Arturo Erdely

On the Construction of Families of Absolutely Continuous Copulas with Given Restrictions

The problem of constructing copulas with a given diagonal section has been studied by Sungur and Yang (1996) and Fredricks and Nelsen (1997a); (b); (2002). The results of Sungur and Yang are especially relevant because, among other results, they have proven that an Archimedean copula is characterized by its diagonal section. The results obtained by Fredricks and Nelsen allow one to build a singular copula with a given a diagonal section. In all cases, the resulting copulas are symmetric. In this article, we provide a family of absolutely continuous copulas with a fixed diagonal, which can differ from another absolutely continuous copula almost everywhere with respect to Lebesgue measure. It is important to mention that the asymmetry in the proposed methodology is not an issue.

Nonparametric and Semiparametric Bivariate Modeling of Petrophysical Porosity-Permeability Dependence from Well Log Data

Assessment of rock formation permeability is a complicated and challenging problem that plays a key role in oil reservoir modeling, production forecast, and the optimal exploitation management. Generally, permeability evaluation is performed using porosity-permeability relationships obtained by integrated analysis of various petrophysical measurements taken from cores and wireline well logs. Dependence relationships between pairs of petrophysical variables, such as permeability and porosity, are usually nonlinear and complex, and therefore those statistical tools that rely on assumptions of linearity and/or normality and/or existence of moments are commonly not suitable in this case. But even expecting a single copula family to be able to model a complex bivariate dependency seems to be still too restrictive, at least for the petrophysical variables under consideration in this work. Therefore, we explore the use of the Bernstein copula, and we also look for an appropriate partition of the data into subsets for which the dependence strucure was simpler to model, and then a conditional gluing copula technique is applied to build the bivariate joint distribution for the whole data set.

A multivariate Bernstein copula model for permeability stochastic simulation

This paper introduces a general nonparametric method for joint stochastic simulation of petrophysical properties using the Bernstein copula. This method consists basically in generating stochastic simulations of a given petrophysical property (primary variable) modeling the underlying empirical dependence with other petrophysical properties (secondary variables) while reproducing the spatial dependence of the first one. This multivariate approach provides a very flexible tool to model the complex dependence relationships of petrophysical properties. It has several advantages over other traditional methods, since it is not restricted to the case of linear dependence among variables, it does not require the assumption of normality and/or existence of moments. In this paper this method is applied to simulate rock permeability using Vugular Porosity and Shear Wave Velocity (S-Waves) as covariates in a carbonate double-porosity formation at well log scale. Simulated permeability values show a high degree of accuracy compared to the actual values.

A nonparametric symmetry test for absolutely continuous bivariate copulas

Based on the works by Klement and Mesiar (Comment Math Univ Carolinae 47:141–148, 2006) and Nelsen (Stat Pap 48:329–336, 2007) on maximal asymmetry of copulas, we define and study the concept of tri-symmetry and we propose a simple statistic to test symmetry of a bivariate copula, given a random sample of an absolutely continuous bivariate random vector. We also make a power comparison against some other well known nonparametric symmetry tests.

Copula-based piecewise regression

Most common parametric families of copulas are totally ordered, and in many cases they are also positively or negatively regression dependent and therefore they lead to monotone regression functions, which makes them not suitable for dependence relationships that imply or suggest a non-monotone regression function. A gluing copula approach is proposed to decompose the underlying copula into totally ordered copulas that once combined may lead to a non-monotone regression function.

Bernstein Copula-Based Spatial Stochastic Simulation of Petrophysical Properties Using Seismic Attributes as Secondary Variable

A novel Bernstein copula-based spatial stochastic co-simulation (BCSCS) method for petrophysical properties using seismic attributes as a secondary variable is presented. The method is fully nonparametric, and it has the advantages of not requiring linear dependence between variables. The methodology is illustrated in a case study from a marine reservoir in the Gulf of Mexico, and the results are compared with sequential Gaussian co-simulation (SGCS) method.

A subcopula based dependence measure

A dependence measure for arbitrary type pairs of random variables is proposed and analyzed, which in the particular case where both random variables are continuous turns out to be a concordance measure. Also, a sample version of the proposed dependence measure based on the empirical subcopula is provided, along with an R package to perform the corresponding calculations.