Factor Copula Models for Replicated Spatial Data

Abstract

We propose a new copula model that can be used with replicated spatial data. Unlike the multivariate normal copula, the proposed copula is based on the assumption that a common factor exists and affects the joint dependence of all measurements of the process. Moreover, the proposed copula can model tail dependence and tail asymmetry. The model is parameterized in terms of a covariance function that may be chosen from the many models proposed in the literature, such as the Matérn model. For some choice of common factors, the joint copula density is given in closed form and therefore likelihood estimation is very fast. In the general case, one-dimensional numerical integration is needed to calculate the likelihood, but estimation is still reasonably fast even with large data sets. We use simulation studies to show the wide range of dependence structures that can be generated by the proposed model with different choices of common factors. We apply the proposed model to spatial temperature data and compare its performance with some popular geostatistics models.

Multivariate Type G Matérn Stochastic Partial Differential Equation Random Fields

“…could be viewed as a generalization of the one-factor copulas in Krupskii and Joe (2015) and Krupskii et al (2018). However, despite the flexibility of the marginal distributions, the model is limited since it is non-ergodic for any non-singular distribution of v, and the sample paths are indistinguishable from sample paths of a Gaussian random field.…”

“…(). Copula‐based modelling is another popular method for non‐Gaussian data, which has been used for creating both univariate (Gräler, ; Bárdossy, ) and multivariate (Krupskii et al ., ) random fields.…”

“…Multivariate versions of this approach were explored by Ma (2013b) and Du et al (2012). Copula-based modelling is another popular method for non-Gaussian data, which has been used for creating both univariate (Gräler, 2014;Bárdossy, 2006) and multivariate (Krupskii et al, 2018) random fields.…”

“…However, creating non-Gaussian multivariate random-field models that allow for likelihoodbased parameter estimation and probabilistic prediction is difficult, especially if they should be able to capture interesting departures from normality within realizations, and not just have non-Gaussian marginal distributions. This requirement excludes fields that are non-Gaussian only in the presence of repeated measurements, such as the factor copula models (Krupskii et al, 2018) and the constructions based on multiplying Gaussian fields with random scalars. Many other copula-based approaches in geostatistics use Gaussian copulas.…”

Gaussian likelihood inference on data from trans‐Gaussian random fields with Matérn covariance function

“…Recently, models for non-Gaussian random fields have received growing attention for real data analyses, for example, precipitation and wind speed data. The main approaches to constructing non-Gaussian random fields include trans-Gaussian random fields (Cressie, 1993), skew-Gaussian processes (Zhang & El-Shaarawi, 2010), scale-mixing Gaussian random fields (Fonseca & Steel, 2011), log-skew-elliptical random fields (Marchenko & Genton, 2010), spatial copula models (Gräler, 2014;Krupskii, Huser, & Genton, 2017), and non-Gaussian Matérn fields based on stochastic partial differential equation (Wallin & Bolin, 2015). These models are usually extensions of their univariate and multivariate counterparts.…”