Waves in the ocean can be as dangerous as they are impressive. In order to study the behaviour of such waves, buoys are commonly deployed to collect recordings of the ocean surface over time. This results in large quantities of high-frequency multivariate time series data. The statistical analysis of such data is of great importance in a variety of engineering and scientific contexts, from the design of coastal flood defences to offshore structures.

We develop methodology for analysing such buoy data, investigating two key questions.
Firstly, how should we perform parameter inference for models of the frequency domain behaviour of the surface, given recorded buoy data?
Secondly, how can we detect statistically significant non-linearities present in these time series?

For parameter inference, we find that pseudo-likelihood approaches greatly outperform state-of-the-art methodologies. As a result, not only can we obtain more reliable parameter estimates, but we can also perform inference for more complicated models, allowing for a more intricate description of the waves. Due to the improved performance of such estimates, we are able to see the evolution of these parameters throughout storm events, using recorded buoy data from both California and the North Sea.

For detecting non-linearities, we develop a robust testing procedure by evaluating the bispectrum of the observed time series against the bispectrum of bootstrap simulated Gaussian processes with similar characteristics. We explore the performance of this technique in simulation studies, and apply the approach to buoy data from California.