2016/03/31 by Ralf Banisch, Péter Koltai · 10 citations
Computer Science · Mathematics · #Time Series Analysis and Forecasting #Topological and Geometric Data Analysis #math.DS #msc:05C81 #msc:37M10 #msc:37N10
paper · pdf · doi:10.1063/1.4971788
arxiv created 2016/04/07 · openalex publication_date 2017/02/22 · arxiv updated 2017/04/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Dynamical systems often exhibit the emergence of long-lived coherent sets, which are regions in state space that keep their geometric integrity to a high extent and thus play an important role in transport. In this article, we provide a method for extracting coherent sets from possibly sparse Lagrangian trajectory data. Our method can be seen as an extension of diffusion maps to trajectory space, and it allows us to construct "dynamical coordinates" which reveal the intrinsic low-dimensional organization of the data. The only a priori knowledge about the dynamics that we require is a locally valid notion of distance, which renders our method highly suitable for automated data analysis. We show convergence of our method to the analytic transfer operator framework of coherence in the infinite data limit, and illustrate its potential on several two- and three-dimensional examples as well as real world data.