Multivariate Gaussian Copula Research Articles

Regression for doubly inflated multivariate Poisson distributions

ABSTRACTDependent multivariate count data occur in several research studies. These data can be modelled by a multivariate Poisson or Negative binomial distribution constructed using copulas. However, when some of the counts are inflated, that is, the number of observations in some cells are much larger than other cells, then the copula-based multivariate Poisson (or Negative binomial) distribution may not fit well and it is not an appropriate statistical model for the data. There is a need to modify or adjust the multivariate distribution to account for the inflated frequencies. In this article, we consider the situation where the frequencies of two cells are higher compared to the other cells and develop a doubly inflated multivariate Poisson distribution function using multivariate Gaussian copula. We also discuss procedures for regression on covariates for the doubly inflated multivariate count data. For illustrating the proposed methodologies, we present real data containing bivariate count observations with inflations in two cells. Several models and linear predictors with log link functions are considered, and we discuss maximum likelihood estimation to estimate unknown parameters of the models.

Identification of Critical Parameters Affecting Voltage and Angular Stability Considering Load-Renewable Generation Correlations

The renewable energy source based generating technologies and flexible demand and storage devices exhibit significant temporal and spatial uncertainties in generating and loading profiles and introduce additional level of uncertainty in network operation. The dynamic behaviors of such a network can be affected and the stable operation may be compromised. This paper proposes a probabilistic analysis approach for the evaluation of the effect of uncertain parameters on power system voltage and angular stability. Load margin, the damping of critical eigenvalues and the transient stability index have been chosen as the relevant stability indices for voltage stability, small-disturbance stability, and transient stability analysis, respectively. The Morris screening sensitivity analysis method coupled with a multivariate Gaussian copula to account for parameter correlations is used for the priority ranking of uncertain parameters. The approach is illustrated on a number of case studies using modified IEEE 68-bus NETS-NYPS test system. The results obtained in this paper reveal that the critical parameters appear as groups if the input dataset is correlated, and, hence even a parameter (which may be uninfluential individually) can have a significant impact on system dynamic behavior due to its correlation with other influential parameters.

A novel method for binarization of scene text images and its application in text identification

The aim of this article is twofold. First, we propose an effective methodology for binarization of scene images. For our present study, we use the publicly available ICDAR 2011 Born Digital Data set. We introduce a new concept of variance map of a gray-level image for detection of text boundary in an image. Based on this boundary information, the image is binarized by means of adaptive thresholding. This binarization procedure produces a number of connected components. Next, these connected components are examined in order to identify possible text components. In this context, a number of shape-based features that distinguish between text and non-text components are proposed. We consider text component identification as an one-class classification problem, i.e., the ground truth information for only the text class is available for the ICDAR 2011 Born Digital Data set. Then, the ground truth text components are used to obtain a certain statistical distribution of the shape-based features. Here, we observe that all the features may not follow a single family of distributions. Therefore, we construct a joint distribution by using multivariate Gaussian copula which allows a coupling of different marginal distributions. As our experiments suggest, the copula-based method is superior to multivariate Gaussian distribution in describing the feature distribution. Finally, a text connected component of an unknown class is subjected to the trained statistical model, and by performing a hypothesis test we successfully identify a possible text component. For a comparative study, we consider a number of state-of-the-art methods. Our proposed approach significantly outperforms most of these methods in terms of recall, precision and F-measure in both the binarization and text identification tasks.

Influence of Stochastic Dependence on Small-Disturbance Stability and Ranking Uncertainties

A high level of stochastic dependence (or correlation) exists between different uncertainties (i.e., loads and renewable generation), which is nonlinear and non-Gaussian and it affects power system stability. Accurate modeling of stochastic dependence becomes more important and influential as the penetration of correlated uncertainties (such as renewable generation) increases in the network. The stochastic dependence between uncertainties can be modeled using 1) copula theory and 2) joint probability distributions. These methods have been implemented in this paper and their performances have been compared in assessing the small-disturbance stability of a power system. The value of modeling stochastic dependence with increased renewables has been assessed. Subsequently, the critical uncertainties that most affect the damping of the most critical oscillatory mode have been identified and ranked in terms of their influence using advanced global sensitivity analysis techniques. This has enabled the quantification and identification of the impact of modeling stochastic dependence on the raking of critical uncertainties. The results suggest that multivariate Gaussian copula is the most suitable approach for modeling correlation as it shows consistently low error even at higher levels of renewable energy penetration into the power system.

