Types Of Copulas Research Articles

Different copula types and reliability applications for a new fisk probability model

In this research endeavor, the authors introduce a novel lifetime probability model. This distribution is meticulously examined and characterized, offering insights into its behavior and applicability in various contexts. The proposed new density function of this distribution has various heavy tail forms that are useful in the field of reliability, insurance and statistical modeling. This allows for the representation and modeling of a wide range of data sets that are diverse in their form and nature. The new distribution is characterized by having different patterns of risk or failure rates. The researchers extend the new distribution to the bivariate domain through different methods, including the Morgenstern-Farley-Gumbel distribution, the modified Morgenstern-Farley-Gumbel distribution, the famous Clayton mathematical versions, and the Rennie versions. These extensions enhance the usefulness of the proposed distribution in modeling multivariate age and reliability data and dependencies between variables. The study presents some statistical modeling experiments on reliability data and some important comparisons are presented within the framework of some statistical comparison criteria.

A copula-based approach for multi-modal demand dependence modeling: Temporal correlation between demand of subway and bike-sharing

As a representative mode of shared mobility, bike-sharing serves not only as a convenient way to conduct short-distance trips in urban areas, but also as a feeder mode to public transit, forming the Bike and Ride (BnR) system. Conducting management for such a hybrid multi-modal system faces various challenges, including the complex interactions between bike-sharing and other modes, highly dynamic passenger demand, and the difficulty of accessing direct transfer data. To overcome such difficulties, our study proposes a framework for assessing the dependency between the two usage modes. Firstly, a Dynamic-Time-Warping-based (DTW) method is utilized to determine the catchment area (CA) between the two modes, allowing the BnR-related tendency similarity under a given time scale to be considered. Then, the patterns of probabilistic dependence between travel demand of the two modes are obtained by a copula-based approach, which separates correlations under specific usage levels from single modal demands. A case study on the multi-modal system formed by docked bike-sharing and subway in New York is conducted to validate the proposed framework. The tendency similarity is found to be most pronounced within 500 m on average under a 4-hour interval. For each formed station group (SG), the best-fitted copula type is selected, capturing the strong tail correlations present only at specific usage levels. The results show a variety of different correlation patterns within SGs, despite the close geographic locations they may share. Areas of potential transfer resistance between the two modes are identified, which is more evident in first-mile-related (FMR) activities. In contrast, the two modes display more weak connections in last-mile-related (LMR) activities. The obtained results can be utilized by bike-sharing service providers to analyze demand distributions and conduct efficient station-level rebalancing. Compared to previous methods, our proposed framework is computationally inexpensive since no direct transfer of data or complex inference network is required. It incorporates statistically significant spatial–temporal information, allowing for a more accurate determination of the bi-modal assessment range. Moreover, considering that single-mode influences are mathematically removed, the resulting correlation in principle links to the strength of the connections between the two modes. Therefore, it can be assessed as an indicator of the reliability of the multi-modal system.

System reliability analysis of Cable-stayed bridge using pair Copula and ICLHS with unequal weights under seismic loading

Considering the correlation between different components, quantifying the failure probability of a high-level hyper-static structure under non-stationary seismic loading is of great significance in guiding earthquake-resistant design and fortification of bridges. In this study, a high-pier and large-span railway cable-stayed bridge in the mountainous region were employed to evaluate the seismic performance. First, the basic theoretical methods of the non-stationary random seismic motion simulation, equivalent extreme value theory based on extreme value distribution (EVD), improved Latin hypercube sampling (ICLHS) with unequal weighted fractional moment estimation and Pair Copula technique are presented. Second, the finite element analysis model of the cable-stayed bridge was established based on the OpenSees platform, the seismic response samples of the bridge were calculated under three levels of seismic ground motion to determine the first passage failure probability of the main components of the bridge under the design failure modes. Finally, the Pair Copula function was introduced to estimate the seismic system reliability of the cable-stayed bridge. The optimal Copula type that can describe the correlation between the bridge components was selected through goodness-of-fit tests such as AIC, BIC, and the minimum distance method. The failure probability of the entire system was obtained by calculating the selected Copula function type under seismic loading.

