Pair-copula Construction Research Articles

Vine Copula-Based Classifiers with Applications

AbstractThe vine pair-copula construction can be used to fit flexible non-Gaussian multivariate distributions to a mix of continuous and discrete variables. With multiple classes, fitting univariate distributions and a vine to each class lead to posterior probabilities over classes that can be used for discriminant analysis. This is more flexible than methods with the Gaussian and/or independence assumptions, such as quadratic discriminant analysis and naive Bayes. Some variable selection methods are studied to accompany the vine copula-based classifier because unimportant variables can make discrimination worse. Simple numerical performance metrics cannot give a full picture of how well a classifier is doing. We introduce categorical prediction intervals and other summary measures to assess the difficulty of discriminating classes. Through extensive experiments on real data, we demonstrate the superior performance of our approaches compared to traditional discriminant analysis methods and random forests when features have different dependent structures for different classes.

A study of one-factor copula models from a tail dependence perspective

AbstractModeling multivariate dependence in high dimensions is challenging, with popular solutions constructing multivariate copula as a composition of lower dimensional copulas. Pair-copula constructions do so by using bivariate linking copulas, but their parametrization, in size, being quadratic in the dimension, is not quite parsimonious. Besides, the number of regular vines grows super-exponentially with the dimension. One parsimonious solution is factor copulas, and in particular, the one-factor copula is touted for its simplicity – with the number of parameters linear in the dimension – while being able to cater to asymmetric non-linear dependence in the tails. In this paper, we add nuance to this claim from the point of view of a popular measure of multivariate tail dependence, the tail dependence matrix (TDM). We focus on the one-factor copula model with the linking copula belonging to the BB1 family, pointing out later the applicability of our results to a wider class of linking copulas. For this model, we derive tail dependence coefficients and study their basic properties as functions of the parameters of the linking copulas. Based on this, we study the representativeness of the class of TDMs supported by this model with respect to the class of all possible TDMs. We establish that since the parametrization is linear in the dimension, it is no surprise that the relative volume is zero for dimensions greater than three, and hence, by necessity, we present a novel manner of evaluating the representativeness that has a combinatorial flavor. We formulate the problem of finding the best representative one-factor BB1 model given a target TDM and suggest an implementation along with a simulation study of its performance across dimensions. Finally, we illustrate the results of the paper by modeling rainfall data, which is relevant in the context of weather-related insurance.

Assessing copula models for mixed continuous-ordinal variables

Abstract Vine pair-copula constructions exist for a mix of continuous and ordinal variables. In some steps, this can involve estimating a bivariate copula for a pair of mixed continuous-ordinal variables. To assess the adequacy of copula fits for such a pair, diagnostic and visualization methods based on normal score plots and conditional Q–Q plots are proposed. The former uses a latent continuous variable for the ordinal variable. The methods are applied to data generated from some existing probability models for a mixed continuous-ordinal variable pair, and for such models, Kullback-Leibler divergence is used to assess whether simple parametric copula families can provide adequate fits. The effectiveness of the proposed visualization and diagnostic methods is illustrated on a dataset.

Multivariate degradation system reliability analysis with multiple sources of uncertainty

As the industrial Internet of Things evolves, multi-sensor monitoring becomes increasingly prevalent, resulting in degrading systems frequently exhibiting multivariate degradation characteristics. Multiple performance characteristics are frequently interdependent due to the consistency of the operating environment, operating conditions, and internal structure of the system. On the other hand, performance characteristic detection is also affected by numerous uncertainties. A multivariate stochastic degradation model that takes dependencies with multiple sources of uncertainty into account is developed for the analysis of multivariately dependent degradation systems. To represent linear or nonlinear degradation qualities, the marginal degradation process is referred to as a generalized stochastic degradation model. By using the D-vine copula function and pair-copula construction, dependencies between numerous variables are formed. The marginal and joint distribution parameters of the degradation model are derived using a two-step parameter estimation procedure. Lastly, the C-MAPSS dataset is used to analyze the applicability of the proposed model for bivariate and multivariate instances. The effectiveness of the proposed method is verified by experiments.