Modeling with a large class of unimodal multivariate distributions

ABSTRACTIn this paper we introduce a new class of multivariate unimodal distributions, motivated by Khintchine's representation for unimodal densities on the real line. We start by introducing a new class of unimodal distributions which can then be naturally extended to higher dimensions, using the multivariate Gaussian copula. Under both univariate and multivariate settings, we provide MCMC algorithms to perform inference about the model parameters and predictive densities. The methodology is illustrated with univariate and bivariate examples, and with variables taken from a real data set.

Roles of load temporal correlation and deterioration-load dependency in structural time-dependent reliability

Aging of structural performance and significant external loads may impair structural safety and serviceability and cause potential economic losses. In the presence of uncertainties associated with both resistance deterioration and external loads, structural safety shall be estimated quantitatively under a probability-based framework. A stochastic load process is often auto-correlated on the temporal scale, with correlations arising from both the occurrence times and intensities. Moreover, a deterioration process is physically dependent on the load magnitudes. This paper investigates the impacts of load temporal correlation and deterioration-load dependency on time-variant structural reliability. The load occurrence process is modeled as a Poisson point process with correlated separation time between two load events. The correlation between the intensities of load events is described by the multi-variate Gaussian copula function. The resistance aging process is considered to be a combination of both gradual and shock deteriorations. Four candidate copula functions, namely Gaussian, Clayton, Gumbel and Frank, are considered to model the dependency of shock deterioration on load intensity. Two types of failure mechanisms are considered: the first is due to the load effect exceeding the resistance, and the second occurs when the cumulative damage within the considered service period reaches the permissible level. A simulation-based method is developed to estimate structural reliability considering the two failure modes. Illustrative examples are presented to demonstrate the applicability of the proposed method. Parametric studies are conducted to investigate the impacts of temporal correlation in loads and deterioration-load dependency on structural failure probability.

Bivariate ensemble model output statistics approach for joint forecasting of wind speed and temperature

Forecast ensembles are typically employed to account for prediction uncertainties in numerical weather prediction models. However, ensembles often exhibit biases and dispersion errors, thus they require statistical post-processing to improve their predictive performance. Two popular univariate post-processing models are the Bayesian model averaging (BMA) and the ensemble model output statistics (EMOS). In the last few years increased interest has emerged in developing multivariate post-processing models, incorporating dependencies between weather quantities, such as for example a bivariate distribution for wind vectors or even a more general setting allowing to combine any types of weather variables. In line with a recently proposed approach to model temperature and wind speed jointly by a bivariate BMA model, this paper introduces a bivariate EMOS model for these weather quantities based on a truncated normal distribution. The bivariate EMOS model is applied to temperature and wind speed forecasts of the eight-member University of Washington mesoscale ensemble and of the eleven-member ALADIN-HUNEPS ensemble of the Hungarian Meteorological Service and its predictive performance is compared to the performance of the bivariate BMA model and a multivariate Gaussian copula approach, post-processing the margins with univariate EMOS. While the predictive skills of the compared methods are similar, the bivariate EMOS model requires considerably lower computation times than the bivariate BMA method.

On Multivariate Log Birnbaum-Saunders Distribution

Univariate Birnbaum-Saunders distribution has received a considerable attention in recent years. Rieck and Nedelman (Technometrics, vol. 33, 51–60, 1991) introduced a log Birnbaum-Saunders distribution. We introduce a multivariate log Birnbaum-Saunders distribution and discuss its different properties. It is observed that the proposed multivariate model can be obtained from the multivariate Gaussian copula. We have proposed the maximum likelihood estimators of the unknown parameters. Since it is a computationally challenging problem, particularly if the dimension is high, we have considered the approximate maximum likelihood estimators based on the Copula structure using two-step procedure. The asymptotic distributions of both these estimators have been obtained. We compare their performances using Monte Carlo simulations, and it is observed that their performances are very similar in nature. One data set has been analyzed for illustrative purposes.

