A review of the application of copulas to improve modelling of non-bigaussian bivariate relationships (with an example using geological data)

Abstract

Correctly modelling bivariate relationships between geological variables is vital in mineral resource estimation. Often these relationships are complex and more simplistic methods of modelling such as Monte-Carlo Simulations (MCS), a bigaussian distribution or linear regression are not suitable. MCS where correlation coefficients are specified are inherently problematic because they can only reproduce the marginal distribution and a specified rank correlation coefficient, they cannot reproduce complex dependency structures. Bigaussian modelling is only appropriate if the data is indeed bigaussian (which is essentially never the case for grade variables). Elementary linear regression models can only model linear relationships and are often used in a deterministic manner. Copulas offer a framework to model and simulate multivariate relationships that go beyond correlation coefficients. They allow the strength of dependence to vary in different quantiles. Copulas have been used extensively in the recent years, but they have only made a limited appearance in the mining industry; for example, they have been used in spatial (geostatistical) simulations (Bardossy and Li, 2008). This paper focuses on the use of copulas to model bivariate relationships in a non-spatial sense. The intended audience for this paper is the non-statistician, such as a resource geologist, or geochemist. We introduce the concept of copulas including fitting, simulation and validation for the unfamiliar reader. A geologically relevant case study is provided where copulas are compared to other traditional methods of bivariate simulations. Copulas offer a framework to model multivariate data structures because the marginal distributions are modelled separately from the dependency structure (which is contained in the copula itself). A copula is a multivariate distribution with uniform margins on the interval [0,1]. There are many types of copulas; each one has a specific dependency structure. For example a Clayton copula has high level of dependence in the lower tail and low dependence in the upper tail. Copulas can be used for simulation purposes; a Bivariate Distribution linked via a Copula (BDC). Figuratively speaking, to simulate from a BDC the marginal distributions are simulated separately from one another and they are joined together in a manner which is consistent with the dependency structure of the copula. The objective of copula fitting is to fit a known copula to the rescaled ranks of the observed data. The case study uses alumina and iron data from an iron ore deposit (figure 1). Several copulas were fitted to the pseudo observations of the bivariate data and as comparison a MCS, linear regression and bigaussian model were also fitted to the raw data. The goodness of fit of the copulas was assessed by a range of statistics and graphical measures. The copulas that provided a reasonable fit were carried through to the fitting of a BDC, in this case the Clayton, Frank, Plackett and normal copulas. The Clayton BDC was chosen as the most suitable to model the data by a selection of statistics and graphical methods. The performance of the comparative (non-copula) modelling techniques was poor; none of them accurately modelled the high strength of dependence in the lower tail. The Clayton BDC modelled the unusual dependency structure better and captured the high level of dependence in the lower tail. Although the Clayton BDC was the most suitable the simulation could have been better. This suggests that a more bespoke copula may give more favourable results. In this case study it was possible to visually discern that the Clayton model was better than the other methods. A quantitative technique for validating across all the methods of bivariate simulations is recommended in future studies. The main conclusions from this case study were that a Clayton copula captured and modelled the high level of dependence in the lower tail better than the comparative modelling techniques. This case study demonstrates that using copulas to model the complex relationships that often exist between geological phenomena could be a promising alternative to other simplistic methods of modelling.