Approximation Of Copulas Research Articles

Stress–strength reliability for electricity systems based on the Bernstein copula approximation

ABSTRACT Electricity is a critical resource for modern society, and its availability is essential for various aspects of daily life, from powering homes to supporting industrial processes and infrastructure. Ensuring a reliable supply of electricity is vital for economic stability and the well-being of a population. As the world moves toward more sustainable and renewable energy sources, understanding the reliability of these sources compared to traditional fossil fuels is crucial. In this study, we develop a methodology that enables the measurement of stress–strength reliability when electricity consumption is considered as the strength variable, positioned between electricity generation from renewable energy sources and fossil fuels as stress variables. We achieve this using the copula-based reliability with Bernstein copula approximation. The methodology is applied to the U.S. monthly electricity data. The result shows that the current rates of electricity output from fossil fuels and renewables will not be sufficient to meet the demand for electricity.

Settling velocity and effective density analysis for aquaculture floc particles: An approach through bivariate parametric copula

The study of flocs' characteristics in aquaculture tanks remains challenging due to their complex composition. Nevertheless, the application of experimental methods, such as particle tracking velocimetry (PTV), has made it possible to measure particle diameters and settling velocities. With the experimental data, bivariate distribution functions through copula modelling were employed to provide a more accurate estimation of the effective density of flocs through the empirical model proposed by Lau and Krishnappan. It was observed that the primary particle density constituting the flocs varied between 1051 kg/m³ and 1426 kg/m³. Moreover, when considering a variable primary particle density, effective floc density values ranging from 980 kg/m³ to 1500 kg/m³ were obtained for floc diameters ranging from 0.068 mm to 1.9 mm, respectively. This variation confirms that the b and c parameters of the Lau-Krishnappan model change for each floc diameter range following the quantiles associated with the conditional probability of diameter and settling velocity. Thus, employing copula approximation, a more accurate fit of the Lau-Krishnappan model was achieved, considering a wide range of particle diameters at the tails. This approach offers better estimates of floc effective density and settling velocity, essential for enhancing the selection and design of aquaculture tanks and settlers to ensure efficient solids removal.

Structured Variational Approximations with Skew Normal Decomposable Graphical Models and Implicit Copulas

Although there is much recent work developing flexible variational methods for Bayesian computation, Gaussian approximations with structured covariance matrices are often preferred computationally in high-dimensional settings. This article considers approximate inference methods for complex latent variable models where the posterior is close to Gaussian, but with some skewness in the posterior marginals. We consider skew decomposable graphical models (SDGMs), which are based on the closed skew normal family of distributions, as variational approximations. These approximations can reflect the true posterior conditional independence structure and capture posterior skewness. To increase flexibility, implicit copula SDGM approximations are also developed, where elementwise transformations of an approximately standardized SDGM random vector are considered. This implicit copula extension is an important contribution of our work, and improves the accuracy of SDGM approximations for only a modest increase in computational cost. Our parameterization of the copula approximation is novel, even in the Gaussian case. Performance of the methods is examined in a number of real examples involving generalized linear mixed models and state space models. Supplemental materials including code and appendix are available online.

On the copula-based reliability of stress-strength model under bivariate stress

In this paper, we consider the stress-strength reliability R = P ( X 1 < Y < X 2 ) where the strength Y of a component lies between the dependent stress variables X 1 and X 2 . We propose a copula-based approach for stress-strength reliability having bivariate stress. We obtain R for Farlie-Gumbel-Morgenstern copula with Burr III marginals. Also, we propose a Bernstein copula approximation for evaluating R under the stress-strength setup. We present empirical and maximum likelihood-based estimation procedures and compare their performances by Monte Carlo simulation. We apply the proposed approach to chemical and overt diabetes data for illustration purpose.

Finite-sample properties of estimators for first and second order autoregressive processes

The class of autoregressive (AR) processes is extensively used to model temporal dependence in observed time series. Such models are easily available and routinely fitted using freely available statistical software like R. A potential problem is that commonly applied estimators for the coefficients of AR processes are severely biased when the time series are short. This paper studies the finite-sample properties of well-known estimators for the coefficients of stationary AR(1) and AR(2) processes and provides bias-corrected versions of these estimators which are quick and easy to apply. The new estimators are constructed by modeling the relationship between the true and originally estimated AR coefficients using weighted orthogonal polynomial regression, taking the sampling distribution of the original estimators into account. The finite-sample distributions of the new bias-corrected estimators are approximated using transformations of skew-normal densities, combined with a Gaussian copula approximation in the AR(2) case. The properties of the new estimators are demonstrated by simulations and in the analysis of a real ecological data set. The estimators are easily available in our accompanying R-package for AR(1) and AR(2) processes of length 10–50, both giving bias-corrected coefficient estimates and corresponding confidence intervals.

