Multivariate Gaussian Copula Research Articles

Bayesian ridge estimators based on a vine copula-based prior in Poisson and negative binomial regression models

Poisson and negative binomial regressions are popular methods for modelling the relationship between a count variable and explanatory variables. In the presence of multicollinearity, ridge regression is an alternative to the maximum likelihood estimation for regression coefficients. Furthermore, ridge estimators are interpreted as the Bayesian posterior mean (or mode) when the regression coefficients follow a multivariate normal prior. However, using the multivariate normal prior may not effectively estimate regression coefficients, especially in the presence of interaction terms. This study proposes vine copula-based priors for Bayesian ridge estimators in Poisson and negative binomial regression models. The simulations and data analysis results indicate that for the Poisson model with equidispersion and the negative binomial model with overdispersion, the Clayton and Gumbel copula priors of the Archimedean family achieve superior performance than the multivariate normal prior and Gaussian copula prior.

Probabilistic steady-state and short-term voltage stability assessment considering correlated system uncertainties

As correlated uncertainties in power grids become more prevalent, modeling of stochastic dependencies is becoming increasingly necessary. Independent probability distributions disregard these correlated uncertainties, potentially leading to significant errors, underscoring the necessity of dependence structures. This paper presents a probabilistic approach for modeling uncertain system parameters with their inherent correlations. The proposed approach is validated through various aspects of system voltage measures, including voltage profile and static and dynamic voltage stability. In total, nine alternative sampling generation techniques are employed, and their accuracy is measured based on real-world data using root mean square error (RMSE) and coefficient of determination (R2) criteria. The result suggests that multivariate (Gaussian and Student's) copula techniques accurately represent the real system data, consistently achieving high accuracy rates of 98 % for voltage profiles, 97 % for static voltage stability, and 93 % for dynamic voltage stability. In contrast, the independent sampling technique failed to follow the real-world system data for the different aspects of system voltage measures.

Fuzzy semi-entropy based downside risk to low-carbon oriented multi-energy dispatch considering multiple dependent uncertainties

Increasing deployment of wind power into integrated energy system becomes a critical pathway for deep decarbonizing the power sector. It is imperative to investigate the impacts of uncertain wind power on integrated power system as the result of the heavy interdependencies among different types of energy in supplying the requirements of economy, low carbon and risk aversion. Here, the concept of fuzzy semi-entropy is presented to quantify the downside risk of uncertain wind power. After that, a mean-semi-entropy model for low-carbon oriented electrical-gas-heat energy system including carbon emission, carbon capture and carbon trading is proposed. In particular, the dependence of wind power outputs from different wind turbines is investigated by the multivariate Gaussian Copula. Results obtained from case studies demonstrate the feasibility of the proposed model in achieving a balanced trade-off among system economy, low carbon emission and operation risk properly. The proposed model enables the reduction in total cost and carbon emission by 2.29% and 5.18% in comparison to these of traditional cost minimization model. In addition, the accommodation level of wind power with multivariate Gaussian Copula improves by 2.17% compared to that without considering coupling.

Stochastic Flood Simulation Method Combining Flood Intensity and Morphological Indicators

The existing flood stochastic simulation methods are mostly applied to the stochastic simulation of flood intensity characteristics, with less consideration for the randomness of the flood hydrograph shape and its correlation with intensity characteristics. In view of this, this paper proposes a flood stochastic simulation method that combines intensity and morphological indicators. Using the Foziling and Xianghongdian reservoirs in the Pi River basin in China as examples, this method utilizes a three-dimensional asymmetric Archimedean M6 Copula to construct stochastic simulation models for peak flow, flood volume, and flood duration. Based on K-means clustering, a multivariate Gaussian Copula is employed to construct a dimensionless flood hydrograph stochastic simulation model. Furthermore, separate two-dimensional symmetric Copula stochastic simulation models are established to capture the correlations between flood intensity characteristics and shape variables such as peak shape coefficient, peak occurrence time, rising inflection point angle, and coefficient of variation. By evaluating the fit between the simulated flood characteristics and the dimensionless flood hydrograph, a complete flood hydrograph is synthesized, which can be applied in flood control dispatch simulations and other related fields. The feasibility and practicality of the proposed model are analyzed and demonstrated. The results indicate that the simulated floods closely resemble natural floods, making the simulation outcomes crucial for reservoir scheduling, risk assessment, and decision-making processes.