A new approach for the assessment of coastal flooding risk. Application in Rhodes island, Greece

A directional, copula-based, Monte Carlo approach is proposed for the quantitative assessment of a new definition of coastal flooding risk, namely a probability-based index due to extreme combinations of wave conditions and sea levels, which could be a key element for coastal zone planning and management. This approach differs from the standard practice, which focuses on the extreme events and studies either their specific return values or their worst-case scenarios. The adopted approach incorporates mean wave direction as a covariate into the probabilistic representation of sea states, and via the use of a variety of probabilistic and statistical tools, represents a total of five stochastic variables, namely the significant wave height, the peak wave period, the mean wave direction, the storm surge, and tide. Two directional, alternative methods have been developed; a trivariate copula- and a bivariate copula-based method, that seemed to produce quite similar results. Four types of copulas have been examined for their goodness-of-fit, i.e., three Archimedean copulas, namely, the Clayton, the Gumbel, and the Frank copulas, as well as the elliptical Gaussian copula. A comparative analysis of the two methods is performed aiming to point out their pros, as well as their accuracy and complexity issues. The application of the proposed method to the natural beaches of the Rhodes Island in Greece, in combination with a recently developed formula for wave run-up, highlighted the most crucial parameters affecting the coastal flooding risk in the study area.

Estimate a nonparametric copula density function based on probit and wavelet transforms

This study employs wavelet transforms to address the issue of boundary effects. Additionally, it utilizes probit transform techniques, which are based on probit functions, to estimate the copula density function. This estimation is dependent on the empirical distribution function of the variables. The density is estimated within a transformed domain. Recent research indicates that the early implementations of this strategy may have been more efficient. Nevertheless, in this work, we implemented two novel methodologies utilizing probit transform and wavelet transform. We then proceeded to evaluate and contrast these methodologies using three specific criteria: root mean square error (RMSE), Akaike information criterion (AIC), and log-likelihood (LogL). The wavelet transform method works better than the probit transform method at all three levels of correlation, as shown by a simulated study with four types of copulas, five sample sizes, and three levels of correlation. Research has demonstrated that probit transformation methods are most appropriate for linkages involving large and medium sample sizes, as indicated by Frank, Joe, and Tawn Copula. On the other hand, for copula functions for all sample sizes, the wavelet transform method was found to be ideal in cases with low

Vine Copula Model: Application to Chemical Elements in Water Samples

Copula can link the bivariate distribution function with marginal distribution functions without requiring specific information about the interdependence among random variables. There are several types of copulas, such as elliptical copulas, Archimedean copulas, and extreme value copulas. However, in multivariate modeling, each type of copula has limitations in modeling complex dependence structures in terms of symmetry and tail dependence properties. The class of vine copulas overcomes these limitations by constructing multivariate models using bivariate copulas in a tree-like structure. The bivariate copulas used in this study include the Clayton, Gumbel, Frank, Gaussian, and Student's t copula families. This study discusses the construction of vine copula models, parameter estimation, and their applications. The construction of vine copulas is done through the decomposition of conditional probability density functions and substituting bivariate copula density functions into the decomposition results. The data used in the study is the logarithm of the concentration of chemical elements in water samples in Colorado. The parameter estimation method used is pseudo-maximum likelihood with sequential estimation. Model selection is then performed using the Akaike information criterion (AIC) to determine the most suitable model. The results indicate that Caesium and Titanium have a dependency relationship with Scandium. Moreover, Scandium and Titanium exhibit the strongest dependence compared to other variable pairs.

Searching for cognitive specializations of neurons using mutual information framework