On Approximation of Copulas, Solution of Elliptic Problem, Using the Finite Difference Method: Generalization

In this paper, we generalize a copula construction method discussed in one of our papers. For this purpose we consider the general form of a linear elliptic PDE. Indeed, a physical interpretation of elliptic equations comes from the notion of conservative flow given by a gradient. This notion provides a mathematical model for equilibrium conservation laws in linear behaviour. This can be applied to many areas of science. Thus, the aim of this paper is to construct a new class of bivariate copulas by solving an elliptic partial differential equation with a Dirichlet condition at the boundary. Copulas belonging to this class allow us to study the stochastic behaviour of the notion of conservative flows. In other words, these copulas will allow us to have an idea on the dependence of those physical phenomena which are governed by elliptic PDEs. For this purpose, we use a discretization method which is the finite difference method which is a common technique for finding approximate solutions of partial differential equations that consists in solving a system of relations (numerical scheme) connecting the values of the unknown functions at some points sufficiently close to each other. For the finite difference method, a mesh is made. This is a set of isolated points called nodes located in the domain of definition of the functions subject to the partial differential equations, a grid on which only the nodes of which the unknowns corresponding to the approximate values of these functions are defined. The mesh also includes nodes located on the boundary of the domain (or at least "close" to this boundary) in order to be able to impose the boundary conditions and/or the initial condition with sufficient accuracy. The primary quality of a mesh is to cover the domain in which it develops as well as possible, to limit the distance between each node and its nearest neighbour. However, the mesh must also allow the discrete formulation of the differentiation operators to be expressed: for this reason, the nodes of the mesh are most often located on a grid whose main directions are the axes of the variables. In the main results of this paper (see section 3), we give a discretization of the solution of the problem followed by a simulation with the MATLAB software of this approximated solution and presenting the discretization errors.

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.

Upper risk bounds in internal factor models with constrained specification sets

For the class of (partially specified) internal risk factor models we establish strongly simplified supermodular ordering results in comparison to the case of general risk factor models. This allows us to derive meaningful and improved risk bounds for the joint portfolio in risk factor models with dependence information given by constrained specification sets for the copulas of the risk components and the systemic risk factor. The proof of our main comparison result is not standard. It is based on grid copula approximation of upper products of copulas and on the theory of mass transfers. An application to real market data shows considerable improvement over the standard method.

On a High-Dimensional Model Representation method based on Copulas

This article provides an alternative to High-Dimensional Model Representation using a Copula approximation of an unknown functional form. We apply our methodology in the context of an extensive Monte Carlo study and to a sample of large US commercial banks. In the Monte Carlo experiment, the approximations errors of the Copula approach are small and behave randomly. In our empirical application, we find that the Copula Approximation performs much better, in terms of Bayes factors for model comparison, compared to High-Dimensional Model Representation, which, in turn, provides better results when compared with standard flexible functional forms, like the translog, the minflex Laurent, and the Generalized Leontief, or a Multilayer Perceptron. Moreover, the choice of approximation has significant implications for productivity and its components (returns to scale, technical inefficiency, technical change, and efficiency change).

Space‐efficient estimation of empirical tail dependence coefficients for bivariate data streams

AbstractThis article proposes a space‐efficient approximation to empirical tail dependence coefficients of an indefinite bivariate stream of data. The approximation, which has stream‐length invariant error bounds, utilizes recent work on the development of a summary for bivariate empirical copula functions. The work in this paper accurately approximates a bivariate empirical copula in the tails of each marginal distribution, therefore modeling the tail dependence between the two variables observed in the data stream. Copulas evaluated at these marginal tails can be used to estimate the tail dependence coefficients. Modifications to the space‐efficient bivariate copula approximation, presented in this paper, allow the error of approximations to the tail dependence coefficients to remain stream‐length invariant. Theoretical and numerical evidence of this, including a case‐study using the Los Alamos National Laboratory netflow data‐set, is provided within this article.

Simplified vine copula models: Approximations based on the simplifying assumption

Vine copulas, or pair-copula constructions, have become an important tool in high-dimensional dependence modeling. Commonly, it is assumed that the data generating copula can be represented by a simplified vine copula (SVC). In this paper, we study the simplifying assumption and investigate the approximation of multivariate copulas by SVCs. We introduce the partial vine copula (PVC) which is a particular SVC where to any edge a $j$-th order partial copula is assigned. The PVC generalizes the partial correlation matrix and plays a major role in the approximation of copulas by SVCs. We investigate to what extent the PVC describes the dependence structure of the underlying copula. We show that, in general, the PVC does not minimize the Kullback-Leibler divergence from the true copula if the simplifying assumption does not hold. However, under regularity conditions, stepwise estimators of pair-copula constructions converge to the PVC irrespective of whether the simplifying assumption holds or not. Moreover, we elucidate why the PVC is often the best feasible SVC approximation in practice.