Costa’s concavity inequality for dependent variables based on the multivariate Gaussian copula

AbstractAn extension of Shannon’s entropy power inequality when one of the summands is Gaussian was provided by Costa in 1985, known as Costa’s concavity inequality. We consider the additive Gaussian noise channel with a more realistic assumption, i.e. the input and noise components are not independent and their dependence structure follows the well-known multivariate Gaussian copula. Two generalizations for the first- and second-order derivatives of the differential entropy of the output signal for dependent multivariate random variables are derived. It is shown that some previous results in the literature are particular versions of our results. Using these derivatives, concavity of the entropy power, under certain mild conditions, is proved. Finally, special one-dimensional versions of our general results are described which indeed reveal an extension of the one-dimensional case of Costa’s concavity inequality to the dependent case. An illustrative example is also presented.

Integrated energy operation considering the dependence of multiple wind turbines

A key pathway to deep decarbonization of the energy sector is the increased use of wind power in integrated energy systems in the situation of the demand for low-carbon development. However, existing work on wind power in the integrated energy system does not adequately take into account the low carbon emissions in the context of multiple wind turbine dependency. In this respect, it is essential to examine a new type of framework for coordinated operations for meeting operational, low carbon and dependence requirements of multiple wind turbines. Here, a low-carbon oriented electrical-gas-heat energy system including carbon emission, carbon capture and carbon trading is proposed. In particular, the multivariate Gaussian copula in the region is used to investigate the dependence of power output from multiple wind turbines. Results from case studies demonstrate the feasibility of the proposed model in achieving a balanced weigh up between economics and low carbon emissions by the dependence on multiple wind turbines.

An approach based on multivariate distribution and Gaussian copulas to predict groundwater quality using DNN models in a data scarce environment

Machine Learning models have become a fruitful tool in water resources modelling. However, it requires a significant amount of datasets for training and validation, which poses challenges in the analysis of data scarce environments, particularly for poorly monitored basins. In such scenarios, using Virtual Sample Generation (VSG) method is valuable to overcome this challenge in developing ML models. The main aim of this manuscript is to introduce a novel VSG based on multivariate distribution and Gaussian Copula called MVD-VSG whereby appropriate virtual combinations of groundwater quality parameters can be generated to train Deep Neural Network (DNN) for predicting Entropy Weighted Water Quality Index (EWQI) of aquifers even with small datasets. The MVD-VSG is original and was validated for its initial application using sufficient observed datasets collected from two aquifers. The validation results showed that from only 20 original samples, the MVD-VSG provided enough accuracy to predict EWQI with an NSE of 0.87. However the companion publication of this Method paper is El Bilali et al. [1].•Development of MVD-VSG to generate virtual combinations of groundwater parameters in data scarce environment.•Training deep neural network to predict groundwater quality.•Validation of the method with sufficient observed datasets and sensitivity analysis.

Joint modelling of longitudinal measurements and survival times via a multivariate copula approach

ABSTRACT Joint modelling of longitudinal and time-to-event data is usually described by a joint model which uses shared or correlated latent effects to capture associations between the two processes. Under this framework, the joint distribution of the two processes can be derived straightforwardly by assuming conditional independence given the random effects. Alternative approaches to induce interdependency into sub-models have also been considered in the literature and one such approach is using copulas to introduce non-linear correlation between the marginal distributions of the longitudinal and time-to-event processes. The multivariate Gaussian copula joint model has been proposed in the literature to fit joint data by applying a Monte Carlo expectation-maximisation algorithm. In this paper, we propose an exact likelihood estimation approach to replace the more computationally expensive Monte Carlo expectation-maximisation algorithm and we consider results based on using both the multivariate Gaussian and t copula functions. We also provide a straightforward way to compute dynamic predictions of survival probabilities, showing that our proposed model is comparable in prediction performance to the shared random effects joint model.