Development of a non-Gaussian copula Bayesian network for safety assessment of metro tunnel maintenance

The operation and maintenance of metro systems play a crucial role in urban development, with a focus on ensuring the serviceability and safety of metro tunnels. Accurate evaluation of the condition of these tunnel states requires investigating the complex interaction of multiple factors that impact their safety state. This study developed a hybrid model that integrates pair copula constructions (PCCs) and Bayesian networks (BN) to assess the safety state of metro tunnels, considering complex dependencies among these factors. First, key performance indicators (KPIs) were selected to assess tunnel safety, based on six failure modes. Then, an improved copula-based PC algorithm was employed to learn the non-Gaussian bayesian network model, which eliminated the normality assumption in the marginal densities of the KPIs. Finally, a combination of forward reasoning and GM(1,1) was utilized to predict the safety state of the metro tunnel in time series, facilitating the formulation of effective operation and maintenance plans. Furthermore, the proposed approach was applied to a real case study of the Wuhan metro system to demonstrate its effectiveness and applicability. The results highlighted the significant influence of some KPIs, such as convergence deformation, dislocation displacement, and stripping area, on metro tunnel safety. These KPIs emerged as key factors requiring focused attention in the operation and maintenance of metro tunnels.

Transformed Pair Copula Construction of Pareto Copula and Applications

The study introduced the transformed copula models and a D-vine structure for three variables. The numerical applications of the models are evaluated using two different sets of real-life data that exhibit nearly no dependence, highly dependent, over, and under dispersed characteristics. We examined only the volatilities of the first data using the exponentiated weighted moving average (EWMA). The parameter estimates of the models were obtained based on the maximum likelihood estimation method for the bivariate copula models and the Dissmann algorithm for sequential top-down estimation for the D-vine structure. The results showed that the introduced copula models outperformed some existing copula models in terms of their fit statistics for both real-life and simulated data sets. In addition, the Gaussian copula model gave a better fit to the D-vine structure than some existing copula models and could be recommended for modeling a D-vine structure comprising of variables that are positively weak correlated and highly correlated.

Predicting binary outcomes based on the pair-copula construction

Predicting binary outcomes based on the pair-copula construction

Copula Regression for Compound Distributions with Endogenous Covariates with Applications in Insurance Deductible Pricing

This article concerns deductible pricing in nonlife insurance contracts. The primary interest of insurers is the effect of the contract deductible on a policyholder’s aggregate loss that is determined by a compound distribution where the sum of individual claim amount is stopped by the number of claims. Policyholders choose the deductible level based on their hidden risks, which makes deductible endogenous in the regressions for both claim frequency and claim severity. To address the endogeneity in the regression for the compound aggregate loss, we introduce a novel approach using pair copula constructions to jointly model the policyholder’s deductible, number of claims, and individual claim amounts, in the context of compound distributions. The proposed method provides insurers an empirical tool to uncover the underlying risk distribution of the potential customers. In the application we consider an insurance portfolio from the property insurance program that provides property coverage for building and contents of local government entities of the Wisconsin. Using the historical data on policyholder and insurance claims, we first provide empirical evidence of the endogeneity of the deductible. Interestingly, we find that the policyholder’s deductible is negatively associated with the claim frequency but positively associated with the claim severity. For the portfolio of policyholders, the endogenous deductible model provides superior prediction for 65% and 71% of policyholders for claim frequency and severity, respectively. The endogeneity of deductible shows significant managerial implications on insurance operations. In particular, the risk score suggested by the proposed method allows the insurer to identify additional profitable underwriting strategies which are quantified by the Gini indices of 0.22 and 0.13 when switching from the exogenous deductible premium and the insurer’s contract premium, respectively. Supplementary materials for this article are available online.