Of particular interest for researching the cognitive specializations of neurons is their correlation with environmental variables and animal behavior. Mutual information (MI) is a preferable method for measuring such correlations, as it allows for the assessment of non-linear relationships between variables, detects synchronization, and provides both significance and strength quantification. However, calculating MI for real data is significantly challenging. In this study, we used updated MI calculation techniques to analyze the connection between calcium fluorescence signaling and behavioral variables. Our approach encompasses novel strategies which we compiled into a software program known as INTENS (Information-Theoretic Evaluation of Neuronal Specializations), and it enabled to identify specialized neurons in mice hippocampal calcium activity data while they explored the arena with varying levels of novelty. Numerous methods exist for analyzing the relationship between neuron spikes and behavioral variables, including information-theoretical approaches [1]. Extracting information about the relationship between calcium fluorescent signals and behavior is of particular interest due to the signal’s ability to provide crucial information about subthreshold activations of the neuron. In this study, we use the GCMI Gaussian copula entropy method to calculate mutual information [2]. This method relies on the fact that mutual information between two random variables is independent of their marginal distributions and only depends on the type of copula used (a multidimensional distribution where each marginal distribution is uniform). The actual MI was compared to its corresponding values computed on the time-shifted signals for assessing the statistical significance of the computed information association between the calcium signal and the behavioral variable. Additionally, we devised a technique for gauging the strength of the coupling effect. This involved normalizing the mutual information between the fluorescence signal and the behavior with the entropy value of both variables, previously calculated as random variables. Importantly, the approach outlined earlier is effective for analyzing continuous variables such as calcium signal and animal speed, as well as pairs of continuous and discrete variables such as calcium signal and the presence or absence of grooming. The analysis of calcium signals recorded from the CA1 region of the hippocampus revealed neuronal specializations related to the animal’s external environment, such as place cells, and specializations related to its behavioral activities, including neurons activated during running, rearing, and freezing. Some neurons selectively activated in response to discrete parameters included the animal’s location within the arena (center, walls, and corners) and its speed (rest, slow, and fast). A total of 781 specializations were detected across 472 neurons throughout all four sessions of the experiment. Notably, a single neuron could have several specializations. However, more than half (55%) of the neurons were found to have only one specialization.

Neurophysiology of creativity and machine learning applications for creative process’ stages differentiation through assessment of EEG/VP signals

Of particular interest for researching the cognitive specializations of neurons is their correlation with environmental variables and animal behavior. Mutual information (MI) is a preferable method for measuring such correlations, as it allows for the assessment of non-linear relationships between variables, detects synchronization, and provides both significance and strength quantification. However, calculating MI for real data is significantly challenging. In this study, we used updated MI calculation techniques to analyze the connection between calcium fluorescence signaling and behavioral variables. Our approach encompasses novel strategies which we compiled into a software program known as INTENS (Information-Theoretic Evaluation of Neuronal Specializations), and it enabled to identify specialized neurons in mice hippocampal calcium activity data while they explored the arena with varying levels of novelty. Numerous methods exist for analyzing the relationship between neuron spikes and behavioral variables, including information-theoretical approaches [1]. Extracting information about the relationship between calcium fluorescent signals and behavior is of particular interest due to the signal’s ability to provide crucial information about subthreshold activations of the neuron. In this study, we use the GCMI Gaussian copula entropy method to calculate mutual information [2]. This method relies on the fact that mutual information between two random variables is independent of their marginal distributions and only depends on the type of copula used (a multidimensional distribution where each marginal distribution is uniform). The actual MI was compared to its corresponding values computed on the time-shifted signals for assessing the statistical significance of the computed information association between the calcium signal and the behavioral variable. Additionally, we devised a technique for gauging the strength of the coupling effect. This involved normalizing the mutual information between the fluorescence signal and the behavior with the entropy value of both variables, previously calculated as random variables. Importantly, the approach outlined earlier is effective for analyzing continuous variables such as calcium signal and animal speed, as well as pairs of continuous and discrete variables such as calcium signal and the presence or absence of grooming. The analysis of calcium signals recorded from the CA1 region of the hippocampus revealed neuronal specializations related to the animal’s external environment, such as place cells, and specializations related to its behavioral activities, including neurons activated during running, rearing, and freezing. Some neurons selectively activated in response to discrete parameters included the animal’s location within the arena (center, walls, and corners) and its speed (rest, slow, and fast). A total of 781 specializations were detected across 472 neurons throughout all four sessions of the experiment. Notably, a single neuron could have several specializations. However, more than half (55%) of the neurons were found to have only one specialization.

Extreme risk spillovers between China and major international stock markets

We examine the complex dependence structure and risk spillovers between the Chinese stock market and twelve major international markets. To this end, we employ three types of vine copulas and tests for the Granger causality in risk of Hong et al. (2009). The results indicate that the R-vine copula is the optimal model to characterize the high-dimensional dependence structure of the markets after China joined the WTO, which suggests obvious structural differences with varying degrees of mainly positive dependences. Moreover, we identify unilateral extreme risk spillovers from China to the United States, France, and Germany, and either from Japan to China. We also detect bilateral spillovers between China and the United States, Japan, as well as Australia.