Multivariate Gaussian Copula Mutual Information to Estimate Functional Connectivity with Less Random Architecture.

Recognition of a brain region’s interaction is an important field in neuroscience. Most studies use the Pearson correlation to find the interaction between the regions. According to the experimental evidence, there is a nonlinear dependence between the activities of different brain regions that is ignored by Pearson correlation as a linear measure. Typically, the average activity of each region is used as input because it is a univariate measure. This dimensional reduction, i.e., averaging, leads to a loss of spatial information across voxels within the region. In this study, we propose using an information-theoretic measure, multivariate mutual information (mvMI), as a nonlinear dependence to find the interaction between regions. This measure, which has been recently proposed, simplifies the mutual information calculation complexity using the Gaussian copula. Using simulated data, we show that the using this measure overcomes the mentioned limitations. Additionally using the real resting-state fMRI data, we compare the level of significance and randomness of graphs constructed using different methods. Our results indicate that the proposed method estimates the functional connectivity more significantly and leads to a smaller number of random connections than the common measure, Pearson correlation. Moreover, we find that the similarity of the estimated functional networks of the individuals is higher when the proposed method is used.

Capital Requirements to Cover Operational Risk in Financial Institutions of Emerging Markets. A Gaussian Copula Model

Context: Advanced Measurement Approach (AMA) has been the umbrella to identify the models used for modeling the capital to cover Operational Risk (Total Operational Value at Risk, OpVaR) in financial institutions in developed countries. The Loss Distribution Approach (LDA) has been the most popular model used by international banks for OpVaR calculation. However, the operational losses frequently have multivariate dependences that are not accounted for in the LDA. This paper applies a Gaussian copula to model the multivariate dependences in operational losses. Method: Two models were compared to estimate capital requirement for operational risk. Model (i) is the standard LDA model (BCBS 2004). Model (ii) incorporates a multivariate Gaussian copula into the LDA to model multivariate dependence between operational losses (severities). This research analyzes an operational loss data set, SAS® Operational Risk Global Data (SAS OpRisk Global Data), in order to model operational risk at financial institutions in emerging markets between 1990 and 2013. Results: The impact of Model (ii) was evaluated on the estimates of the total regulatory capital for operational risk and compared with the one predicted by (i). The results confirm the existence of diversification benefit up to 33%. Conclusions: Modeling explicitly the multivariate dependence between operational losses has a clear impact on capital requirement for institutions in emerging markets. The implementation of a Gaussian copula into the LDA model provides a sophisticated tool to estimate operational risk capital in emerging markets, as well as the possibility for diversification benefit. Acknowledgements: To SAS for providing the database used in this research.

A novel generative approach for modal frequency probabilistic prediction under varying environmental condition using incomplete information

Modal frequency (MF) prediction under varying environmental condition (EC) has received tremendous attention. Most existing methods for MF prediction, belonging to discriminative approaches, collapse when information of EC is incomplete in the prediction stage. Generative approaches attempt to construct a joint PDF of uncertain MF and EC, and then predict MF under varying EC by conducting operations of marginalization and conditioning using the joint PDF. There are two fundamental difficulties in this process. The first difficulty is joint PDF modelling of multivariate MF and EC as they are too irregular to be depicted by traditional multivariate distributions. The second difficulty is predictive PDF derivation of MF conditional on incomplete information of EC as operations of marginalization and conditioning in the high-dimensional case require huge computational cost. Aiming to resolve these two difficulties of generative approaches, this paper proposes BAyeSIan Copula-based Uncertainty Quantification (BASIC-UQ). BASIC-UQ contains three stages. Stage I introduces probabilistic model class candidates. Stage II conducts Bayesian inference on parameters and model class candidates with embedding the idea of inference functions for margins. The optimal parameters are determined in a decoupled way, and the most plausible univariate marginal PDF of each random variable (RV) along with the most plausible copula are selected. Stage III derives predictive PDF of MF conditional on incomplete information of EC. The exact solutions are derived for the multivariate Gaussian copula and the multivariate Student’s t copula. Examples using simulated data and SHM data are presented to illustrate the capability of BASIC-UQ in joint PDF modelling of MF and EC, and predictive PDF derivation of MF using incomplete information of EC.