Bivariate nonparametric estimation of the Pickands dependence function using Bernstein copula with kernel regression approach

In this study, a new nonparametric approach using Bernstein copula approximation is proposed to estimate Pickands dependence function. New data points obtained with Bernstein copula approximation serve to estimate the unknown Pickands dependence function. Kernel regression method is then used to derive an intrinsic estimator satisfying the convexity. Some extreme-value copula models are used to measure the performance of the estimator by a comprehensive simulation study. Also, a real-data example is illustrated. The proposed Pickands estimator provides a flexible way to have a better fit and has a better performance than the conventional estimators.

Vine copula approximation: a generic method for coping with conditional dependence

Pair-copula constructions (or vine copulas) are structured, in the layout of vines, with bivariate copulas and conditional bivariate copulas. The main contribution of the current work is an approach to the long-standing problem: how to cope with the dependence structure between the two conditioned variables indicated by an edge, acknowledging that the dependence structure changes with the values of the conditioning variables. The changeable dependence problem, though recognized as crucial in the field of multivariate modelling, remains widely unexplored due to its inherent complication and hence is the motivation of the current work. Rather than resorting to traditional parametric or nonparametric methods, we proceed from an innovative viewpoint: approximating a conditional copula, to any required degree of approximation, by utilizing a family of basis functions. We fully incorporate the impact of the conditioning variables on the functional form of a conditional copula by employing local learning methods. The attractions and dilemmas of the pair-copula approximating technique are revealed via simulated data, and its practical importance is evidenced via a real data set.

Extending approximate Bayesian computation methods to high dimensions via a Gaussian copula model

Approximate Bayesian computation (ABC) refers to a family of inference methods used in the Bayesian analysis of complex models where evaluation of the likelihood is difficult. Conventional ABC methods often suffer from the curse of dimensionality, and a marginal adjustment strategy was recently introduced in the literature to improve the performance of ABC algorithms in high-dimensional problems. The marginal adjustment approach is extended using a Gaussian copula approximation. The method first estimates the bivariate posterior for each pair of parameters separately using a 2-dimensional Gaussian copula, and then combines these estimates together to estimate the joint posterior. The approximation works well in large sample settings when the posterior is approximately normal, but also works well in many cases which are far from that situation due to the nonparametric estimation of the marginal posterior distributions. If each bivariate posterior distribution can be well estimated with a low-dimensional ABC analysis then this Gaussian copula method can extend ABC methods to problems of high dimension. The method also results in an analytic expression for the approximate posterior which is useful for many purposes such as approximation of the likelihood itself. This method is illustrated with several examples.

Dependence structure between LIBOR rates by copula method

This paper discusses the correlation structure between London Interbank Offered Rates (LIBOR) by using the copula function. We start from one simplified model of A. Brace, D. Gatarek, and M. Musiela (1997) and find out that the copula function between two LIBOR rates can be expressed as a sum of an infinite series, where the main term is a distribution function with Gaussian copula. Partial differential equation method is used for deriving the copula expansion. Numerical results show that the copula of the LIBOR rates and Gaussian copula are very close in the central region and differ in the tail, and the Gaussian copula approximation to the copula function between the LIBOR rates provides satisfying results in the normal situation.

On the approximation of copulas via shuffles of Min

We study a multivariate extension of shuffles of Min that has a probabilistic interpretation in terms of mutually completely dependent process. The closure properties of the class of such copulas under different types of convergence is investigated.

Approximation of bivariate copulas by patched bivariate Fréchet copulas

Bivariate Fréchet (BF) copulas characterize dependence as a mixture of three simple structures: comonotonicity, independence and countermonotonicity. They are easily interpretable but have limitations when used as approximations to general dependence structures. To improve the approximation property of the BF copulas and keep the advantage of easy interpretation, we develop a new copula approximation scheme by using BF copulas locally and patching the local pieces together. Error bounds and a probabilistic interpretation of this approximation scheme are developed. The new approximation scheme is compared with several existing copula approximations, including shuffle of min, checkmin, checkerboard and Bernstein approximations and exhibits better performance, especially in characterizing the local dependence. The utility of the new approximation scheme in insurance and finance is illustrated in the computation of the rainbow option prices and stop-loss premiums.

Multivariate shuffles and approximation of copulas

We present and study a method for constructing multivariate copulas, which includes both the shuffles of Min and the ordinal sums. Such a method has been used in order to show that suitable transformations of a given copula constitute a dense set in the class of all copulas with respect to the L ∞ norm.

On approximation of copulas

New sufficient conditions for strong approximation of copulas, generated by sequences of partitions of unity, are given. Results are applied to the checkerboard and Bernstein approximations.

Strong Approximation of Copulas

We introduce strong convergence with regard to approximation of copulas. This new type of convergence is useful in dealing with the ∗-product of Darsow, Nguyen, and Olsen for copulas. We also provide tools for constructing strong approximations of copulas by using partitions of unity.