Shape investigation and probabilistic representation of coastal storms. Applications to Mykonos and Barcelona

The main objective of the present work is the probabilistic representation of coastal storms’ parameters through non-parametric probability distributions that can give satisfactory estimates in the whole range of the variables’ values. Their proper probabilistic representation can provide crucial information for many applications including the probabilistic design of marine and coastal structures. Different non-parametric univariate distributions in combination with different copula types or the conditional model are applied, to check the closeness of their fit to the available bivariate coastal storm data (e.g., the storms’ maximum wave height and the associated peak wave period). Moreover, except for the classical kernel distribution, a new, recently developed Box-Cox transformed kernel method is also examined and applied. The latter has been already evaluated so far to sea states data but not to storm data. Therefore, the adopted methodology is implemented to real coastal storms in two different locations, derived from wave recordings from Mykonos (Greece) and Barcelona (Spain). Furthermore, some of the significant statistical properties of the observed coastal storms, such as the calm period, the duration, and the shape (triangular or trapezoid) are also detected. From the present analysis, the derived methodology for representing accurately coastal storms is clarified, giving thus a useful tool to engineers and researchers to consider extreme events probability of occurrence, which is essential for the design of coastal structures and consequently for the protection of coastal communities.

Partitioning the Uncertainties in Compound Hot and Dry Precipitation, Soil Moisture, and Runoff Extremes Projections in CMIP6

AbstractA comprehensive assessment of compound hot and dry extremes based on different drought conditions (low precipitation, runoff, or soil moisture) and associated uncertainties is necessary to fully understand the possible risks. Here, we analyze changes in the likelihood of compound hot and dry conditions associated with low precipitation, runoff, and soil moisture using Coupled Model Intercomparison Project Phase6 (CMIP6) simulations for present‐day climate (+1°C) and additional global warming levels (+1.5°C, +2°C, +3°C). Further, we investigate the contributions of different components (e.g., global warming levels, climate models, copula types) to the total spread in their future projections. Results show the significance of global warming levels in governing risks of rising compound hot and dry extremes. The hotspot regions include the Mediterranean, South Central America, Amazonia, and Sahara. The rising risks are also accompanied by rising uncertainty as the spread in changing likelihood is significantly contributed by Earth System Models (ESMs), global warming levels, their interactions, and the statistical estimation error. The uncertainty due to ESMs spread was observed to be most significant in the case of compound hot and low soil moisture extremes, which also corresponds to some of the most impactful conditions. It was observed that the estimation error dominates the uncertainty in compound hot and low precipitation extremes as compared to the two other combinations. Our findings indicate that the regional likelihood and associated uncertainties of compound hot‐dry events in CMIP6 projections are functions of both the selection of drought types and the methodology of deriving the joint extremes.

Extreme linkages of carbon futures, energy markets, and economic indicators: A copula approach

ABSTRACT The interdependence of carbon allowances and different energy sources in extreme market behavior is still unsettled in the literature. Using different types of static and time-varying copulas, this piece of research aims to quantify the dependence structures of Europe-based carbon future returns and selected energy future returns (i.e. coal, electricity, oil, and natural gas), and to investigate whether or not these dependence structures are influenced by economic indicators. Our results show strong evidence that time-varying parameter copulas with extreme tails are the best fit to the dependence structure. We also find that the speculation activity and the uncertainty of the state of the global economy are two important components of this robust dependence structure in the period of oil price crises. These findings are relevant for the implementation of effective policies to make the carbon market operate more efficiently and stably.

Review of Copula for Bivariate Distributions of Zero-Inflated Count Time Series Data

The class of bivariate integer-valued time series models, described via copula theory, is gaining popularity in the literature because of applications in health sciences, engineering, financial management and more. Each time series follows a Markov chain with the serial dependence captured using copula-based distribution functions from the Poisson and the zero-inflated Poisson margins. The copula theory is again used to capture the dependence between the two series.
 