Influence of dependence of directional extreme wind speeds on wind load effects with various mean recurrence intervals

This study offers an improved understanding of the influence of statistical dependence between directional extreme wind speeds when estimating the wind load effects with various mean recurrence intervals (MRIs). Existing multivariate approaches that concern the wind directionality effect are first reviewed. Several factors that influence the prediction with and without consideration of the statistical dependence between directional extreme wind speeds are discussed by using Gaussian copula model. The influence of wind speed masking on the wind effect estimation is discussed. The influence of use of different joint probability distribution models for directional extreme wind speeds is illustrated through a comparison between multivariate Gaussian and Gumbel copula models. The necessity of using multivariate approach is discussed and a simplified method is proposed to account for directional dependence, which not only provides accurate prediction but also reduces calculation effort. Examples with real wind climate model and generic wind tunnel test results are shown to illustrate the influences brought by directional dependence, model difference, and wind speed masking. Also discussion is made on the partition of directional sectors which concerns the balance of number of sectors and modeling uncertainty.

Sequential Bayesian Model Selection of Regular Vine Copulas

Regular vine copulas can describe a wider array of dependency patterns than the multivariate Gaussian copula or the multivariate Student's t copula. This paper presents two contributions related to model selection of regular vine copulas. First, our pair copula family selection procedure extends existing Bayesian family selection methods by allowing pair families to be chosen from an arbitrary set of candidate families. Second, our method represents the first Bayesian model selection approach to include the regular vine density construction in its scope of inference. The merits of our approach are established in a simulation study that benchmarks against methods suggested in current literature. A real data example about forecasting of portfolio asset returns for risk measurement and investment allocation illustrates the viability and relevance of the proposed scheme.

Dependence Modelling with Regular Vine Copula Models: A Case-Study for Car Crash Simulation Data

SummaryThe analysis of car crash output parameters such as firewall intrusion points assist the overall engineering process. Such data are nowadays collected from many numerical simulations and it is not possible for the engineer to analyse this growing amount of data by hand. Therefore, data mining and statistical methods are needed. Here, we propose to use the flexible class of regular vine (R-vine) copulas for modelling the dependence between such output variables. R-vine copulas are multivariate copulas constructed hierarchically from bivariate copulas as building blocks. We introduce the concept of such constructions and their graphical tree representation. Applied to simulated frontal crash data of a Ford Taurus such graphs help us to illustrate the dependence structure among different firewall intrusion locations. The big advantage of R-vines compared with standard approaches such as the multivariate normal distribution or the multivariate Gaussian copula is the ability to model asymmetries and dependence in the tails. Our application demonstrates the strong potential of R-vines in the engineering context and opens further application areas.

Approximate Uncertainty Modeling in Risk Analysis with Vine Copulas.

Many applications of risk analysis require us to jointly model multiple uncertain quantities. Bayesian networks and copulas are two common approaches to modeling joint uncertainties with probability distributions. This article focuses on new methodologies for copulas by developing work of Cooke, Bedford, Kurowica, and others on vines as a way of constructing higher dimensional distributions that do not suffer from some of the restrictions of alternatives such as the multivariate Gaussian copula. The article provides a fundamental approximation result, demonstrating that we can approximate any density as closely as we like using vines. It further operationalizes this result by showing how minimum information copulas can be used to provide parametric classes of copulas that have such good levels of approximation. We extend previous approaches using vines by considering nonconstant conditional dependencies, which are particularly relevant in financial risk modeling. We discuss how such models may be quantified, in terms of expert judgment or by fitting data, and illustrate the approach by modeling two financial data sets.

Estimating Risk of Natural Gas Portfolios by Using GARCH-EVT-Copula Model.

