Depth Patterns and their Applications in Animal Tracking

TL;DR

The paper introduces depth patterns as a multivariate analogue of ordinal patterns by using Tukey's halfspace depth to obtain a center-outward ordering in $\mathbb{R}^d$. It proves limit theorems for depth-pattern probabilities under both known and unknown reference distributions, establishing consistency and, under weak dependence, asymptotic normality of the pattern-frequency estimators. The authors apply the framework to Antarctic fur seal movement data, showing that depth-patterns can distinguish movement models and concluding that a biased antipersistent random walk best explains the observed patterns. The approach is robust to perturbations due to its ordinal-depth basis and offers a practical, model-selection oriented tool for analyzing multivariate time series.

Abstract

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 \emph{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 statistical properties of depth pattern probabilities, establishing consistency and asymptotic normality under the assumption of weakly dependent time series. Since our concept only depends 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.

Figures (7)

• Figure 1: The six univariate ordinal patterns of order 3 (not allowing for ties).
• Figure 2: As $Q^{(m)}$ approaches $Q$, the depth pattern of $(x_1,x_2,x_3)$ changes: $\Pi_{Q^{(3)}}(x_1,x_2,x_3)=(1,3,3)$, $\Pi_{Q^{(6)}}(x_1,x_2,x_3)=(2,3,2)$ and $\Pi_{Q^{(7)}}(x_1,x_2,x_3)=(2,3,1)$. The black dots are the values of $x_i$ while the white dots denote the point-masses of $Q^{(m)}$.
• Figure 3: Motion paths of 15 individual seal pubs at Freshwater Beach (FWB).
• Figure 4: Histogram of step lengths of 15 individual seal pubs at Freshwater Beach (FWB) and density of the exponential distribution with parameter $\lambda=\frac{1}{\hat{\theta}}$, where $\hat{\theta}=48.44$ derives from Maximum-Likelihood estimation based on the step lengths of all 15 individuals.
• Figure 5: Illustration of a step in a biased (anti)persistent random walk. In blue the step from the current to the next location, i.e. from $(x_1(t), x_2(t))$ to $(x_1(t+1), x_2(t+1))$. The red arrows indicate the bearing to the center $(c_1, c_2)$ and in direction of the previous turning angel $a(t-1)$ standardized to a steplength of 1. In green the corresponding steps taken in direction of $x$- and $y$-axis.

Theorems & Definitions (31)

• Definition 2.1
• Definition 2.2: Halfspace Depth
• Definition 2.3
• Definition 2.4
• Proposition 3.1
• Definition 3.2
• Definition 3.3
• Definition 3.4
• Definition 3.5
• Theorem 3.6