Probabilistic Forecasting Based Sizing and Control of Hybrid Energy Storage for Wind Power Smoothing

With the increasing wind power integration, the security and economy of the power system operations are greatly influenced by the intermittency and fluctuation of wind power. Due to the flexible operational modes for charging/discharging, the hybrid energy storage system (HESS) is composed of battery energy storage system and super-capacitor can effectively mitigate the wind power uncertainty. This paper proposes a probabilistic forecasting-based HESS sizing and control scheme to cost-effectively smooth wind power fluctuations. First, probabilistic wind power forecasting is combined with multivariate Gaussian copula to generate temporally correlated wind power scenarios. Then, an adaptive variational mode decomposition (VMD) method is proposed to extract the frequency components of each wind scenario and determine the pre-scheduled power of wind-HESS system adaptively. Finally, a two-stage stochastic optimization model is constructed to determine the capacity and correct the pre-scheduled power of HESS with the objective of minimizing total cost. Case studies based on actual data from a Danish wind farm demonstrate that the proposed HESS sizing and control scheme can significantly reduce the installation cost and operation cost of HESS and prominently smooth wind power fluctuations.

Copula-based joint modeling of crash count and conflict risk measures with accommodation of mixed count-continuous margins

This current study proposes to model crash count and conflict risk measures jointly by developing a multivariate copula-based modeling framework. As conflict risk measures can either be event counts or continuous random variables, the proposed framework is devised to accommodate mixed count-continuous margins. Specifically, three longitudinal conflict risk measures extracted from real-world connected vehicle data collected in Ann Arbor, Michigan as well as rear-end crash count are modeled via the multivariate Gaussian copula. The presence of stronger dependences among conflict risk measures than those between conflict risk measures and crash count are revealed and the dependency structure is relatively stable across different conflict risk measure threshold values and sample sizes. Comparing to the existing crash count and conflict risk measures modeling approaches, the proposed modeling framework also contributes to the transportation safety literature by (a) better reflecting safety by treating conflict risk measures and crashes equally; (b) better accounting for the impact of potential unobserved factors on crash count and conflict risk measures simultaneously; and (c) better accounting for the exposure and traffic risk factors and their heterogenous impact on crash count and conflict risk measures. For practical applications, the proposed copula-based approach not only achieves better in-sample crash count prediction accuracies across different sample sizes of conflict risk measure comparing to several classical crash frequency models in literature, but also connects with the commonly used high-risk location ranking measure, namely potential for safety improvement, for high-risk location identification as there is great similarity between the marginal cumulative distribution functions that constitute the copula and the potential for safety improvement. Based on the total rank differences test, the copula-based high-risk location ranking measure shows the potential to identify high-risk locations that are risky by both crash count and conflict risk measures simultaneously.

Multivariate Bayesian hierarchical Gaussian copula modeling of the non-stationary traffic conflict extremes for crash estimation

