Persistent Homology, Regimes and Climate Data

The concept of weather or climate 'regimes' have been studied since the 70s, to a large extent because of the possibility they offer of truncating complicated dynamics to vastly simpler, Markovian, dynamics. Despite their attraction, detecting them in data is often problematic, and a unified definition remains nebulous. We argue that the crucial common feature across different dynamical systems with regimes is the non-trivial topology of the underlying phase space. Such non-trivial topology can be detected in a robust and explicit manner using persistent homology, a powerful new tool to compute topological invariants in arbitrary datasets. We show some state of the art examples of the application of persistent homology to various non-linear dynamical systems, including real-world climate data, and show how these techniques can shed light on questions such as how many regimes there really are in e.g. the Euro-Atlantic sector. Future directions are also discussed.