Identification of Anomalous Geospatial Trajectories via Persistent Homology

TL;DR

The paper addresses the detection of crop-circle–like anomalies in AIS geospatial trajectories by embedding each track into $\mathbb{R}^3$ via Takens' embedding and applying persistent homology to identify long-lived $1$-dimensional features. It introduces a velocity-scaled metric to construct a Rips filtration and uses the longest $H_1$ lifetime to discriminate anomalous from normal tracks, validating the approach with synthetic Rio Bus data and real AIS data near San Francisco. The main contributions are a track-level, parallelizable TDA framework that treats loops as obstructions to normality, robustness to perturbations and coordinate projections, and a data-driven method for calibrating the metric. The work demonstrates the practicality of topological methods for geospatial anomaly detection and points to avenues for richer topological features, improved embeddings, and scalable pre-processing in maritime surveillance and environmental monitoring contexts.

Abstract

We present a novel method for analyzing geospatial trajectory data using topological data analysis (TDA) to identify a specific class of anomalies, commonly referred to as crop circles, in AIS data. This approach is the first of its kind to be applied to spatiotemporal data. By embedding $2+1$-dimensional spatiotemporal data into $\mathbb{R}^3$, we utilize persistent homology to detect loops within the trajectories in $\mathbb{R}^2$. Our research reveals that, under normal conditions, trajectory data embedded in $\mathbb{R}^3$ over time do not form loops. Consequently, we can effectively identify anomalies characterized by the presence of loops within the trajectories. This method is robust and capable of detecting loops that are invariant to small perturbations, variations in geometric shape, and local coordinate projections. Additionally, our approach provides a novel perspective on anomaly detection, offering enhanced sensitivity and specificity in identifying atypical patterns in geospatial data. This approach has significant implications for various applications, including maritime navigation, environmental monitoring, and surveillance.

Figures (12)

• Figure 1: (Left: Example of crop circles found in Shanghai Harris. Right: Example of continuous circles found near San Francisco, a region with known anomalies skytruth.
• Figure 2: Cargo ship identified using features derived from persistent homology of AIS trajectory data near Point Reyes
• Figure 3: An example construction of a filtration and its associated persistence diagram. Disks (red) are inflated around five points (black) in the plane, and edges (white) are drawn between their centers according to the intersections of the disks to mark the Rips complexes $\Delta_r$ constructed at each stage. A persistence diagram notes the births and deaths of the topological features of $\Delta_r$ as the radius increases to infinity. Noteworthy events A, B, C, and D occur high above the diagonal in this graph. The stacked crosses in event B are located at the same point since they all occur simultaneously. Notionally, features that never die have a death time of $\infty$, as is the case with event C.
• Figure 4: Example of non-anomalous AIS track around Point Reyes 2020 (left). The individual latitudinal and longitudinal components of the track over time are shown (center) along with their dimension-3 Takens embeddings (right). Note that there are loop features present in the Takens embeddings of the individual latitude and longitude signals, but no anomalous behavior is observed in the track data.
• Figure 5: An unaugmented track (left) from the Rio Bus Dataset viewed in three dimensions and an augmented track (right) synthesized by inserting an anomaly in the middle third of the original track.

Theorems & Definitions (1)

• Definition 1