Recent studies have demonstrated that single conflict indicators represent only a fractional aspect of the severity of a traffic interaction. As such, integrating several conflict indicators in a unified model can improve conflict-based crash estimation. This study develops a multivariate Bayesian hierarchical Gaussian copula modeling approach, which comprises a multivariate Gaussian copula and a Bayesian hierarchical structure. The former has generalized extreme value marginals and captures multivariate dependence among several conflict indicators, while the latter combines traffic conflicts from different sites, incorporating several covariates and site-specific unobserved heterogeneity. The copula approach offers the flexibility in modeling multivariate structures including the selection of margins from various parametric families of univariate distribution and the construction of parametric copulas which satisfies different types of dependence structure. A model estimation approach for the multivariate Bayesian hierarchical Gaussian copula model is proposed and applied to estimate rear-end crashes from four signalized intersections in the city of Surrey, British Columbia. The modified time to collision (MTTC), post encroachment time (PET), and deceleration rate to avoid a crash (DRAC) were employed as conflict indicators. Three covariates including traffic volume, shock wave area, and platoon ratio were considered to account for non-stationarity in conflict extremes. The results show that in terms of crash estimation accuracy, the multivariate Bayesian hierarchical Gaussian copula model outperforms both the bivariate Bayesian hierarchical Gaussian copula models and the recently proposed multivariate Bayesian hierarchical model in which multiple traffic conflict indicators were combined by one dependence parameter.

Developing a hybrid modeling and multivariate analysis framework for storm surge and runoff interactions in urban coastal flooding

The formation of urban coastal flooding is mainly ruled by the interaction between rainfall-runoff and storm surge. This study aims to advance the understanding of coastal urban flood mechanism by developing an integrated modeling and multivariate analysis framework, which involves a hydrologic model (Storm Water Management Model (SWMM)), as the core model, coupled with a coastal hydrodynamic model (Delft3D). The uncertainty associated with the flood depth prediction by integrated models is analyzed using the multivariate Gaussian Copula. The performance of the integrated modeling framework is evaluated for the Chittagong City of Bangladesh, which has experienced extreme and frequent coastal urban floods. Results from modeling indicate that changes in the tidal phase of coastal urban flooding alter the flood’s duration and depth. The intensity of compound flooding is higher for the co-occurrence of rainfall and surge peaks than the occurrence of both events in succession. The average flood duration and depth can be increased by about 2.5 h and 0.24 m, respectively, during compound events. When the storm surge occurs during the transition phase, between high/low tides (2–4 h before peak low/high tide), the duration of flood extends due to longer surge duration (4–4.5 h). Finally, the multivariate Gaussian Copula model adjusts the integrated modeling outputs and enhances the skill to predict the inundation depth by 4.6–24.3%. The findings of this study are critical for a better understanding of coastal urban flood processes and enhancing the informed decision-making for emergency management and planning in low-lying coastal regions.

Probability-space surrogate modeling for fast multidisciplinary optimization under uncertainty

This paper proposes a probability-space surrogate modeling approach for computationally efficient multidisciplinary design optimization under uncertainty. This paper uses a probability-space surrogate as opposed to an algebraic surrogate so that the probability distributions of the required outputs at a given design input can naturally be obtained without repeated Monte Carlo runs of an algebraic surrogate at different realizations of the uncertain variables. We consider three probability-space surrogates with analytical solutions for prediction and inference - Multivariate Gaussian, Gaussian Copula, and Gaussian Mixture Model, and investigate their applicability to perform multidisciplinary design optimization under uncertainty. All the input design and random variables, coupling variables, objective and constraint functions are incorporated within the probability-space surrogate, which helps analytically obtain the distributions of coupling variables, objective and constraint functions at desired design inputs while enforcing multidisciplinary compatibility. The training points for the probability-space surrogates are obtained by performing one-pass analysis through the disciplinary models at different realizations of the input variables. The proposed methodology is demonstrated for reliability-based design optimization (RBDO) and reliability-based robust design optimization (RBRDO) in an aircraft design example. The performance of the probability-space surrogates is compared against a Kriging algebraic surrogate and fully coupled Monte Carlo analysis.