Stability of spatial dependence structure of extreme precipitation and the concurrent risk over a nested basin

Extreme low and high precipitation can cause natural disasters (e.g., drought and flood events) which have devastatingeffects on natural environment and the human society. In this study, we present a framework for assessing the stability of spatial dependence of extreme precipitation and their concurrence probability in a nested catchment. The Pearl River basin was used as a case study to test the applicability of the framework. This framework is threefold. This first part involves modeling spatial dependence of both summer maxima and winter minima precipitation between pairs of scattered gauges over the basin by employing the pair-copula constructions as a baseline model. The second part of this framework aims to identify the variability of spatial dependence as a result of the effects of external drivers. Specifically, variations in the spatial dependence of extreme precipitation in the Pearl River basin over different periods associated with the influence of large-scale climate signals were examined. The final part quantifies the probability of occurring extreme precipitation-related events simultaneously over broader areas related to a specific event at a given gauge and the possible spatial extents being affected. The results indicate that the presented framework is able to capture the large-scale spatial dependence structures of both summer maxima and winter minima precipitation. It also allows for explicitly estimating the pairwise concurrence probability of extreme high and low precipitation events and event-impacted areas across the basin.

Agricultural Drought Prediction Based on Conditional Distributions of Vine Copulas

AbstractMonitoring and prediction of agricultural drought are paramount to food security at the global and regional scales, particularly under the influence of climate change and anthropogenic activities. Soil moisture is an effective indicator for monitoring and characterizing agricultural drought. Soil moisture (agricultural drought) is mainly affected by precipitation (meteorological drought) and temperature (hot conditions). Owing to the flexibility of vine copulas in handling multidimensional variables by decomposing them into pair copula constructions (PCCs), we propose a novel drought prediction method using three predictors, namely antecedent meteorological drought, previous hot conditions, and persistent agricultural drought, based on the conditional distributions of C‐vine copulas in a four‐dimensional scenario. The proposed model was applied to agricultural drought (characterized by the standardized soil moisture index (SSI)) prediction with 1–2‐months lead time for the summer season (i.e., August at a 6‐months timescale) in China. Taking two severe agricultural drought events that occurred in many regions across China in August of 2006 and 2014 as validation cases, the SSI predictions with 1–2‐months lead time using the conditional C‐vine copulas model were found to be generally consistent with the corresponding historical SSI observations in most parts of China. Performance evaluation using the Nash‐Sutcliffe efficiency (NSE), coefficient of determination (R2), and F1 score (F1S) for different climate regions also indicated that this model provided a reliable prediction of agricultural drought for most areas of China. The outcome of this study can serve as a guidance for drought prediction, early warning, and drought mitigation.

How simplifying and flexible is the simplifying assumption in pair-copula constructions – analytic answers in dimension three and a glimpse beyond

Motivated by the increasing popularity and the seemingly broad applicability of pair-copula constructions underlined by numerous publications in the last decade, in this contribution we tackle the unavoidable question on how flexible and simplifying the commonly used ‘simplifying assumption’ is from an analytic perspective and provide answers to two related open questions posed by Nagler and Czado in 2016. Aiming at a simplest possible setup for deriving the main results we first focus on the three-dimensional setting. We prove that the family of simplified copulas is flexible in the sense that it is dense in the set of all three-dimensional copulas with respect to the uniform metric d∞ – considering stronger notions of convergence like the one induced by the metric D1, by weak conditional convergence, by total variation, or by Kullback-Leibler divergence, however, the family even turns out to be nowhere dense and hence insufficient for any kind of flexible approximation. Furthermore, returning to d∞ we show that the partial vine copula is never the optimal simplified copula approximation of a given, non-simplified copula C, and derive examples illustrating that the corresponding approximation error can be strikingly large and extend to more than 28% of the diameter of the metric space. Moreover, the mapping ψ assigning each three-dimensional copula its unique partial vine copula turns out to be discontinuous with respect to d∞ (but continuous with respect to D1 and to weak conditional convergence), implying a surprising sensitivity of partial vine copula approximations. The afore-mentioned main results concerning d∞ are then extended to the general multivariate setting.

