Recent advances in technology have led to large sets of tracking data, which brings new challenges in statistical modeling and prediction. Built on recent developments in Gaussian process modeling for spatio-temporal data and stochastic differential equations (SDEs), we develop a sequence of new models and corresponding inferential methods to meet these challenges. We first propose Bayesian Melding (BM) and downscaling frameworks to combine observations from different sources. To use BM for big tracking data, we exploit the properties of the processes along with approximations to the likelihood to break a high dimensional problem into a series of lower dimensional problems. To implement the downscaling approach, the integrated nested Laplace approximation (INLA) is applied to fit a linear mixed effect model that connects the two sources of observations. We apply these two approaches in a case study involving the tracking of marine mammals. Both of our frameworks are shown to have superior predictive performance compared with traditional approaches in both cross-validation and simulation studies.
We further develop the BM frameworks with stochastic processes that can reflect the time varying features of the tracks. We propose a linear SDE with splines as its coefficients and call it a generalized Ornstein-Ulhenbeck (GOU) process. The GOU achieves flexible modeling of the tracks in both mean and covariance with a reasonably parsimonious parameterization. Its inferences and predictions can be computed via the Kalman filter and smoother. BM with the GOU achieves a smaller prediction error and better credibility intervals in cross-validation comparisons to the basic BM and downscaling models. Following the success with the GOU, we further study a special class of SDEs called the potential field (PF) models, which formulates the drift term as the gradient of another function. A mixture of PF model is shown to provide good interpretation of the animal’s track.