflect many hydrometeorological variables (precipitation, evapotranspiration, soil moisture and stream flow) and different kinds of drought (meteorological, agricultural and hydrological) simultaneously. Vine copula was used to solve nonlinear and asymmetric relationships among drought indices due to its flexibility over the free selection of copula(s) in each step of hierarchical structure in high dimensional modelling. The IDI was constructed for 1‐ and 4‐month timescales for the upper Tapti basin of the central region in India. The performance of IDI was tested with dependence measures (Pearson's correlation coefficient, Mutual Information) and evaluated against the Terrestrial Water Storage Anomaly data derived from the Gravity Recovery and Climate Experiment (GRACE) mission. Spatial analysis of drought was carried out by fuzzy c‐means (FCM) clustering algorithm with IDI. IDI based on vine copula solved the nonlinear and asymmetric relationships among different variables associated with the occurrence of droughts effectively with a reduction of uncertainty as compared to the single drought indices for different kinds of droughts. Analysis revealed spatially different drought risks in the upper and lower river basins. In general, the vine copula addresses nonlinear and asymmetric relationships that exist between the variables associated with natural hazards like drought.

Uncertainty Quantification of Rainfall-runoff Simulations Using the Copula-based Bayesian Processor: Impacts of Seasonality, Copula Selection and Correlation Coefficient

The outputs of Rainfall-runoff models are inherently uncertain and quantifying the associated uncertainty is crucial for water resources management activities. This study presents the uncertainty quantification of rainfall-runoff simulations using the copula-based Bayesian processor (CBP) in Danjiangkou Reservoir basin, China. The seasonality of uncertainty in rainfall-runoff modeling is explored, and impacts of copula selection and correlation coefficient on uncertainty quantification results are investigated. Results show that the overall performance of the CBP is satisfactory, which provides a useful tool for estimating the uncertainty of rainfall-runoff simulations. It is also demonstrated that the dry season has higher reliability and greater resolution compared with wet season, which illustrates that the CBP captures the actual uncertainty of rainfall-runoff simulations more accurately in dry season. Moreover, the performance the CBP highly depends on the selected Copula function and considered Kendall tau correlation coefficient. As a result, great attention should be paid to selecting the appropriate Copula function and effectively capturing the actual dependence between observed and simulated flows in the CBP-based uncertainty quantification of rainfall-runoff simulations practice.

Correcting for selection bias in HIV prevalence estimates: an application of sample selection models using data from population-based HIV surveys in seven sub-Saharan African countries.

IntroductionPopulation‐based biomarker surveys are the gold standard for estimating HIV prevalence but are susceptible to substantial non‐participation (up to 30%). Analytical missing data methods, including inverse‐probability weighting (IPW) and multiple imputation (MI), are biased when data are missing‐not‐at‐random, for example when people living with HIV more frequently decline participation. Heckman‐type selection models can, under certain assumptions, recover unbiased prevalence estimates in such scenarios.MethodsWe pooled data from 142,706 participants aged 15–49 years from nationally representative cross‐sectional Population‐based HIV Impact Assessments in seven countries in sub‐Saharan Africa, conducted between 2015 and 2018 in Tanzania, Uganda, Malawi, Zambia, Zimbabwe, Lesotho and Eswatini. We compared sex‐stratified HIV prevalence estimates from unadjusted, IPW, MI and selection models, controlling for household and individual‐level predictors of non‐participation, and assessed the sensitivity of selection models to the copula function specifying the correlation between study participation and HIV status.ResultsIn total, 84.1% of participants provided a blood sample to determine HIV serostatus (range: 76% in Malawi to 95% in Uganda). HIV prevalence estimates from selection models diverged from IPW and MI models by up to 5% in Lesotho, without substantial precision loss. In Tanzania, the IPW model yielded lower HIV prevalence estimates among males than the best‐fitting copula selection model (3.8% vs. 7.9%).ConclusionsWe demonstrate how HIV prevalence estimates from selection models can differ from those obtained under missing‐at‐random assumptions. Further benefits include exploration of plausible relationships between participation and outcome. While selection models require additional assumptions and careful specification, they are an important tool for triangulating prevalence estimates in surveys with substantial missing data due to non‐participation.

Systemic risk measurement: A limiting threshold copula approach to CoVaR

This article proposes a novel copula-based approach to estimate CoVaR. Copula selection is based on multivariate extreme value theory rather than model fitting results. The Clayton copula as a limiting lower threshold copula is found to be able to provide a useful dependence structure for modelling the joint tail behaviour of financial returns. The out-of-sample testing of one-step quantile forecasts shows that the dynamic Clayton copula significantly outperforms the alternatives. It also shows the GJR-GARCH-Skew t model is a powerful competitor of the GPD-based models. Hence, the method provides a simple and effective way for estimating CoVaR and VaR.

