Improving Topological Detection of Weather Regimes in climate dynamical systems

Weather regimes provide a useful framework for describing large-scale atmospheric variability and its impacts on regional weather. Despite extensive study, there is still no universally accepted definition or method for identifying weather regimes. Recent work has shown that weather regimes can be interpreted geometrically as topological structures in the phase space of the atmospheric system. In this approach, regimes are identified using a density–radius bifiltration combined with persistent homology, a well-established tool from Topological Data Analysis (TDA). This topological perspective provides a unifying view of regimes and, unlike traditional methods, does not require the number of regimes to be specified in advance. However, the method relies on density estimation techniques (typically Gaussian kernel density estimation), which can over–smooth weakly populated but dynamically important regions of the phase space.

💡 Research Summary

The paper addresses a key limitation of the density‑radius bifiltration introduced by Strommen et al. (2023) for detecting weather regimes in high‑dimensional climate data. The original method relies on Gaussian kernel density estimation (KDE) to rank points by local density before applying a radius‑based filtration and persistent homology (PH). While effective for dense regions, KDE tends to over‑smooth low‑density but dynamically important structures such as thin loops or trimodal features, leading to missed regimes in several benchmark datasets (e.g., the southern jet regime in the JetLat data and low‑density loops in the Charney‑De Vore model).

To overcome this, the authors replace the density measure with a centrality measure derived from the k‑distance‑to‑measure (k‑DTM) function. For each data point, the average distance to its k nearest neighbours is computed; points with small k‑DTM values are deemed highly central because they are tightly connected to their neighbourhood. By thresholding on centrality rather than density, a “centrality‑radius bifiltration” is constructed: points passing the centrality cut are retained, and a radius parameter is varied to build a Vietoris–Rips filtration on this subset. The parameter k controls the spatial scale of locality: small k highlights fine‑scale structures, while larger k provides robustness against noise.

The method is evaluated on four datasets that were also used in the original study: (1) the Lorenz‑63 chaotic system, (2) the multiscale Lorenz‑96 model, (3) the six‑variable Charney‑De Vore (CdV) model, and (4) the observational North Atlantic jet latitude (JetLat) dataset derived from ERA20C reanalysis. In each case, the centrality‑based approach either matches or surpasses the KDE‑based results. For Lorenz‑63 and Lorenz‑96, additional thin loops are recovered, offering a richer description of regime transitions. In the CdV model, the previously hidden low‑density spiralling loops become clearly visible, confirming the presence of two distinct regimes (blocked and zonal) and their transition pathways. Most strikingly, in the JetLat data the centrality‑radius bifiltration successfully isolates all three jet regimes (southern, central, northern) across a broad range of k values, whereas the KDE‑based method failed to separate the southern regime.

Statistical significance is assessed via bootstrap resampling of persistence diagrams and confidence interval construction. The centrality‑based bifiltration consistently yields a larger number of high‑persistence points, indicating that the identified topological features are robust and likely reflect genuine dynamical structures. Computationally, k‑nearest‑neighbour searches scale as O(N log N), making the centrality calculation more efficient than KDE, especially in high‑dimensional settings.

The authors integrate the new pipeline into the existing software framework from Strommen et al., providing modular code, documentation, and a practical guide for selecting k based on data characteristics (sample size, dimensionality, expected regime scale). They acknowledge that k selection remains data‑dependent and that very high‑dimensional applications may still face distance‑computation costs. Future work is proposed on automated k‑optimization (e.g., cross‑validation or Bayesian optimisation), hybrid schemes combining dimensionality reduction with centrality filtering, and embedding the topological regime information directly into predictive climate models.

In summary, the centrality‑radius bifiltration resolves the over‑smoothing problem of KDE, captures weakly populated yet dynamically crucial regions of phase space, and offers a scalable, parameter‑light framework for systematic detection of weather regimes across both idealized models and real‑world climate datasets. This advancement positions topological data analysis as a more reliable and versatile tool for climate dynamics research.