Trajectory data are generated by tracking the position of objects that move through space, for example using GPS technology. This thesis focuses on the statistical modelling of thousands of trajectories to infer various oceanographic statistics of interest. The motivating example used throughout the thesis is the Global Drifter Program which provides over 25,000 trajectories of free-floating buoys known as drifters, the motion of which track near-surface currents.

In the first piece of work we propose a novel multi-output probabilistic prediction model. The motivation for this arises because drifter trajectories are often used by practitioners to calibrate and estimate oceanic models which predict northward and eastward velocities using two independent models. This independence assumption is seldom realistic. As such, we extend an existing framework known as Natural Gradient Boosting (NGBoost) to predict multivariate outputs. The model is applied to predict a conditional distribution of drifter velocities.

We then develop a novel method to compute the most likely path taken by drifters between arbitrary fixed locations in the ocean. We also provide an estimate of the travel time associated with this path. Our method, which utilises Markov transition matrices, is purely data driven and requires no simulations of drifter trajectories, in contrast to existing approaches.

Finally, we propose an alternative method of estimating travel times in a modelfree way. Specifically, we investigate how long it takes a collection of drifters to travel to and from locations of interest. However, this direct approach, when applied naively, results in estimates which can be biased, missing, and noisy. Hence we use transformed multidimensional scaling to lessen these undesirable properties. We discuss the merits and disadvantages of our two proposed methods for estimating travel times, and we provide real-world studies to motivate and justify each methodological approach.