Magnitude-Based Features for Multispecies Spatial Data

Multispecies spatial data arise in many applications where interactions between different entities are central to system behaviour, including biomedical imaging, geospatial analysis, and species ecology. Despite their importance, relatively few quantitative tools exist to capture such interactions. In this work, we propose magnitude-based features for the analysis of multispecies spatial data. Magnitude is a real-valued invariant of finite metric spaces that can be interpreted as an effective number of points, incorporating both spatial configuration and scale. We develop global and local magnitude feature vectors and demonstrate their utility on synthetic tumour microenvironment data, and in tissue microarray data from human colorectal cancer samples. Locally, the method identifies distinct neighbourhood types and reveals spatial heterogeneity; in the model, this includes radial patterns associated with different qualitative outcomes of the simulations, while in the real-world data it reflects the importance of tertiary lymphoid structure-like interactions between B and T cell populations. Globally, the approach recovers known classifications of long-term simulation outcomes across parameter regimes in synthetic data, and suggests important roles for CD4+ T cells and CD163+ macrophages in distinguishing patients with favourable Crohn's like reactions from unfavourable diffuse immune infiltration. Together, these results suggest that magnitude-based features provide a powerful and flexible tool for the analysis of multispecies spatial data.

Persistent Homology via Ellipsoids

Persistent homology is one of the most popular methods in Topological Data Analysis. An initial step in any analysis with persistent homology involves constructing a nested sequence of simplicial complexes, called a filtration, from a point cloud. There is an abundance of different complexes to choose from, with Rips, Alpha, and witness complexes being popular choices. In this manuscript, we build a different type of a geometrically-informed simplicial complex, called an ellipsoid complex. This complex is based on the idea that ellipsoids aligned with tangent directions better approximate the data compared to conventional (Euclidean) balls centered at sample points that are used in the construction of Rips and Alpha complexes, for instance. We use Principal Component Analysis to estimate tangent spaces directly from samples and present algorithms as well as an implementation for computing ellipsoid barcodes, i.e., topological descriptors based on ellipsoid complexes. Furthermore, we conduct extensive experiments and compare ellipsoid barcodes with standard Rips barcodes. Our findings indicate that ellipsoid complexes are particularly effective for estimating homology of manifolds and spaces with bottlenecks from samples. In particular, the persistence intervals corresponding to a ground-truth topological feature are longer compared to the intervals obtained when using the Rips complex of the data. Furthermore, ellipsoid barcodes lead to better classification results in sparsely-sampled point clouds. Finally, we demonstrate that ellipsoid barcodes outperform Rips barcodes in classification tasks.