Copula-based spatio-temporal modelling for count data

Abstract

Modelling of spatio-temporal count data has received considerable attention in recent statistical research. However, the presence of massive correlation between locations, time points and variables imposes a great computational challenge. In existing literature, latent models under the Bayesian framework are predominately used. Despite numerous theoretical and practical advantages, likelihood analysis of spatio-temporal modelling on count data is less wide spread, due to the difficulty in identifying the general class of multivariate distributions for discrete responses.

In this thesis, we propose a Gaussian copula regression model (copSTM) for the analysis of multivariate spatio-temporal data on lattice. Temporal effects are modelled through the conditional marginal expectations of the response variables using an observation-driven time series model, while spatial and cross-variable correlations are captured in a block dependence structure, allowing for both positive and negative correlations. The proposed copSTM model is flexible and sufficiently generalizable to many situations. We provide pairwise composite likelihood inference tools. Numerical examples suggest that the proposed composite likelihood estimator produces satisfactory estimation performance.

While variable selection of generalized linear models is a well developed topic, model subsetting in applications of Gaussian copula models remains a relatively open research area. The main reason is the computational burden that is already quite heavy for simply fitting the model. It is therefore not computationally affordable to evaluate many candidate sub-models. This makes penalized likelihood approaches extremely inefficient because they need to search through different levels of penalty strength, apart from the fact suggested by our numerical experience that optimization of penalized composite likelihoods with many popular penalty terms (e.g LASSO and SCAD) usually does not converge in copula models. Thus, we propose to use a criterion-based selection approach that borrows strength from the Gibbs sampling technique.The methodology guarantees to converge to the model with the lowest criterion value, yet without searching through all possible models exhaustively.

Finally, we present an R package implementing the estimation and selection of the copSTM model in C++. We show examples comparing our package to many available R packages (on some special cases of the copSTM), confirming the correctness and efficiency of the package functions. The package copSTM provides a competitive toolkit option for the analysis spatio-temporal count data on lattice in terms of both model flexibility and computational efficiency.