TRAKNN: Efficient Trajectory Aware Spatiotemporal kNN for Rare Meteorological Trajectory Detection

Deep learning weather models now match numerical weather prediction accuracy while running orders of magnitude faster, but produce deterministic forecasts without uncertainty estimates, a critical gap for high-stakes decisions during extreme weather events. This paper proposes Neural Tangent Kernel-based uncertainty quantification (NTK-UQ) using last-layer empirical features. Theoretical analysis predicts that UQ quality is architecture-dependent through two mechanisms. First, a variance collapse mechanism explains when UQ fails: when the eigenvalue truncation rank approaches the effective rank of the feature space, the GP correction term consumes nearly all prior variance, destroying discrimination between tropical cyclones and routine conditions; architectures with concentrated spectra (spectral operators) require aggressive truncation ($k \leq 10$), while attention-based models tolerate full-rank computation. Second, decomposition performance depends on the non-Gaussian, heavy-tailed structure of extreme weather: Independent Component Analysis exploits higher-order statistics (kurtosis, negentropy) to isolate heavy-tailed extreme-event features, achieving higher discrimination than singular value decomposition, which captures only second-order variance. A data-driven selection rule chooses ICA or SVD from the feature eigenspectrum concentration ratio, correctly prescribing the superior decomposition for all four evaluated architectures. Compared to split conformal prediction (the natural post-hoc baseline), NTK-UQ achieves 31--37% sharper prediction intervals at 90% coverage, and uniquely produces adaptive intervals that scale with extreme event severity, which conformal prediction cannot achieve by construction. The framework requires no retraining; inference-time uncertainty requires only a single matrix-vector product per sample.

Trajectory anomaly detection underpins applications from fraud detection to urban mobility analysis. Dense GPS preserves fine-grained evidence such as abnormal speeds and short-duration events, but its quadratic cost makes multi-month analysis intractable; sparse stay-point methods scale by discarding that evidence and require a separate modeling regime. We argue that this bottleneck is unnecessary: dense and sparse trajectories share a natural two-dimensional cyclic structure along within-day and across-day axes. TITAnD (Trajectory Image Transformer for Anomaly Detection) is the first framework to cast both in a single representation, a Hyperspectral Trajectory Image (HTI), a day $\times$ time-of-day grid whose channels encode spatial, semantic, temporal, and kinematic information. Under this formulation, agent-level detection reduces to image classification and temporal localization to semantic segmentation. The Cyclic Factorized Transformer (CFT) models the two temporal axes directly, reducing attention cost by up to two orders of magnitude. With the HTI, multi-month dense anomaly detection becomes feasible for the first time, and CFT keeps it accurate and fast. Empirically, TITAnD matches or beats every evaluated baseline in AUC-PR on all four benchmarks. It surpasses vision models such as U-Net while using as few as 6.5M parameters, and it runs 11--75$\times$ faster than a capacity-matched flat Transformer on the same trajectory images.

We present a scalable framework for unsupervised clustering of maritime trajectories derived from terabyte-scale Automatic Identification System (AIS) archives. Variable-length trajectories are encoded with a custom BERT-based model trained via masked token modeling and clustered using CURE hierarchical clustering, producing physically interpretable trajectory groups without requiring a predefined number of clusters. An intrinsic unsupervised anomaly detection method based on reconstruction loss and clustering noise assignment identifies irregular navigation patterns. The framework is demonstrated on a national-scale AIS dataset comprising billions of messages spanning one year, yielding stable trajectory clusters and a clear separation between nominal and anomalous vessel behavior.