This paper concentrates on estimating the risk of Title Transfer Facility (TTF) Hub natural gas portfolios by using the GARCH-EVT-copula model. We first use the univariate ARMA-GARCH model to model each natural gas return series. Second, the extreme value distribution (EVT) is fitted to the tails of the residuals to model marginal residual distributions. Third, multivariate Gaussian copula and Student t-copula are employed to describe the natural gas portfolio risk dependence structure. Finally, we simulate N portfolios and estimate value at risk (VaR) and conditional value at risk (CVaR). Our empirical results show that, for an equally weighted portfolio of five natural gases, the VaR and CVaR values obtained from the Student t-copula are larger than those obtained from the Gaussian copula. Moreover, when minimizing the portfolio risk, the optimal natural gas portfolio weights are found to be similar across the multivariate Gaussian copula and Student t-copula and different confidence levels.

Surrogacy assessment using principal stratification with multivariate normal and Gaussian copula models.

The validation of intermediate markers as surrogate markers (S) for the true outcome of interest (T) in clinical trials offers the possibility for trials to be run more quickly and cheaply by using the surrogate endpoint in place of the true endpoint. Working within a principal stratification framework, we propose causal quantities to evaluate surrogacy using a Gaussian copula model for an ordinal surrogate and time-to-event final outcome. The methods are applied to data from four colorectal cancer clinical trials, where S is tumor response and T is overall survival. For the Gaussian copula model, a Bayesian estimation strategy is used and, as some parameters are not identifiable from the data, we explore the use of informative priors that are consistent with reasonable assumptions in the surrogate marker setting to aid in estimation. While there is some bias in the estimation of the surrogacy quantities of interest, the estimation procedure does reasonably well at distinguishing between poor and good surrogate markers. Some of the parameters of the proposed model are not identifiable from the data, and therefore, assumptions must be made in order to aid in their estimation. The proposed quantities can be used in combination to provide evidence about the validity of S as a surrogate marker for T.

Identification of topology changes in power grids using phasor measurements

Phasor measurement units (PMUs) are increasingly important for monitoring the state of an electrical power grid and quickly detecting topology changes caused by events such as lines going down or large loads being dropped. Phasors are complex‐valued measurements of voltage and current at various points of generation and consumption. If a line goes down or a load is removed, power flows change throughout the grid according to known physical laws, and the probability distribution of phasor measurements changes accordingly. This paper develops a method to estimate the current topology of a power grid from phasor measurements and considers the design goal of placing PMUs at strategic points in a distribution system to achieve good sensitivity to single‐line outages. From a vector of phasor measurements, probabilities are computed corresponding to the scenario that all power lines are operational and to alternate scenarios in which each line goes down individually. These probabilities are functions of the joint distributions of phasor measurements under each possible scenario, obtained through Monte Carlo simulations with random load profiles. We use log‐spline densities to estimate marginal distributions of phasor measurements and fold these into a multivariate Gaussian copula to capture important correlations. Sensitivity to outages varies according to which line goes down and where PMUs are placed on the grid. A greedy search algorithm is demonstrated for placing PMUs at locations that provide good sensitivity to single‐line outages. Published 2014. This article is a U.S. Government work and is in the public domain in the USA.

Semiparametric Gaussian copula models: Geometry and efficient rank-based estimation

We propose, for multivariate Gaussian copula models with unknown margins and structured correlation matrices, a rank-based, semiparametrically efficient estimator for the Euclidean copula parameter. This estimator is defined as a one-step update of a rank-based pilot estimator in the direction of the efficient influence function, which is calculated explicitly. Moreover, finite-dimensional algebraic conditions are given that completely characterize efficiency of the pseudo-likelihood estimator and adaptivity of the model with respect to the unknown marginal distributions. For correlation matrices structured according to a factor model, the pseudo-likelihood estimator turns out to be semiparametrically efficient. On the other hand, for Toeplitz correlation matrices, the asymptotic relative efficiency of the pseudo-likelihood estimator can be as low as 20%. These findings are confirmed by Monte Carlo simulations. We indicate how our results can be extended to joint regression models.

