Gaussian processes on the support of cylindrical surfaces, with application to periodic spatio-temporal data

Abstract

There is a need to construct valid covariance structures for modeling spatial data on the support of cylinder, for applications such as colon or esophagus cancer, heat mass transfer on cylindrical surfaces, sonar response on cylindrical surfaces or disease mapping on tree-trunks. Such processes could also be used to model spatio-temporal periodic processes, such as monthly CO2 emissions, rainfall or average temperature, recorded over one year, or daily sleep patterns. If the underlying spatial support is k-dimensional then the support of these spatio-temporal processes conforms to k+1-dimensional cylindrical surface, viewing time as a periodic and hence circular dimension.Our contribution is to introduce nonseparable covariance structures for Gaussian spatial and spatio-temporal processes where the underlying support of these processes can be viewed as a cylindrical surface. This is accomplished by first determining the complete set of stationary autocovariance functions on the support of the circle S, and next, using nonseparable forms, constructing valid covariance structures on Rk×S, i.e, the k+1-dimensional cylinder. We note that while such models could be applied to periodic spatio-temporal processes, they could easily be generalized to the case where the circle denotes a physical, rather than temporal dimension. Examples span physics, medical applications, ecology, and others.To illustrate model performance in a case where we know the truth, we use simulation examples. We also work with a monthly (periodic) temperature data from the Carbon Dioxide Information Analysis Center and use a Bayesian approach and MCMC to address the problems of estimation, kriging, and model comparison.