 However, the efficiency and adaptability of the copula are being challenged because of the discrete nature of data and also in the case of zero-inflation of count time series. Likelihood-based inference is used to estimate the model parameters for simulated and real data with the bivariate integral of copula functions. While such copula functions offer great flexibility in capturing dependence, there remain challenges related to identifying the best copula type for a given application.  This paper presents a survey of the literature on bivariate copula for discrete data with an emphasis on the zero-inflated nature of the modelling. We demonstrate additional experiments on to confirm that the copula has potential as greater research area.

Identification and Risk Characteristics of Agricultural Drought Disaster Events Based on the Copula Function in Northeast China

Accurate feature identification of drought disaster events is required for proper risk management in agriculture. This study improved the crop water deficit index (CWDI) by including the daily meteorological, crop development stage, soil moisture content, and yield data for 1981–2020 in northeastern China. Two drought characteristic variables (drought duration and intensity) were extracted using the theory of runs to produce the improved crop water deficit index (CWDIwp). Thresholds for the bivariate indicators were also determined for agricultural drought events of varying severity. A joint distribution model for drought variables was constructed based on five types of Archimedean copulas. The joint probability and the joint recurrence period for agricultural drought events were analyzed for drought events with varying intensities in northeast China. The results suggest that the CWDIwp can reliably identify the onset, duration, and intensity of drought events over the study area and can be used to monitor agricultural drought events. The conditional probability of drought intensity (duration) decreased as the drought duration (intensity) threshold increased, whereas the drought recurrence period increased as the threshold for drought duration and intensity rose. In the period (1981–2020), drought intensity in the three Northeastern provinces showed an increasing trend in the order Jilin Province > Liaoning Province > Heilongjiang Province. The spatial distribution of the joint probability and the joint recurrence period was obvious, and the joint probability showed a decreasing distribution trend from west to east. The distribution trend for the joint probability was opposite to that of the joint recurrence period. Furthermore, the areas with high drought probability values corresponded to the areas with low values for the recurrence period, indicating that the drought occurrence probability was higher, and the recurrence period value was lower in the drought-prone areas. The high-risk drought areas (60–87%) were in western Liaoning and western Jilin, with a recurrence period of 1–3 years, whereas the low-risk areas (<40%) were located in the mountainous areas of eastern Liaoning and eastern Jilin. The joint probability and joint recurrence period for agricultural drought events of varying severity were quite different, with the probability following the order light drought > moderate drought > severe drought > extreme drought. The order for the recurrence period was light drought < moderate drought < severe drought < extreme drought. The results provide technical support for disaster prevention and mitigation in drought risk management.

Copula Modelling to Analyse Financial Data

Copula modelling is a popular tool in analysing the dependencies between variables. Copula modelling allows the investigation of tail dependencies, which is of particular interest in risk and survival applications. Copula modelling is also of specific interest to economic and financial modelling as it can help in the prediction of financial contagion and periods of “boom” or “bust”. Bivariate copula modelling has a rich variety of copulas that may be chosen to represent the modelled dataset dependencies and possible extreme events that may lie within the dataset tails. Financial copula modelling tends to diverge as this richness of copula types within the literature may not be well realised with the two different types of modelling, one being non-time-series and the other being time-series, being undertaken differently. This paper investigates standard copula modelling and financial copula modelling and shows why the modelling strategies in using time-series and non-time-series copula modelling is undertaken using different methods. This difference, apart from the issues surrounding the time-series component, is mostly due to standard copula modelling having the ability to use empirical CDFs for the probability integral transformation. Financial time-series copula modelling uses pseudo-CDFs due to the standardized time-series residuals being centred around zero. The standardized residuals inhibit the estimation of the possible distributions required for constructing the copula model in the usual manner.