Conditional quantile reproducibility of multivariate distributions and simplified pair copula construction

В работе дается эквивалентное определение свойства воспроизводимости условных квантилей многомерных вероятностных распределений с использованием понятия копулы. На основе этого определения устанавливается связь между свойством воспроизводимости условных квантилей и упрощающим предположением в упрощенной конструкции из парных копул (simplified pair-copula construction). Кроме того, показывается, что свойство воспроизводимости условных квантилей сохраняется при переходе к копуле многомерного распределения.

Conditional Quantile Reproducibility of Multivariate Distributions and Simplified Pair Copula Construction

We give an equivalent definition of the conditional quantile reproducibility property for multivariate probability distributions using the concept of copula. Based on this definition, we establish ...

Control of key performance indicators of manufacturing production systems through pair-copula modeling and stochastic optimization

Key performance indicators (KPIs) modeling and control is important for efficient design and operation of complex manufacturing production systems. This paper proposes to implement the KPI control based on KPI modeling and stochastic optimization. The KPI relationship is first approximated using ordered block model and pair-copula construction (OBM-PCC) model, which is a non-parametric model that facilitates a flexible surrogate of the KPI relationship. Then, the KPI control is framed into a stochastic optimization problem, where the randomness in the cost function depends on the decision variables. To solve this stochastic optimization problem, the standard uniform distribution is employed to link the OBM-PCC model and the cost function to transform the problem into an ordinary stochastic optimization problem. The proposed method is efficient in KPI control and the performance is robust to the cost function. Extensive numerical studies and comparisons, together with a case study, are presented to demonstrate the effectiveness of the proposed KPI control framework.

Pair-copula Models for Analyzing Family Data

The construction of multivariate models for analyzing family data can be challenging due to the various dependencies between family members. These models typically have parameters that are subject to intricate and stringent constraints, which make it more difficult to arrive at a proper estimation of these model parameters. Recently, vines and pair-copula constructions have been utilized in order to simplify construction of multivariate models for both continuous and discrete data. The pair-copula models break down the hierarchy of dependence into vines and employ bivariate or pair-copulas as “building bricks” to generate appropriate multivariate distributions. This paper discusses pair-copula methods to model the convoluted dependence structure and proposes using a C-Vine type copula to model the dependence among family members. Analysis of actual, real-world family data is presented to illustrate our models and estimation procedures.

Pair copula construction for longitudinal data with zero-inflated power series marginal distributions

ABSTRACT Assessing the temporal dependency among outcomes under investigation is critical in many fields. One complication in the modeling process of the discrete longitudinal data is the presence of excess zeros. We propose a framework for modeling count repeated measurements using members of power series family of distributions. The framework accommodates count outcomes having extra zeros. The longitudinal observations of response variable is modeled using pair copula constructions with a D-vine structure. The maximum likelihood estimates of parameters are obtained using a two-stage approach. Some simulation studies are performed for illustration of the proposed methods, for comparing its performance with that of a generalized linear mixed effects (GLME) model and for assessing the robustness of D-vine and GLME models with respect to the distribution of random effects. In the empirical analysis, the proposed method is applied for analysing a real data set of a kidney allograft rejection study.

Assessment of generation adequacy by modeling a joint probability distribution model

The socio-politically motivated energy transition leads to expansion of renewable energy power plants (REPP) and simultaneous shutdowns of conventional power plants. As the electric load always needs to be balanced by sufficient generation and the energy production by REPP cannot be steered adequate to the demand but depends on weather situations, new methods are needed to assess generation adequacy. These results could be used afterwards to define measures to ensure reliable energy supply.To evaluate the generation adequacy a joint probability distribution model is proposed. It models the dependencies of weather characteristics, the electrical load and unavailabilities of conventional power plants using pair-copula constructions. The model is, then, applied to determine the probability of generation imbalance. The implementation is shown for Germany in the scenario year 2030.

