Alexandra Schmidt, McGill University, Canada
- Thais Fonseca, UFRJ, Brazil.
Bayesian Cross-Validation of Geostatistical Models
The problem of validating or criticising models for georeferenced data is challenging as the conclusions may be sensitive to which partition of the data into training and validation cases is utilized. This is an obvious problem with the basic model validation scheme, since only a few out-of-sample locations are usually selected to validate the model. On the other hand, the cross-validation approach which considers several possible configurations of data divided into training and validation observations is an appealing alternative, but it could be computationally demanding, as estimation of parameters usually requires computationally intensive methods.
This work proposes the use of cross-validation techniques to choose between competing models and to assess the goodness of fit of spatial models in different regions of the spatial domain. In particular, we consider uncertainty in the locations by assigning a probability distribution to them. To deal with the computational burden of cross-validation we estimated discrepancy functions in a computationally efficient manner based on importance weighting posterior samples. Furthermore, we propose a stratified cross-validation scheme to take into account spatial heterogeneity, reducing the total variance of estimated predictive discrepancy measures. We illustrate the advantages of our proposal with simulated examples of homogeneous and inhomogeneous spatial processes and with an application to rainfall in Rio de Janeiro.
Using a non-homogenous Poisson model with spatial anisotropy and change-points to study rate of ozone exceedances in Mexico City
We consider a non-homogeneous Poisson model to analyse the rate of ozone exceedances. Besides its dependence on time, the rate function of this Poisson model will also depend on some parameters. We also allow the presence of change-points in the model. An anisotropic spatial component is imposed on the vector of the parameters of the Poisson rate function as well as on the vector of change-points. The parameters of the rate function and of the spatial model as well as the location of the change-points will be estimated using the Bayesian point of view via a Metropolis-Hastings algorithm within the Gibbs sampling. The model is applied to ozone data obtained from ten stations which are part of the monitoring network of Mexico City, Mexico. Results suggest that two change-points are necessary to have a good fit of the accumulated observed and estimated mean functions in each station. They also indicate that, in general, the behaviour of the rate function is decreasing with smaller rates as we go towards the end of the observational period. This is a joint work with Geoff Nicholls, Mario H. Tarumoto and Guadalupe Tzintzun.
Spatio-temporal models for heavy tailed skewed processes
In the analysis of most spatio-temporal processes in environmental studies, observations present skewed distributions frequently with heavy tails. Usually, data transformations are used to approximate normality, and stationary Gaussian processes are assumed to model the transformed data. We propose a spatio-temporal model for skewed processes that accommodates heavier tails than the ones based on skew normal distributions. For each time t, the resultant process is decomposed as the sum between a temporal structure and two independent processes: a log-Gaussian and a Gaussian process. We discuss important properties of the resultant process, such as the covariance structure, the kurtosis and skewness. The proposed model is applied to simulated data and to environmental data.