Dynamic reliability prediction for the steel box girder based on multivariate Bayesian dynamic Gaussian copula model and SHM extreme stress data

This article presents the dynamic reliability prediction method of the existing steel box girder considering the time-variant nonlinear correlations among the performance functions for the failure modes at the multiple control monitoring points. Firstly, the multivariate Bayesian dynamic linear model (MBDLM) considering the nonlinear correlations among the multiple variables is built to predict the extreme stresses at the different control monitoring points; secondly, based on the predicted covariance matrix of MBDLM, the dynamic correlation coefficients between any two performance functions can be accurately predicted; and finally, multivariate Bayesian dynamic Gaussian copula model through combining MBDLM with Gaussian copula technique is proposed to predict the dynamic reliability of the steel box girder, and the monitoring extreme stress data of an actual bridge are provided to illustrate the feasibility and application of the proposed method. The results show that predicted dynamic reliability of the bridge girder with considering the time-variant nonlinear correlation of failure modes at the multiple control monitoring points is bigger than that without considering the time-dependent nonlinear correlation. It is illustrated that the predicted results without considering the dynamic nonlinear correlation of failure modes at the multiple control monitoring points are more conservative.

Locally Private High-Dimensional Crowdsourced Data Release Based on Copula Functions

With the increasing popularity of crowdsourcing services, high-dimensional crowdsourced data provides a wealth of knowledge. Nonetheless, unprecedented privacy threats to participants have emerged, due to complex correlations among multiple attributes and the vulnerabilities of untrusted crowdsourcing servers. Differential privacy-based paradigms have been proposed to release privacy-preserving datasets with statistical approximation. Nonetheless, most existing schemes are limited when facing highly correlated attributes, and cannot prevent privacy threats from untrusted crowdsourcing servers. To address this issue, we propose two novel solutions, namely <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">LoCop</i> and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">DR_LoCop</i> , which guarantee local differential privacy based on the randomized response technique while synthesizing and releasing high-dimensional crowdsourced data with high data utility. Particularly, <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">LoCop</i> leverages copula theory to synthesize high-dimensional crowdsourced data via univariate marginal distribution and attribute dependence. Univariate marginal distribution is estimated by the Lasso-based regression algorithm from aggregated privacy-preserving bit strings. Dependencies among attributes are modeled as multivariate Gaussian copula. Based on <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">LoCop</i> , the enhanced solution <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">DR_LoCop</i> not only takes advantage of C-vine copula to reflect conditional dependencies among high-dimensional attributes, but also achieves dimension reduction. Extensive experiments on real-world datasets demonstrate that our solutions substantially outperform the state-of-the-art techniques in terms of both data utility and computational overhead.

M-Vine decomposition and VAR(1) models

The M-Vine decomposition of high-dimensional first-order Vector AutoRegressive[VAR(1)] models is detailed by representing a VAR(1) with a Multivariate Gaussian Copula (MGC), building VAR(1) models with MGCs, decomposing MGCs following M-Vine structures, and reconstructing an MGC from an M-Vine structure.

A general framework for coupled hydro-mechanical modelling of rainfall-induced instability in unsaturated slopes with multivariate random fields

An accurate estimation of rainfall-induced instability of slopes for extremely nonhomogeneous materials such as lignite mine spoils is a major challenge. This paper investigates the stability of nonhomogeneous soil slopes with respect to slip surface development, size of sliding volume, and determination of safety factor. Specified dependent random variables are cross-correlated using a multivariate Gaussian copula, the use of which provides a faster and more accurate representation of the inter-dependent properties of randomly-distributed soil. A Monte-Carlo simulation is used to generate a series of multivariate random fields for slopes. These are then implemented in Abaqus and analysed under constant rainfall conditions using a fully coupled hydro-elasto-plastic model. The resulting stress, strain, pore pressure, and displacement data are further processed in MATLAB to evaluate critical slip surfaces and safety factors. Results indicate that the factor of safety in a homogenous case is overestimated compared to the nonhomogeneous condition, while the sliding volume is underestimated. Moreover, the factor of safety decreases as the rainfall simulation continues and the probability of failure increases to nearly 100% after 10 days of rainfall. The framework developed in this paper can provide guidance for conducting relatively inexpensive probabilistic analyses.