Depth patterns and their applications in animal tracking.

We establish a definition of ordinal patterns for multivariate data sets based on the concept of Tukey's halfspace depth. Given the definition of these depth patterns, we are interested in the probabilities of observing specific patterns in time series. For this, we consider the relative frequency of depth patterns as natural estimators for their occurrence probabilities. Depending on the choice of reference distribution and the relation between reference and data distribution, we distinguish different settings that are considered separately. Within these settings, we study the statistical properties of depth pattern probabilities, establishing consistency and asymptotic normality under the assumption of weakly dependent time series. Since our concept depends only on ordinal depth information, the resulting values are robust under small perturbations and measurement errors. We emphasize the applicability of our method by analyzing the depth patterns, which are found in seal pubs' movement. We use our approach in order to choose an appropriate model out of a range of two-dimensional random walks, which are commonly used in mathematical biology.