We develop continuous-time probabilistic models to study trajectory data consisting of times and locations of user “check-ins”. We model the data as realizations of a marked point process, with intensity and mark-distribution modulated by a latent Markov jump process (MJP). We also include user-heterogeneity in our model by assigning each user a vector of “preferred locations”. Our model extends latent Dirichlet allocation by dropping the bag-of-words assumption and operating in continuous time. We show how an appropriate choice of priors allows efficient posterior inference. Our experiments demonstrate the usefulness of our approach by comparing with various baselines on a variety of tasks.

Ecologists often seek to understand patterns and processes across multiple spatial and temporal scales ranging from centimeters to hundreds of meters and from seconds to years. Hierarchical statistical models offer a framework for sampling design and analysis that can be used to incorporate the information collected at finer scales while allowing comparison at coarser scales. In this study we use a Hierarchical Bayesian model to assess the relationship between measurements collected on the ground at centimeter scales nested within 2 × 3 m quadrats, which are in turn nested within much larger (0.1–12 ha) plots. We compare these measurements with the Normalized Difference Vegetation Index (NDVI) derived from radiometrically and geometrically corrected 30-m resolution LANDSAT ETM+ data to assess the NDVI–Biomass relationship in the Cape Floristic Region of South Africa. Our novel modeling approach allows the data observed at submeter scales to be incorporated directly into the model and thus all the data (and variability) collected at finer scales are represented in the estimates of biomass at the LANDSAT scale. The model reveals that there is a strong correlation between NDVI and biomass, which supports the use of NDVI in spatiotemporal analysis of vegetation dynamics in Mediterranean shrubland ecosystems. The methods developed here can be easily generalized to other ecosystems and ecophysiological parameters.