A new mixture copula model for spatially correlated multiple variables with an environmental application

Abstract

In environmental monitoring, multiple spatial variables are often sampled at a geographical location that can depend on each other in complex ways, such as non-linear and non-Gaussian spatial dependence. We propose a new mixture copula model that can capture those complex relationships of spatially correlated multiple variables and predict univariate variables while considering the multivariate spatial relationship. The proposed method is demonstrated using an environmental application and compared with three existing methods. Firstly, improvement in the prediction of individual variables by utilising multivariate spatial copula compares to the existing univariate pair copula method. Secondly, performance in prediction by utilising mixture copula in the multivariate spatial copula framework compares with an existing multivariate spatial copula model that uses a non-linear principal component analysis. Lastly, improvement in the prediction of individual variables by utilising the non-linear non-Gaussian multivariate spatial copula model compares to the linear Gaussian multivariate cokriging model. The results show that the proposed spatial mixture copula model outperforms the existing methods in the cross-validation of actual and predicted values at the sampled locations.

Figure 1

A diagram for spatial mixture copula construction.

Figure 2

Example plot to show the possible pairs with four locations.

Figure 3

An example correlogram. The blue dashed line indicates the upper limit of cut-off distance at which pairs of points are no longer considered to be spatially dependent. Empirical $\tau$ values (black dots) overlaid with theoretical cubic smooth line.

Figure 4

BEF data. Spatial distributions of (a) $Z_1$, (b) $Z_2$, and (c) scatter plot between $Z_1$ and $Z_2$.

Figure 5

Reproduction of bivariate relationship using various methods. Actual (red), predicted (black), $Z_1$ given $Z_2$ (green), and $Z_2$ given $Z_1$ (blue).