Bivariate bias correction of the regional climate model ensemble over the Adriatic region

AbstractThe main goal of this analysis is to provide an assessment of the fine grid regional climate model (RCM) ensemble over the broader Adriatic region, which is the domain with orographically complex terrain and a developed coastline. The results of this study are a beneficial complement to earlier work on using modified bivariate bias correction methods. This analysis was based on an ensemble of 12 combinations of three RCMs and four global climate models (GCMs) from 1971 to 2004, divided into calibration and validation parts. A comparison of one univariate and three versions of bivariate bias correction methods of summer and winter monthly temperature and precipitation was performed. To examine the effect of various methods, we used different marginal distributions and distributions of dependence (copula). Bivariate bias correction was conducted with parametric and empirical marginal distributions. In the quantile mapping (univariate) and the parametric bivariate method, gamma (precipitation) and normal (temperature) probability distributions were used. We documented the bias and the influence of bias correction methods on precipitation and temperature, including their climate change signals in historical data. The considered methods maintained the spatial distribution of trends from the native ensemble. If the relationship between variables was not distinct and the correlation coefficient was low, bivariate bias correction methods tended to weaken the correlation coefficient from the model. Two experiments on how the bias correction methods influence statistical measures of the considered variables and their relationships showed that the bivariate method with empirical distributions (eeG) performed slightly better than other methods. Although the eeG method performed better than the other methods, none of the considered methods was able to fully preserve the correlation coefficient between variables detected from observations. Considering two copulas (Gaussian and Student's t), we conclude that the effect of different copula types on method performance is small.

Modeling asymmetrically dependent multivariate ocean data using truncated copulas

Characterizing multivariate ocean parameters is quite important for offshore engineering reliability design and risk assessment. To fully understand ocean conditions, a robust and accurate multivariate model is essential for the analysis and estimation of the ocean state. Therefore, advanced simulation of the ocean parameters helps to improve practices in offshore engineering. In this work, the principle of a new type of copula, namely truncated copula, is developed and adopted for modeling the multivariate ocean data. Unlike previous studies on modeling asymmetric ocean data by purely mathematical fitting techniques, this study proposes a truncated method based on physical limits to study asymmetrically dependent ocean data. The truncated copula method is contrasted with the conventional symmetric and existing asymmetric copula from the literature using real environmental observations for the demonstration. Various commonly used traditional copula models are modified by the proposed truncation technique and applied to fit multivariate ocean data collected in buoys off the US coast. Based on the fitting of ocean data, this paper compares the advantages and disadvantages of different copula models. The properties of different copula models for data simulation and extreme value prediction are also discussed.

A generalized computationally efficient copula-polynomial chaos framework for probabilistic power flow considering nonlinear correlations of PV injections

This paper develops a general computationally efficient copula-polynomial chaos expansion (copula-PCE) framework for power system probabilistic power flow that can handle both linear and nonlinear correlations of uncertain power injections, such as wind and PVs. A data-driven copula statistical model is used to capture both nonlinear and linear correlations of uncertain power injections. This allows us to resort to the Rosenblatt transformation to transform correlated variables into independent ones while preserving the dependence structure. It paves the way for leveraging PCE for surrogate modeling and uncertainty quantification of power flow results. To further improve the computational efficiency of the proposed method without sacrificing accuracy, two strategies are developed and unified together. Specifically, the branch flow sensitivity analysis (BFSA) is used to select an appropriate number of outputs that need PCE surrogate modeling while the analysis of variance (ANOVA) scheme enables us to reduce the number of basis functions and coefficients evaluations for each PCE surrogate. Simulations carried out on the IEEE 57-bus and 118-bus and PEGASE 1354-bus systems show that the proposed framework can obtain better accuracy and computational efficiency than the existing PCE-based methods with linear correlation assumption and other Monte Carlo-based methods. The sensitivities to different copula types and parameters are also investigated.

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 Mixture of Regular Vines for Multiple Dependencies

To uncover complex hidden dependency structures among variables, researchers have used a mixture of vine copula constructions. To date, these have been limited to a subclass of regular vine models, the so-called drawable vine, fitting only one type of bivariate copula for all variable pairs. However, the variation of complex hidden correlations from one pair of variables to another is more likely to be present in many real datasets. Single-type bivariate copulas are unable to deal with such a problem. In addition, the regular vine copula model is much more capable and flexible than its subclasses. Hence, to fully uncover and describe complex hidden dependency structures among variables and provide even further flexibility to the mixture of regular vine models, a mixture of regular vine models, with a mixed choice of bivariate copulas, is proposed in this paper. The model was applied to simulated and real data to illustrate its performance. The proposed model shows significant performance over the mixture of R-vine densities with a single copula family fitted to all pairs.