Characterization of Stationary and Nonstationary Random Fields with Different Copulas on Undrained Shear Strength of Soils: Probabilistic Analysis of Embankment Stability on Soft Ground

Soil properties are known to have high spatial variability and often fluctuate with depth. The objective of this study was to investigate the effects of using different models to simulate the spatial variability of undrained shear strength (su) to calculate the failure probability of an embankment on soft ground. Two-dimensional random fields of su were generated based on one Gaussian and two non-Gaussian copulas, with stationary and nonstationary assumptions. Statistical parameters of su variation—mean, coefficient of variance, and scale fluctuation (correlation length)—were estimated from simulated and field data. Monte Carlo probabilistic analyses were performed on embankment stability based on both stationary and nonstationary random fields and all copula approaches; results showed more frequent embankment failures at low water levels in the embankment ditch. In particular, the nonstationary random field (su increases with depth) simulations more closely reflected real observed data, with higher probabilities of slope failure and lower mean factor of safety than the stationary random field simulations. Additionally, the non-Gaussian copulas provided simulated data that more accurately reflected observed field data, highlighting the importance of copula selection when characterizing soil parameter random fields.

Influence of copula approaches on reliability analysis of slope stability using random adaptive finite element limit analysis

AbstractRecently, random field copulas have been deployed to characterise the dependent structures of spatially variable soil parameters. This study expands on this concept by implementing finite element limit analysis (FELA) with adaptive meshing alongside the copula approach to perform reliability assessment of a geotechnical problem. A hypothetical slope with drained and undrained soil conditions was used as an example. First, adaptive FELA was carried out on the hypothetical slope. Second, single and multiple dependency soil parameter structures were represented by several selected copulas (Gaussian, Frank and Clayton). Finally, random adaptive FELA was implemented to explore the effects of copula selection on slope stability and reliability analyses. Results show that considerably different slope failure probabilities are produced by different copula approaches, and that failure probabilities reduce for simulating multiple dependency structures. In addition, failure probability exhibits different tendencies between drained and undrained soil conditions as soil parameter cross‐correlations become stronger. Failure probabilities are also different when performing lower and upper bound (UB) random adaptive FELA with each of the selected copulas.

Selection of mixed copula for association modeling with tied observations

Selection of mixed copula for association modeling with tied observations

Risk assessment of soil slope failure considering copula-based rotated anisotropy random fields

In natural slopes, apart from isotropic and transversely anisotropic soil fabric, rotated anisotropic soil fabric is also commonly observed due to deposition and weathering. Previous studies mainly focused on the effects of copula selection on transversely anisotropic soil slope stability. However, the influence of copula selection on rotated anisotropically deposited soil slope stability is still not clear. This study develops a new method to generate copula-based rotated anisotropy random fields for characterizing the spatial variability of soil properties. The index associated with the failure probability and consequence is employed to assess the risk of failure of two cohesive-frictional slopes in spatially variable soils (i.e., a single- and a two-layer slope). The results reveal that the commonly-used Gaussian copula may underestimate the risk of failure of both the single- and two-layer slopes at any soil deposition orientation. The risk of slope failure calculated by four candidate copulas shows a great difference. This difference is mainly affected by slope failure probability. For a single-layer cohesive-frictional slope, the slope with the highest risk of failure occurs at a rotational angle β = 150°. The corresponding soil deposition orientation (30°) is slightly smaller than the slope inclination (45°), independent of copula types. Additionally, when a slope fails, it can be found that the Gaussian copula corresponds to the largest sliding volume for some combination anisotropy cases, which is in reverse to the failure probability. Therefore, for a cohesive-frictional slope, it is suggested to consider both failure probability and failure consequence to assess the potential slope failure risk. Furthermore, the No.16 copula can give a more conservative risk assessment when dependence structure between cohesion and friction angle is uncertain.

A Bayesian nonparametric mixture model for grouping dependence structures and selecting copula functions

The demand for advanced dependence modeling arises in a variety of fields, including finance, insurance and health science. When analyzing dependent data, it is important but challenging to properly model the dependence structure in order to carry out valid and efficient inferences. Grouping the data according to the similarity in the dependence structure is necessary, especially for data of a small size. A copula-based model, indexed by copula selection indicators and dependence parameters, is introduced to delineate dependent data and group similar dependence structures. To conduct inference, a Bayesian nonparametric method with the prior distributions specified as a Dirichlet Process is proposed as a mixture of Dirichlet process mixture copula model (M-DPM-CM). Extensive simulation studies have been conducted to evaluate the performance of the proposed procedure, and the results show that the proposed M-DPM-CM can recover the true grouping structure and achieve high accuracy in copula model selection under various finite sample settings. The M-DPM-CM is applied to analyze the Vertebral Column dataset from UCI Machine Learning Repository.

Modelling mortality dependence: An application of dynamic vine copula

Vine copula, constructed from bivariate copulas, provides great flexibility in modelling complex high-dimensional dependence. When applied to multi-population mortality modelling, vine copula yields significant improvement over traditional multivariate copulas. In this paper, we propose to capture time-varying features in mortality dependence with dynamic regular vine (R-vine) copula which is built from bivariate copulas with time-varying dependence parameters. We develop two dependence dynamics for R-vine copulas and illustrate the selection and estimation of dynamic R-vine copulas using mortality data from eight populations. The estimated R-vine copulas using the proposed dependence dynamics are shown to yield better goodness of fit than both static and regime-switching vine copulas. We further demonstrate the simulation of mortality paths using dynamic R-vine copulas and examine the impact of vine copula choice on the assessed effectiveness of longevity hedge.