Trajectory analysis has been a central problem in applications of location tracking systems. Recently, the (discrete) Fréchet distance becomes a popular approach for measuring the similarity of two trajectories because of its high feature extraction capability. Despite its importance, the Fréchet distance has several limitations: (i) sensitive to noise as a trade-off for its high feature extraction capability; and (ii) it cannot be incorporated into machine learning frameworks due to its non-smooth functions. To address these problems, we propose the Fréchet kernel (FRK), which is associated with a smoothed Fréchet distance using a combination of two approximation techniques. FRK can adaptively acquire appropriate extraction capability from trajectories while retaining robustness to noise. Theoretically, we find that FRK has a positive definite property, hence FRK can be incorporated into the kernel method. We also provide an efficient algorithm to calculate FRK. Experimentally, FRK outperforms other methods, including other kernel methods and neural networks, in various noisy real-data classification tasks.

Extracting the interaction rules of biological agents from movement sequences pose challenges in various domains. Granger causality is a practical framework for analyzing the interactions from observed time-series data; however, this framework ignores the structures and assumptions of the generative process in animal behaviors, which may lead to interpretational problems and sometimes erroneous assessments of causality. In this paper, we propose a new framework for learning Granger causality from multi-animal trajectories via augmented theory-based behavioral models with interpretable data-driven models. We adopt an approach for augmenting incomplete multi-agent behavioral models described by time-varying dynamical systems with neural networks. For efficient and interpretable learning, our model leverages theory-based architectures separating navigation and motion processes, and the theory-guided regularization for reliable behavioral modeling. This can provide interpretable signs of Granger-causal effects over time, i.e., when specific others cause the approach or separation. In experiments using synthetic datasets, our method achieved better performance than various baselines. We then analyzed multi-animal datasets of mice, flies, birds, and bats, which verified our method and obtained novel biological insights.