Analysis of multivariate dependent accelerated degradation data using a random-effect general Wiener process and D-vine Copula

A modern product usually shows multiple performance characteristics that degrade simultaneously. It is quite common that these degradation processes are dependent due to some common factors such as internal structure, operating conditions, and working history. Technically, the essence of modeling such dependent degradation processes is to find appropriate marginal degradation models and capture the dependence structure among the multiple performance characteristics. This paper develops a multivariate dependent accelerated degradation test model based on a random-effect general Wiener process and D-vine copula. The proposed model and statistical inference method analyze nonlinear accelerated degradation data considering three sources of uncertainty and construct the marginal distributions. To overcome the lack of flexibility and parameter restrictions of standard multivariate copulas, the pair-copula constructions and their vine graphical representations are applied to reveal and fully understand the complex and hidden dependence patterns in the multivariate degradation data. A simulation example and a real application on the accelerated degradation data of a tuner are provided to illustrate the performance and benefits of the proposed model and statistical estimation method.

Parametric Vine Copula Construction for Flood Analysis for Kelantan River Basin in Malaysia

The multivariate approach of flood characteristics such as flood peak flow (P), volume (V), and duration (D) is much beneficial in recognizing the critical behaviour of flood episodes at a river basin scale. The incorporation of 2-dimensional copulas for establishing bivariate flood dependency frequently appears, but it could be more comprehensive if we focus all the three flood characteristic simultaneously. In such circumstances, incorporation of vine or Pair-Copula Construction (PCC) could produce a better approximation of joint probability density and much practical approach in the uncertainty analysis, in comparison with conventional trivariate copula distribution. This study demonstrated the efficacy of parametric vine copula in the modelling of trivariate flood characteristics for the Kelantan River basin in Malaysia. The D-vine tree structure is selected where the Gaussian and Frank copula is recognized for bivariate flood pairs (P-V) and (P-D) pairs in the first stage, using the maximum-pseudo-likelihood (MPL) estimation procedure. Similarly, the Gumbel copula is selected in the modelling of conditioned data obtained through the conditional distribution function of bivariate copulas selected in the previous stage based on the partial differentiation, also called h-function. Finally, the full density function of the 3-dimension structure is derived and compared with the observed flood characteristics. Furthermore, tail dependence properties and behaviour of D-vine copula are also investigated, which reveals for well capturing the general behaviour of Gaussian and Frank copula fitted to flood pair (P-V) and (V-D) and reproduces the overall flood correlation structure fairely well. Both the primary ‘OR’ and ‘AND’ joint return periods for trivariate flood events are estimated which pointing that ‘AND’ joint case produces higher return value than ‘OR’ case.

Constructing gene regulatory networks from microarray data using non-Gaussian pair-copula Bayesian networks.

Many biological and biomedical research areas such as drug design require analyzing the Gene Regulatory Networks (GRNs) to provide clear insight and understanding of the cellular processes in live cells. Under normality assumption for the genes, GRNs can be constructed by assessing the nonzero elements of the inverse covariance matrix. Nevertheless, such techniques are unable to deal with non-normality, multi-modality and heavy tailedness that are commonly seen in current massive genetic data. To relax this limitative constraint, one can apply copula function which is a multivariate cumulative distribution function with uniform marginal distribution. However, since the dependency structures of different pairs of genes in a multivariate problem are very different, the regular multivariate copula will not allow for the construction of an appropriate model. The solution to this problem is using Pair-Copula Constructions (PCCs) which are decompositions of a multivariate density into a cascade of bivariate copula, and therefore, assign different bivariate copula function for each local term. In fact, in this paper, we have constructed inverse covariance matrix based on the use of PCCs when the normality assumption can be moderately or severely violated for capturing a wide range of distributional features and complex dependency structure. To learn the non-Gaussian model for the considered GRN with non-Gaussian genomic data, we apply modified version of copula-based PC algorithm in which normality assumption of marginal densities is dropped. This paper also considers the Dynamic Time Warping (DTW) algorithm to determine the existence of a time delay relation between two genes. Breast cancer is one of the most common diseases in the world where GRN analysis of its subtypes is considerably important; Since by revealing the differences in the GRNs of these subtypes, new therapies and drugs can be found. The findings of our research are used to construct GRNs with high performance, for various subtypes of breast cancer rather than simply using previous models.