Uncertainty analysis of bias from satellite rainfall estimates using copula method

The aim of this study is to develop a copula-based ensemble simulation method for analyzing the uncertainty and adjusting the bias of two high resolution satellite precipitation products (PERSIANN and TMPA-3B42). First, a set of sixty daily rainfall events that each of them occurs concurrently over twenty 0.25°×0.25° pixels (corresponding to both PERSIANN and TMPA spatial resolution) is determined to perform the simulations and validations. Next, for a number of fifty-four out of sixty (90%) selected events, the differences between rain gauge measurements as reference surface rainfall data and satellite rainfall estimates (SREs) are considered and termed as observed biases. Then, a multivariate Gaussian copula constructed from the multivariate normal distribution is fitted to the observed biases. Afterward, the copula is employed to generate multiple bias fields randomly based on the observed biases. In fact, copula is invariant to monotonic transformations of random variables and thus the generated bias fields have the same spatial dependence structure as that of the observed biases. Finally, the simulated biases are imposed over the original satellite rainfall estimates in order to obtain an ensemble of bias-adjusted rainfall realizations of satellite estimates. The study area selected for the implementation of the proposed methodology is a region in the southwestern part of Iran. The reliability and performance of the developed model in regard to bias correction of SREs are examined for a number of six out of those sixty (10%) daily rainfall events. Note that these six selected events have not participated in the steps of bias generation. In addition, three statistical indices including bias, root mean square error (RMSE), and correlation coefficient (CC) are used to evaluate the model. The results indicate that RMSE is improved by 35.42% and 36.66%, CC by 17.24% and 14.89%, and bias by 88.41% and 64.10% for bias-adjusted PERSIANN and TMPA-3B42 estimates, respectively.

Semiparametric Gaussian Copula Models: Geometry and Efficient Rank-Based Estimation

For multivariate Gaussian copula models with unknown margins and structured correlation matrices, a rank-based, semiparametrically efficient estimator is proposed for the Euclidean copula parameter. This estimator is defined as a one-step update of a rank-based pilot estimator in the direction of the efficient influence function, which is calculated explicitly. Moreover, finite-dimensional algebraic conditions are given that completely characterize adaptivity of the model with respect to the unknown marginal distributions and of efficiency of the pseudo-likelihood estimator. For correlation matrices structured according to a factor model, the pseudo-likelihood estimator turns out to be semiparametrically efficient. On the other hand, for Toeplitz correlation matrices, the asymptotic relative efficiency of the pseudo-likelihood estimator with respect to our one-step estimator can be as low as 20%. These findings are confirmed by Monte Carlo simulations.

Simulating dependent discrete data

This article describes a method for simulating n-dimensional multivariate non-normal data, with emphasis on count-valued data. Dependence is characterized by either Pearson correlations or Spearman correlations. The simulation is accomplished by simulating a vector of correlated standard normal variates. The elements of this vector are then transformed to achieve the target marginal distributions. We prove that the method corresponds to simulating data from a multivariate Gaussian copula. The simulation method does not restrict pairwise dependence beyond the limits imposed by the marginal distributions and can achieve any Pearson or Spearman correlation within those limits. Two examples are included. In the first example, marginal means, variances, Pearson correlations, and Spearman correlations are estimated from the epileptic seizure data set of Diggle et al. [P. Diggle, P. Heagerty, K.Y. Liang, and S. Zeger, Analysis of Longitudinal Data, Oxford University Press, Oxford, 2002]. Data with these means and variances are simulated to first achieve the estimated Pearson correlations and then achieve the estimated Spearman correlations. The second example is of a hypothetical time series of Poisson counts with seasonal mean ranging between 1 and 9 and an autoregressive(1) dependence structure.

The planting real option in cash rent valuation

After entering into a farmland cash rent contract in the fall, a tenant farmer has flexibility over the spring crop choice and the input application level. Failure to account for these options will bias estimates of what farmers should pay to rent land. Applying Monte Carlo simulation methods, this study investigates the option values for these choices. A Multivariate Gaussian Copula (MGC) is employed to account for dependence among yields and prices. Results show that the average cash rent valuation for the real option approach is $33.6 higher than that for the conventional Net Present Value (NPV) method, in which the input intensity option is $0.9. Crop planting sequence is shown to impact the real option value.